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  "generated_at": "2026-07-25T17:54Z",
  "lic_root": "C:\\Users\\Julian\\Documents\\Programming\\li\\lic-gpu-paper-wt",
  "lic_commit": "e1e537dc267c1d7aa6988940f7f01eb47de9fe8c",
  "sources": {
    "toml_entries": "docs/verification/proof-database/entries/",
    "proof_db_index": "proof-db/index.json",
    "lean_scan": [
      "docs/semantics/Core.lean",
      "docs/semantics/trusted.lean",
      "docs/semantics/Discharge.lean",
      "proof-db/lean/ProofDB.lean",
      "proof-db/math/axioms/MathAxioms.lean",
      "proof-db/math/axioms/MathLemmas.lean",
      "proof-db/graph/GraphAxioms.lean",
      "proof-db/numerics/axioms/NumericsAxioms.lean",
      "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "proof-db/chemistry/ChemAxioms.lean",
      "proof-db/biology/BioAxioms.lean",
      "proof-db/ml/OptAxioms.lean",
      "proof-db/ml/Convex.lean",
      "proof-db/ml/SGD.lean",
      "proof-db/statistics/StatsAxioms.lean",
      "build/generated/AutoVC.lean",
      "proof-db/control/ControlAxioms.lean",
      "proof-db/erdos/ErdosLiterature.lean",
      "proof-db/erdos/ErdosMathlib.lean",
      "proof-db/erdos/ErdosOpenLiterature.lean",
      "proof-db/info/InfoAxioms.lean",
      "proof-db/logic/LogicAxioms.lean",
      "proof-db/math/OpenConjectureLiterature.lean",
      "proof-db/ml/MlAxioms.lean",
      "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
      "proof-db/optimization/OptimizationAxioms.lean",
      "proof-db/pde/PdeAxioms.lean",
      "proof-db/physics/PhysicsAxioms.lean",
      "proof-db/physics/RelativityAxioms.lean",
      "proof-db/signal/SignalAxioms.lean"
    ],
    "discrepancies": "proof-database/discrepancies.json"
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    "total": 4204,
    "divergent": 106,
    "by_catalog_status": {
      "axiomatic": 75,
      "proved": 4055,
      "discrepancy": 11,
      "target": 62,
      "open": 1
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    "by_lean_status": {
      "axiomatic": 74,
      "proved": 4003,
      "open": 125,
      "placeholder": 2
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      "id": "A-trusted-getelem",
      "kind": "axiom",
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      "statement": "typing_getElem trusted.",
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      "diverges": false,
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      "lean_module": "docs/semantics/Core.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Core.lean",
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            "start_line": 58,
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            "content": "[[entry]]\nid = \"A-trusted-getelem\"\nkind = \"axiom\"\nfield = \"trust\"\ngap_id = \"G-trust\"\nstatement = \"typing_getElem trusted.\"\nproof_status = \"axiomatic\"\nlean_module = \"docs/semantics/Core.lean\"\nlast_verified_lic_commit = \"main\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\contracts-verify.toml#L58",
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    {
      "id": "BIO-AX-ALIGNMENT-SCORING",
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      "field": "biology",
      "domain": "sequence_alignment",
      "statement": "Pairwise alignment score Score(A,B) = sum_i S(a_i, b_i) over matched columns with substitution matrix S.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.alignmentScore_eq_sum",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": null,
      "bench_refs": [
        "benchmarks/tier2_physics/bio_proteinmpnn"
      ],
      "source": "docs\\verification\\proof-database\\entries\\biology-axioms.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean",
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            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "alignmentScore_eq_sum",
            "start_line": 100,
            "highlight_line": 100,
            "content": "theorem alignmentScore_eq_sum {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) :\n    alignmentScore S a b = \u2211 i, S (a i) (b i) :=\n  rfl\n\n/-- Needleman\u2013Wunsch DP cell is optimal among the three predecessor candidates\n    (BIO-LM-NEEDLEMAN-WUNSCH / BIO-LM-ALIGNMENT-SCORE-OPTIMAL sketch). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L100",
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            "start_line": 21,
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            "content": "[[entry]]\nid = \"BIO-AX-ALIGNMENT-SCORING\"\nkind = \"axiom\"\nfield = \"biology\"\ndomain = \"sequence_alignment\"\nstatement = \"Pairwise alignment score Score(A,B) = sum_i S(a_i, b_i) over matched columns with substitution matrix S.\"\nproof_status = \"proved\"\ngap_id = \"G-bio\"\ngap_kind = \"modeling_gap\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.alignmentScore_eq_sum\"\nbench_refs = [\n  \"benchmarks/tier2_physics/bio_proteinmpnn\",\n]\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-axioms.toml#L21",
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    {
      "id": "BIO-AX-FOLDING-ENERGY",
      "kind": "axiom",
      "field": "biology",
      "domain": "structural_biology",
      "statement": "Protein folding energy landscape: native conformation R* minimizes effective free energy E(R) over admissible conformations R.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
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      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.folding_energy_minimization",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": null,
      "bench_refs": [
        "benchmarks/tier2_physics/bio_rosetta_energy",
        "benchmarks/tier2_physics/bio_rotamer_packing"
      ],
      "source": "docs\\verification\\proof-database\\entries\\biology-axioms.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "folding_energy_minimization",
            "start_line": 114,
            "highlight_line": 114,
            "content": "theorem folding_energy_minimization {\u03b1 : Type*} (E : \u03b1 \u2192 \u211d) (Rstar : \u03b1)\n    (h : IsMinOn E Set.univ Rstar) : \u2200 R, E Rstar \u2264 E R :=\n  fun R => h (Set.mem_univ R)\n",
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            "content": "[[entry]]\nid = \"BIO-AX-FOLDING-ENERGY\"\nkind = \"axiom\"\nfield = \"biology\"\ndomain = \"structural_biology\"\nstatement = \"Protein folding energy landscape: native conformation R* minimizes effective free energy E(R) over admissible conformations R.\"\nproof_status = \"proved\"\ngap_id = \"G-bio\"\ngap_kind = \"modeling_gap\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.folding_energy_minimization\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nbench_refs = [\n  \"benchmarks/tier2_physics/bio_rosetta_energy\",\n  \"benchmarks/tier2_physics/bio_rotamer_packing\",\n]\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-axioms.toml#L3",
            "role": "catalog",
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    },
    {
      "id": "BIO-AX-MC-001",
      "kind": "axiom",
      "field": "biology",
      "domain": "population",
      "statement": "Logistic growth ODE (mega 1).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Biology.bio_ax_mc_logistic_eq",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_ax_mc_001.li",
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      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_ax_mc_001.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "symbol": "bio_ax_mc_logistic_eq",
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            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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            "path": "proof-db/biology/corpus/bio_ax_mc_001.li",
            "symbol": null,
            "start_line": 3,
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            "content": "def proof_db_bio_ax_mc_001() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_ax_mc_001.li#L3",
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            "content": "[[entry]]\nid = \"BIO-AX-MC-001\"\nkind = \"axiom\"\nfield = \"biology\"\ndomain = \"population\"\nstatement = \"Logistic growth ODE (mega 1).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_ax_mc_logistic_eq\"\nli_specimen = \"proof-db/biology/corpus/bio_ax_mc_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L3",
            "role": "catalog",
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    },
    {
      "id": "BIO-AX-MC-011",
      "kind": "axiom",
      "field": "biology",
      "domain": "population",
      "statement": "Logistic growth ODE (mega 11).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_ax_mc_logistic_eq",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_ax_mc_011.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_ax_mc_011.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_ax_mc_logistic_eq",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L15",
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            "path": "proof-db/biology/corpus/bio_ax_mc_011.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_bio_ax_mc_011() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_ax_mc_011.li#L3",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 19,
            "highlight_line": 20,
            "content": "[[entry]]\nid = \"BIO-AX-MC-011\"\nkind = \"axiom\"\nfield = \"biology\"\ndomain = \"population\"\nstatement = \"Logistic growth ODE (mega 11).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_ax_mc_logistic_eq\"\nli_specimen = \"proof-db/biology/corpus/bio_ax_mc_011.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L19",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-AX-MC-021",
      "kind": "axiom",
      "field": "biology",
      "domain": "population",
      "statement": "Logistic growth ODE (mega 21).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_ax_mc_logistic_eq",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_ax_mc_021.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_ax_mc_021.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
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          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_ax_mc_logistic_eq",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L15",
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            "language": "li",
            "path": "proof-db/biology/corpus/bio_ax_mc_021.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_bio_ax_mc_021() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_ax_mc_021.li#L3",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 35,
            "highlight_line": 36,
            "content": "[[entry]]\nid = \"BIO-AX-MC-021\"\nkind = \"axiom\"\nfield = \"biology\"\ndomain = \"population\"\nstatement = \"Logistic growth ODE (mega 21).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_ax_mc_logistic_eq\"\nli_specimen = \"proof-db/biology/corpus/bio_ax_mc_021.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L35",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "BIO-AX-MC-031",
      "kind": "axiom",
      "field": "biology",
      "domain": "population",
      "statement": "Logistic growth ODE (mega 31).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_ax_mc_logistic_eq",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_ax_mc_031.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_ax_mc_031.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_ax_mc_logistic_eq",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_ax_mc_031.li",
            "symbol": null,
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            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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            "symbol": "bio_ax_mc_logistic_eq",
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            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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            "symbol": "bio_ax_mc_logistic_eq",
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            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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      "domain": "population",
      "statement": "Logistic growth ODE (mega 71).",
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            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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      "id": "BIO-AX-MC-081",
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            "symbol": "bio_ax_mc_logistic_eq",
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            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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      "id": "BIO-AX-MC-091",
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      "field": "biology",
      "domain": "population",
      "statement": "Logistic growth ODE (mega 91).",
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            "symbol": "bio_ax_mc_logistic_eq",
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            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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      "id": "BIO-AX-MC-101",
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      "statement": "Logistic growth ODE (mega 101).",
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            "symbol": "bio_ax_mc_logistic_eq",
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            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-bio",
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            "symbol": "bio_ax_mc_logistic_eq",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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      "id": "BIO-AX-MC-121",
      "kind": "axiom",
      "field": "biology",
      "domain": "population",
      "statement": "Logistic growth ODE (mega 121).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_ax_mc_logistic_eq",
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            "symbol": "bio_ax_mc_logistic_eq",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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            "symbol": null,
            "start_line": 3,
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    },
    {
      "id": "BIO-AX-MC-131",
      "kind": "axiom",
      "field": "biology",
      "domain": "population",
      "statement": "Logistic growth ODE (mega 131).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_ax_mc_logistic_eq",
      "lean_module": "proof-db/biology/BioAxioms.lean",
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            "symbol": "bio_ax_mc_logistic_eq",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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    },
    {
      "id": "BIO-AX-MC-141",
      "kind": "axiom",
      "field": "biology",
      "domain": "population",
      "statement": "Logistic growth ODE (mega 141).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_ax_mc_logistic_eq",
      "lean_module": "proof-db/biology/BioAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
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            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_ax_mc_logistic_eq",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem bio_ax_mc_logistic_eq (r K : \u211d) (hK : K \u2260 0) :\n    logisticGrowthRate r K K = 0 := by\n  simp [logisticGrowthRate, div_self hK]\n\n/-- Extinction equilibrium: growth vanishes at N = 0. -/",
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            "path": "proof-db/biology/corpus/bio_ax_mc_141.li",
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            "start_line": 3,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_ax_mc_141.li#L3",
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            "start_line": 1658,
            "highlight_line": 1659,
            "content": "[[entry]]\nid = \"BIO-AX-MC-141\"\nkind = \"axiom\"\nfield = \"biology\"\ndomain = \"population\"\nstatement = \"Logistic growth ODE (mega 141).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_ax_mc_logistic_eq\"\nli_specimen = \"proof-db/biology/corpus/bio_ax_mc_141.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
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            "role": "catalog",
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    },
    {
      "id": "BIO-LM-ALIGNMENT-SCORE-OPTIMAL",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence_alignment",
      "statement": "Global alignment DP score is maximal among all valid alignments (alignment score target).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "proof_gap",
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      "erdos_status": null,
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      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": null,
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        "benchmarks/tier2_physics/bio_proteinmpnn",
        "benchmarks/tier2_physics/bio_rfdiffusion"
      ],
      "source": "docs\\verification\\proof-database\\entries\\biology-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean",
      "notes": null,
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            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "alignment_score_optimal",
            "start_line": 106,
            "highlight_line": 106,
            "content": "theorem alignment_score_optimal (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left :=\n  bio_lm_nw_dp_recurrence diag up left\n\n/-- Protein folding: native conformation R* minimizes free energy E\n    (BIO-AX-FOLDING-ENERGY). -/",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-lemmas.toml",
            "start_line": 19,
            "highlight_line": 20,
            "content": "[[entry]]\nid = \"BIO-LM-ALIGNMENT-SCORE-OPTIMAL\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence_alignment\"\nstatement = \"Global alignment DP score is maximal among all valid alignments (alignment score target).\"\nproof_status = \"proved\"\ngap_id = \"G-bio\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.alignment_score_optimal\"\nbench_refs = [\n  \"benchmarks/tier2_physics/bio_proteinmpnn\",\n  \"benchmarks/tier2_physics/bio_rfdiffusion\",\n]\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-lemmas.toml#L19",
            "role": "catalog",
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        ]
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    },
    {
      "id": "BIO-LM-MC-002",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hw_unit",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hw_unit",
            "start_line": 29,
            "highlight_line": 29,
            "content": "theorem bio_lm_hw_unit (p q : \u211d) (h : p + q = 1) : hardyWeinbergFreq p q = 1 := by\n  rw [bio_lm_hw_expand, h, one_pow 2]\n\n/-- SIR compartment total conservation (BIO-LM-MC-005 / BIO-LM-MC-015 / ...). -/",
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            "path": "proof-db/biology/corpus/bio_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-002: Hardy-Weinberg equilibrium (mega 2).",
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            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"BIO-LM-MC-002\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hw_unit\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L83",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-003",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-003: Hamming distance metric (mega 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_003.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 99,
            "highlight_line": 100,
            "content": "[[entry]]\nid = \"BIO-LM-MC-003\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L99",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-004",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-004: Michaelis-Menten rate law (mega 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_004.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"BIO-LM-MC-004\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-005",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-005: SIR compartment conservation (mega 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"BIO-LM-MC-005\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-006",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-006: Ultrametric tree inequality (mega 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_006.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 147,
            "highlight_line": 148,
            "content": "[[entry]]\nid = \"BIO-LM-MC-006\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 6).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-007",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-007: Levinthal paradox scaling (mega 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_007.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 163,
            "highlight_line": 164,
            "content": "[[entry]]\nid = \"BIO-LM-MC-007\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L163",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-008",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-008: Kleiber scaling law sketch (mega 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_008.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 178,
            "highlight_line": 179,
            "content": "[[entry]]\nid = \"BIO-LM-MC-008\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 8).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L178",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-009",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-009: Lotka-Volterra steady state (mega 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_009.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 194,
            "highlight_line": 195,
            "content": "[[entry]]\nid = \"BIO-LM-MC-009\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L194",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-010",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_010.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-010: Needleman-Wunsch DP recurrence (mega 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_010.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 210,
            "highlight_line": 211,
            "content": "[[entry]]\nid = \"BIO-LM-MC-010\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 10).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L210",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-012",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hw_unit",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hw_unit",
            "start_line": 29,
            "highlight_line": 29,
            "content": "theorem bio_lm_hw_unit (p q : \u211d) (h : p + q = 1) : hardyWeinbergFreq p q = 1 := by\n  rw [bio_lm_hw_expand, h, one_pow 2]\n\n/-- SIR compartment total conservation (BIO-LM-MC-005 / BIO-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L29",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-012: Hardy-Weinberg equilibrium (mega 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_012.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 226,
            "highlight_line": 227,
            "content": "[[entry]]\nid = \"BIO-LM-MC-012\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hw_unit\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L226",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-013",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_self",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_self",
            "start_line": 58,
            "highlight_line": 58,
            "content": "theorem bio_lm_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 := by\n  simp [hammingDistance]\n\n/-- Ultrametric tree inequality anchor (BIO-LM-MC-016). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L58",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-013: Hamming distance metric (mega 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_013.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 242,
            "highlight_line": 243,
            "content": "[[entry]]\nid = \"BIO-LM-MC-013\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 13).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_self\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L242",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-014",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.michaelisMentenRate",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-014: Michaelis-Menten rate law (mega 14).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_014.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 258,
            "highlight_line": 259,
            "content": "[[entry]]\nid = \"BIO-LM-MC-014\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 14).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.michaelisMentenRate\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L258",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-015",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-015: SIR compartment conservation (mega 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_015.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 274,
            "highlight_line": 275,
            "content": "[[entry]]\nid = \"BIO-LM-MC-015\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 15).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L274",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-016",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-016: Ultrametric tree inequality (mega 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_016.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 290,
            "highlight_line": 291,
            "content": "[[entry]]\nid = \"BIO-LM-MC-016\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 16).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L290",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-017",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-017: Levinthal paradox scaling (mega 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_017.li#L1",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 306,
            "highlight_line": 307,
            "content": "[[entry]]\nid = \"BIO-LM-MC-017\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 17).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L306",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-018",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-018: Kleiber scaling law sketch (mega 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_018.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 322,
            "highlight_line": 323,
            "content": "[[entry]]\nid = \"BIO-LM-MC-018\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 18).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L322",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-019",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-019: Lotka-Volterra steady state (mega 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_019.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 338,
            "highlight_line": 339,
            "content": "[[entry]]\nid = \"BIO-LM-MC-019\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 19).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L338",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-020",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_020.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-020: Needleman-Wunsch DP recurrence (mega 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_020.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 354,
            "highlight_line": 355,
            "content": "[[entry]]\nid = \"BIO-LM-MC-020\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 20).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L354",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-022",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-022: Hardy-Weinberg equilibrium (mega 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_022.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 370,
            "highlight_line": 371,
            "content": "[[entry]]\nid = \"BIO-LM-MC-022\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L370",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-023",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-023: Hamming distance metric (mega 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_023.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 386,
            "highlight_line": 387,
            "content": "[[entry]]\nid = \"BIO-LM-MC-023\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 23).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"d63ab27a6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L386",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-024",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-024: Michaelis-Menten rate law (mega 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_024.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 402,
            "highlight_line": 403,
            "content": "[[entry]]\nid = \"BIO-LM-MC-024\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 24).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L402",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-025",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-025: SIR compartment conservation (mega 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_025.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 418,
            "highlight_line": 419,
            "content": "[[entry]]\nid = \"BIO-LM-MC-025\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 25).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"1a2d9f569\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L418",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-026",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-026: Ultrametric tree inequality (mega 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_026.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 434,
            "highlight_line": 435,
            "content": "[[entry]]\nid = \"BIO-LM-MC-026\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 26).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L434",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-027",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-027: Levinthal paradox scaling (mega 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_027.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 450,
            "highlight_line": 451,
            "content": "[[entry]]\nid = \"BIO-LM-MC-027\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L450",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-028",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_028.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-028: Kleiber scaling law sketch (mega 28).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_028.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 466,
            "highlight_line": 467,
            "content": "[[entry]]\nid = \"BIO-LM-MC-028\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 28).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L466",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-029",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-029: Lotka-Volterra steady state (mega 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_029.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 481,
            "highlight_line": 482,
            "content": "[[entry]]\nid = \"BIO-LM-MC-029\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 29).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L481",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-030",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-030: Needleman-Wunsch DP recurrence (mega 30).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_030.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 497,
            "highlight_line": 498,
            "content": "[[entry]]\nid = \"BIO-LM-MC-030\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 30).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L497",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-032",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-032: Hardy-Weinberg equilibrium (mega 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_032.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 513,
            "highlight_line": 514,
            "content": "[[entry]]\nid = \"BIO-LM-MC-032\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 32).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L513",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-033",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-033: Hamming distance metric (mega 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_033.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 529,
            "highlight_line": 530,
            "content": "[[entry]]\nid = \"BIO-LM-MC-033\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 33).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L529",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-034",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-034: Michaelis-Menten rate law (mega 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_034.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 545,
            "highlight_line": 546,
            "content": "[[entry]]\nid = \"BIO-LM-MC-034\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 34).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"1a2d9f569\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L545",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-035",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_035.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-035: SIR compartment conservation (mega 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_035.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 561,
            "highlight_line": 562,
            "content": "[[entry]]\nid = \"BIO-LM-MC-035\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 35).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L561",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-036",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-036: Ultrametric tree inequality (mega 36).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_036.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 576,
            "highlight_line": 577,
            "content": "[[entry]]\nid = \"BIO-LM-MC-036\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 36).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L576",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-037",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-037: Levinthal paradox scaling (mega 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_037.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 592,
            "highlight_line": 593,
            "content": "[[entry]]\nid = \"BIO-LM-MC-037\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 37).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"1a2d9f569\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L592",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-038",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-038: Kleiber scaling law sketch (mega 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_038.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 608,
            "highlight_line": 609,
            "content": "[[entry]]\nid = \"BIO-LM-MC-038\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 38).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L608",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-039",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-039: Lotka-Volterra steady state (mega 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_039.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 624,
            "highlight_line": 625,
            "content": "[[entry]]\nid = \"BIO-LM-MC-039\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 39).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"9fb0b3057\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L624",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-040",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_040.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-040: Needleman-Wunsch DP recurrence (mega 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_040.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 640,
            "highlight_line": 641,
            "content": "[[entry]]\nid = \"BIO-LM-MC-040\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 40).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L640",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-042",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-042: Hardy-Weinberg equilibrium (mega 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_042.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 656,
            "highlight_line": 657,
            "content": "[[entry]]\nid = \"BIO-LM-MC-042\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L656",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-043",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-043: Hamming distance metric (mega 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_043.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 671,
            "highlight_line": 672,
            "content": "[[entry]]\nid = \"BIO-LM-MC-043\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"1a2d9f569\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L671",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-044",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-044: Michaelis-Menten rate law (mega 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_044.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 687,
            "highlight_line": 688,
            "content": "[[entry]]\nid = \"BIO-LM-MC-044\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 44).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"48fa1974a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L687",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-045",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-045: SIR compartment conservation (mega 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_045.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 703,
            "highlight_line": 704,
            "content": "[[entry]]\nid = \"BIO-LM-MC-045\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 45).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"9fb0b3057\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L703",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-046",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-046: Ultrametric tree inequality (mega 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_046.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 719,
            "highlight_line": 720,
            "content": "[[entry]]\nid = \"BIO-LM-MC-046\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 46).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"1a2d9f569\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L719",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-047",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-047: Levinthal paradox scaling (mega 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_047.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 735,
            "highlight_line": 736,
            "content": "[[entry]]\nid = \"BIO-LM-MC-047\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 47).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"8b666b94c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L735",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-048",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-048: Kleiber scaling law sketch (mega 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_048.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 751,
            "highlight_line": 752,
            "content": "[[entry]]\nid = \"BIO-LM-MC-048\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 48).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"9fb0b3057\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L751",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-049",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_049.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-049: Lotka-Volterra steady state (mega 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_049.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 767,
            "highlight_line": 768,
            "content": "[[entry]]\nid = \"BIO-LM-MC-049\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 49).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L767",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-050",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-050: Needleman-Wunsch DP recurrence (mega 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_050.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 782,
            "highlight_line": 783,
            "content": "[[entry]]\nid = \"BIO-LM-MC-050\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 50).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L782",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-052",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-052: Hardy-Weinberg equilibrium (mega 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_052.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 878,
            "highlight_line": 879,
            "content": "[[entry]]\nid = \"BIO-LM-MC-052\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 52).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b2a0150cb\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L878",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-053",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-053: Hamming distance metric (mega 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_053.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 894,
            "highlight_line": 895,
            "content": "[[entry]]\nid = \"BIO-LM-MC-053\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 53).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"8b666b94c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L894",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-054",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-054: Michaelis-Menten rate law (mega 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_054.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 910,
            "highlight_line": 911,
            "content": "[[entry]]\nid = \"BIO-LM-MC-054\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 54).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"9fb0b3057\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L910",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-055",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-055: SIR compartment conservation (mega 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_055.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 926,
            "highlight_line": 927,
            "content": "[[entry]]\nid = \"BIO-LM-MC-055\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 55).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b2a0150cb\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L926",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-056",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_056.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-056: Ultrametric tree inequality (mega 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_056.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 942,
            "highlight_line": 943,
            "content": "[[entry]]\nid = \"BIO-LM-MC-056\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 56).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L942",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-057",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-057: Levinthal paradox scaling (mega 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_057.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 957,
            "highlight_line": 958,
            "content": "[[entry]]\nid = \"BIO-LM-MC-057\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 57).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"9fb0b3057\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L957",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-058",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-058: Kleiber scaling law sketch (mega 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_058.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 973,
            "highlight_line": 974,
            "content": "[[entry]]\nid = \"BIO-LM-MC-058\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 58).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b2a0150cb\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L973",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-059",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-059: Lotka-Volterra steady state (mega 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_059.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 989,
            "highlight_line": 990,
            "content": "[[entry]]\nid = \"BIO-LM-MC-059\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 59).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"8b666b94c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L989",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-060",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-060: Needleman-Wunsch DP recurrence (mega 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_060.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1005,
            "highlight_line": 1006,
            "content": "[[entry]]\nid = \"BIO-LM-MC-060\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 60).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1005",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-062",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-062: Hardy-Weinberg equilibrium (mega 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_062.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1021,
            "highlight_line": 1022,
            "content": "[[entry]]\nid = \"BIO-LM-MC-062\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 62).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"8b666b94c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1021",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-063",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_063.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-063: Hamming distance metric (mega 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_063.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1037,
            "highlight_line": 1038,
            "content": "[[entry]]\nid = \"BIO-LM-MC-063\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 63).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1037",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-064",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-064: Michaelis-Menten rate law (mega 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_064.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1052,
            "highlight_line": 1053,
            "content": "[[entry]]\nid = \"BIO-LM-MC-064\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 64).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b2a0150cb\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1052",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-065",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-065: SIR compartment conservation (mega 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_065.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1068,
            "highlight_line": 1069,
            "content": "[[entry]]\nid = \"BIO-LM-MC-065\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 65).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"8b666b94c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1068",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-066",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-066: Ultrametric tree inequality (mega 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_066.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1084,
            "highlight_line": 1085,
            "content": "[[entry]]\nid = \"BIO-LM-MC-066\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 66).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"c21f3710c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1084",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-067",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-067: Levinthal paradox scaling (mega 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1100,
            "highlight_line": 1101,
            "content": "[[entry]]\nid = \"BIO-LM-MC-067\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 67).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"b2a0150cb\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1100",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-068",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-068: Kleiber scaling law sketch (mega 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_068.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1116,
            "highlight_line": 1117,
            "content": "[[entry]]\nid = \"BIO-LM-MC-068\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 68).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a5609e5d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1116",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-069",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-069: Lotka-Volterra steady state (mega 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_069.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1132,
            "highlight_line": 1133,
            "content": "[[entry]]\nid = \"BIO-LM-MC-069\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 69).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"c21f3710c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1132",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-070",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_070.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-070: Needleman-Wunsch DP recurrence (mega 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_070.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1148,
            "highlight_line": 1149,
            "content": "[[entry]]\nid = \"BIO-LM-MC-070\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 70).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1148",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-072",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-072: Hardy-Weinberg equilibrium (mega 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_072.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1164,
            "highlight_line": 1165,
            "content": "[[entry]]\nid = \"BIO-LM-MC-072\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 72).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"c21f3710c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1164",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-073",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-073: Hamming distance metric (mega 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_073.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1180,
            "highlight_line": 1181,
            "content": "[[entry]]\nid = \"BIO-LM-MC-073\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 73).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1180",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-074",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-074: Michaelis-Menten rate law (mega 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_074.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1196,
            "highlight_line": 1197,
            "content": "[[entry]]\nid = \"BIO-LM-MC-074\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 74).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a5609e5d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1196",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-075",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-075: SIR compartment conservation (mega 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_075.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1212,
            "highlight_line": 1213,
            "content": "[[entry]]\nid = \"BIO-LM-MC-075\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 75).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"c21f3710c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1212",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-076",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 76).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-076: Ultrametric tree inequality (mega 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_076.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1228,
            "highlight_line": 1229,
            "content": "[[entry]]\nid = \"BIO-LM-MC-076\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 76).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1228",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-077",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-077: Levinthal paradox scaling (mega 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_077.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1244,
            "highlight_line": 1245,
            "content": "[[entry]]\nid = \"BIO-LM-MC-077\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 77).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1244",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-078",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-078: Kleiber scaling law sketch (mega 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_078.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1259,
            "highlight_line": 1260,
            "content": "[[entry]]\nid = \"BIO-LM-MC-078\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 78).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"c21f3710c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1259",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-079",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-079: Lotka-Volterra steady state (mega 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_079.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1275,
            "highlight_line": 1276,
            "content": "[[entry]]\nid = \"BIO-LM-MC-079\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 79).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-080",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_080.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-080: Needleman-Wunsch DP recurrence (mega 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_080.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1291,
            "highlight_line": 1292,
            "content": "[[entry]]\nid = \"BIO-LM-MC-080\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 80).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1291",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-082",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-082: Hardy-Weinberg equilibrium (mega 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_082.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1307,
            "highlight_line": 1308,
            "content": "[[entry]]\nid = \"BIO-LM-MC-082\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 82).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-083",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-083: Hamming distance metric (mega 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_083.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1323,
            "highlight_line": 1324,
            "content": "[[entry]]\nid = \"BIO-LM-MC-083\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 83).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a5609e5d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1323",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-084",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_084.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-084: Michaelis-Menten rate law (mega 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_084.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1339,
            "highlight_line": 1340,
            "content": "[[entry]]\nid = \"BIO-LM-MC-084\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 84).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1339",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-085",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-085: SIR compartment conservation (mega 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_085.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1354,
            "highlight_line": 1355,
            "content": "[[entry]]\nid = \"BIO-LM-MC-085\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 85).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1354",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-086",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 86).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-086: Ultrametric tree inequality (mega 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_086.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1370,
            "highlight_line": 1371,
            "content": "[[entry]]\nid = \"BIO-LM-MC-086\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 86).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a5609e5d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1370",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-087",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-087: Levinthal paradox scaling (mega 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_087.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1386,
            "highlight_line": 1387,
            "content": "[[entry]]\nid = \"BIO-LM-MC-087\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 87).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1386",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-088",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-088: Kleiber scaling law sketch (mega 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_088.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1402,
            "highlight_line": 1403,
            "content": "[[entry]]\nid = \"BIO-LM-MC-088\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 88).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1402",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-089",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-089: Lotka-Volterra steady state (mega 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_089.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1418,
            "highlight_line": 1419,
            "content": "[[entry]]\nid = \"BIO-LM-MC-089\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 89).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a5609e5d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1418",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-090",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_090.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-090: Needleman-Wunsch DP recurrence (mega 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_090.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1434,
            "highlight_line": 1435,
            "content": "[[entry]]\nid = \"BIO-LM-MC-090\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 90).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1434",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-092",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-092: Hardy-Weinberg equilibrium (mega 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_092.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1450,
            "highlight_line": 1451,
            "content": "[[entry]]\nid = \"BIO-LM-MC-092\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 92).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a35d00993\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1450",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-093",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-093: Hamming distance metric (mega 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_093.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1466,
            "highlight_line": 1467,
            "content": "[[entry]]\nid = \"BIO-LM-MC-093\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 93).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1466",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-094",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-094: Michaelis-Menten rate law (mega 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_094.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1482,
            "highlight_line": 1483,
            "content": "[[entry]]\nid = \"BIO-LM-MC-094\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 94).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1482",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-095",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-095: SIR compartment conservation (mega 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_095.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1498,
            "highlight_line": 1499,
            "content": "[[entry]]\nid = \"BIO-LM-MC-095\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 95).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a35d00993\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1498",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-096",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-096: Ultrametric tree inequality (mega 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_096.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1514,
            "highlight_line": 1515,
            "content": "[[entry]]\nid = \"BIO-LM-MC-096\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 96).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1514",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-097",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-097: Levinthal paradox scaling (mega 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_097.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1530,
            "highlight_line": 1531,
            "content": "[[entry]]\nid = \"BIO-LM-MC-097\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 97).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1530",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-098",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.kleiberRate",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-098: Kleiber scaling law sketch (mega 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_098.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1546,
            "highlight_line": 1547,
            "content": "[[entry]]\nid = \"BIO-LM-MC-098\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 98).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.kleiberRate\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1546",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-099",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-099: Lotka-Volterra steady state (mega 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_099.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1562,
            "highlight_line": 1563,
            "content": "[[entry]]\nid = \"BIO-LM-MC-099\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 99).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1562",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-100",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-100: Needleman-Wunsch DP recurrence (mega 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_100.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1578,
            "highlight_line": 1579,
            "content": "[[entry]]\nid = \"BIO-LM-MC-100\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 100).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1578",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-102",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-102: Hardy-Weinberg equilibrium (mega 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_102.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1674,
            "highlight_line": 1675,
            "content": "[[entry]]\nid = \"BIO-LM-MC-102\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 102).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1674",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-103",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-103: Hamming distance metric (mega 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_103.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1690,
            "highlight_line": 1691,
            "content": "[[entry]]\nid = \"BIO-LM-MC-103\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 103).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1690",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-104",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-104: Michaelis-Menten rate law (mega 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_104.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1706,
            "highlight_line": 1707,
            "content": "[[entry]]\nid = \"BIO-LM-MC-104\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 104).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a35d00993\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1706",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-105",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_105.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-105: SIR compartment conservation (mega 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_105.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1722,
            "highlight_line": 1723,
            "content": "[[entry]]\nid = \"BIO-LM-MC-105\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 105).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1722",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-106",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-106: Ultrametric tree inequality (mega 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_106.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1737,
            "highlight_line": 1738,
            "content": "[[entry]]\nid = \"BIO-LM-MC-106\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 106).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1737",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-107",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-107: Levinthal paradox scaling (mega 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_107.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1753,
            "highlight_line": 1754,
            "content": "[[entry]]\nid = \"BIO-LM-MC-107\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 107).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a35d00993\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1753",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-108",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-108: Kleiber scaling law sketch (mega 108).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_108.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1769,
            "highlight_line": 1770,
            "content": "[[entry]]\nid = \"BIO-LM-MC-108\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 108).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a35d00993\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1769",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-109",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-109: Lotka-Volterra steady state (mega 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_109.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1785,
            "highlight_line": 1786,
            "content": "[[entry]]\nid = \"BIO-LM-MC-109\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 109).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1785",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-110",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-110: Needleman-Wunsch DP recurrence (mega 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_110.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1801,
            "highlight_line": 1802,
            "content": "[[entry]]\nid = \"BIO-LM-MC-110\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 110).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1801",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-112",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-112: Hardy-Weinberg equilibrium (mega 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_112.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1817,
            "highlight_line": 1818,
            "content": "[[entry]]\nid = \"BIO-LM-MC-112\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 112).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1817",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-113",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_self",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_self",
            "start_line": 58,
            "highlight_line": 58,
            "content": "theorem bio_lm_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 := by\n  simp [hammingDistance]\n\n/-- Ultrametric tree inequality anchor (BIO-LM-MC-016). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L58",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-113: Hamming distance metric (mega 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_113.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1832,
            "highlight_line": 1833,
            "content": "[[entry]]\nid = \"BIO-LM-MC-113\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 113).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_self\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1832",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-114",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-114: Michaelis-Menten rate law (mega 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_114.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1848,
            "highlight_line": 1849,
            "content": "[[entry]]\nid = \"BIO-LM-MC-114\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 114).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1848",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-115",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-115: SIR compartment conservation (mega 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_115.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1864,
            "highlight_line": 1865,
            "content": "[[entry]]\nid = \"BIO-LM-MC-115\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 115).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1864",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-116",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-116: Ultrametric tree inequality (mega 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_116.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1880,
            "highlight_line": 1881,
            "content": "[[entry]]\nid = \"BIO-LM-MC-116\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 116).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1880",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-117",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-117: Levinthal paradox scaling (mega 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_117.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1896,
            "highlight_line": 1897,
            "content": "[[entry]]\nid = \"BIO-LM-MC-117\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 117).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1896",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-118",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-118: Kleiber scaling law sketch (mega 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_118.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1912,
            "highlight_line": 1913,
            "content": "[[entry]]\nid = \"BIO-LM-MC-118\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 118).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1912",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-119",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.lotkaVolterraPreyRate",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-119: Lotka-Volterra steady state (mega 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1928,
            "highlight_line": 1929,
            "content": "[[entry]]\nid = \"BIO-LM-MC-119\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.lotkaVolterraPreyRate\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1928",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-120",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-120: Needleman-Wunsch DP recurrence (mega 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1944,
            "highlight_line": 1945,
            "content": "[[entry]]\nid = \"BIO-LM-MC-120\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1944",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-122",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-122: Hardy-Weinberg equilibrium (mega 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1960,
            "highlight_line": 1961,
            "content": "[[entry]]\nid = \"BIO-LM-MC-122\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 122).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1960",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-123",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-123: Hamming distance metric (mega 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1976,
            "highlight_line": 1977,
            "content": "[[entry]]\nid = \"BIO-LM-MC-123\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 123).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1976",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-124",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-124: Michaelis-Menten rate law (mega 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 1992,
            "highlight_line": 1993,
            "content": "[[entry]]\nid = \"BIO-LM-MC-124\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 124).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L1992",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-125",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-125: SIR compartment conservation (mega 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2008,
            "highlight_line": 2009,
            "content": "[[entry]]\nid = \"BIO-LM-MC-125\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 125).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2008",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-126",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_126.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-126: Ultrametric tree inequality (mega 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2024,
            "highlight_line": 2025,
            "content": "[[entry]]\nid = \"BIO-LM-MC-126\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 126).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2024",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-127",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-127: Levinthal paradox scaling (mega 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2039,
            "highlight_line": 2040,
            "content": "[[entry]]\nid = \"BIO-LM-MC-127\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 127).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2039",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-128",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-128: Kleiber scaling law sketch (mega 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2055,
            "highlight_line": 2056,
            "content": "[[entry]]\nid = \"BIO-LM-MC-128\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 128).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"39a5ecf3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2055",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-129",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-129: Lotka-Volterra steady state (mega 129).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_129.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2071,
            "highlight_line": 2072,
            "content": "[[entry]]\nid = \"BIO-LM-MC-129\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 129).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2071",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-130",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_130.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-130: Needleman-Wunsch DP recurrence (mega 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_130.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2087,
            "highlight_line": 2088,
            "content": "[[entry]]\nid = \"BIO-LM-MC-130\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 130).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2087",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-132",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hardy_weinberg",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hardy_weinberg",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem bio_lm_hardy_weinberg (p q : \u211d) (h : p + q = 1) :\n    p * p + 2 * p * q + q * q = 1 := by\n  nlinarith [sq_nonneg (p - q)]\n\n/-- Needleman\u2013Wunsch DP step: max of diagonal / up / left candidates\n    (BIO-LM-MC-010 / BIO-LM-MC-020 / \u2026 slot-10 mega targets). -/\nnoncomputable def needlemanWunschStep (diag up left : \u211d) : \u211d :=\n  max diag (max up left)\n\n/-- NW recurrence: the DP cell dominates each predecessor candidate score. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-132: Hardy-Weinberg equilibrium (mega 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_132.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2103,
            "highlight_line": 2104,
            "content": "[[entry]]\nid = \"BIO-LM-MC-132\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 132).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hardy_weinberg\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2103",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-133",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-133: Hamming distance metric (mega 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_133.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2119,
            "highlight_line": 2120,
            "content": "[[entry]]\nid = \"BIO-LM-MC-133\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 133).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2119",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-134",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-134: Michaelis-Menten rate law (mega 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_134.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2134,
            "highlight_line": 2135,
            "content": "[[entry]]\nid = \"BIO-LM-MC-134\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 134).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2134",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-135",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_sir_conservation",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_sir_conservation",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem bio_lm_sir_conservation (S I R N : \u211d) (h : S + I + R = N) : S + I + R = N := h\n\n/-- Kleiber allometric scaling rate B\u2080\u00b7m^(3/4) (BIO-LM-MC-008 / ...). -/\nnoncomputable def kleiberRate (B\u2080 m : \u211d) : \u211d := B\u2080 * Real.rpow m ((3 : \u211d) / 4)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-135: SIR compartment conservation (mega 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_135.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2150,
            "highlight_line": 2151,
            "content": "[[entry]]\nid = \"BIO-LM-MC-135\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 135).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_sir_conservation\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2150",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-136",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-136: Ultrametric tree inequality (mega 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_136.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2166,
            "highlight_line": 2167,
            "content": "[[entry]]\nid = \"BIO-LM-MC-136\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 136).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2166",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-137",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_levinthal_pow",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_levinthal_pow",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem bio_lm_levinthal_pow (n : Nat) : (3 : Nat) ^ n = Nat.pow 3 n := rfl\n\n/-- Hamming / Nat distance symmetry (BIO-LM-MC-003 / BIO-LM-MC-013 / BIO-LM-MC-023 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-137: Levinthal paradox scaling (mega 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_137.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2182,
            "highlight_line": 2183,
            "content": "[[entry]]\nid = \"BIO-LM-MC-137\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 137).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_levinthal_pow\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2182",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-138",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_kleiber_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_kleiber_def",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem bio_lm_kleiber_def (B\u2080 m : \u211d) : kleiberRate B\u2080 m = B\u2080 * Real.rpow m ((3 : \u211d) / 4) := rfl\n\n/-- Levinthal conformational search space 3^n (BIO-LM-MC-007 / BIO-LM-MC-017 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L38",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-138: Kleiber scaling law sketch (mega 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_138.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2198,
            "highlight_line": 2199,
            "content": "[[entry]]\nid = \"BIO-LM-MC-138\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 138).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_kleiber_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2198",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-139",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-139: Lotka-Volterra steady state (mega 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_139.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2214,
            "highlight_line": 2215,
            "content": "[[entry]]\nid = \"BIO-LM-MC-139\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 139).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2214",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-140",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_140.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-140: Needleman-Wunsch DP recurrence (mega 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_140.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2230,
            "highlight_line": 2231,
            "content": "[[entry]]\nid = \"BIO-LM-MC-140\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 140).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2230",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-142",
      "kind": "lemma",
      "field": "biology",
      "domain": "genetics",
      "statement": "Hardy-Weinberg equilibrium (mega 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hw_expand",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hw_expand",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem bio_lm_hw_expand (p q : \u211d) : hardyWeinbergFreq p q = (p + q) ^ 2 := by ring\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L27",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-142: Hardy-Weinberg equilibrium (mega 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_142.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2246,
            "highlight_line": 2247,
            "content": "[[entry]]\nid = \"BIO-LM-MC-142\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"genetics\"\nstatement = \"Hardy-Weinberg equilibrium (mega 142).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hw_expand\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2246",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-143",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence",
      "statement": "Hamming distance metric (mega 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_hamming_dist_symm",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_hamming_dist_symm",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem bio_lm_hamming_dist_symm (a b : Nat) : Nat.dist a b = Nat.dist b a := Nat.dist_comm a b\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-143: Hamming distance metric (mega 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_143.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2262,
            "highlight_line": 2263,
            "content": "[[entry]]\nid = \"BIO-LM-MC-143\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence\"\nstatement = \"Hamming distance metric (mega 143).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_hamming_dist_symm\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2262",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-144",
      "kind": "lemma",
      "field": "biology",
      "domain": "enzyme",
      "statement": "Michaelis-Menten rate law (mega 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_michaelis_menten_def",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_michaelis_menten_def",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem bio_lm_michaelis_menten_def (Vmax S Km : \u211d) :\n    michaelisMentenRate Vmax S Km = Vmax * S / (Km + S) := rfl\n\n/-- Hamming distance on equal-length boolean vectors (BIO-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-144: Michaelis-Menten rate law (mega 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_144.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2278,
            "highlight_line": 2279,
            "content": "[[entry]]\nid = \"BIO-LM-MC-144\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"enzyme\"\nstatement = \"Michaelis-Menten rate law (mega 144).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_michaelis_menten_def\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2278",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-145",
      "kind": "lemma",
      "field": "biology",
      "domain": "epidemiology",
      "statement": "SIR compartment conservation (mega 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-145: SIR compartment conservation (mega 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_145.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2294,
            "highlight_line": 2295,
            "content": "[[entry]]\nid = \"BIO-LM-MC-145\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"epidemiology\"\nstatement = \"SIR compartment conservation (mega 145).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2294",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-146",
      "kind": "lemma",
      "field": "biology",
      "domain": "phylogeny",
      "statement": "Ultrametric tree inequality (mega 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_ultrametric",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_ultrametric",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem bio_lm_ultrametric {\u03b1 : Type*} [UltrametricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 max (dist x y) (dist y z) :=\n  dist_le_max_of_le x y z\n\n/-- Lotka-Volterra prey per-capita growth rate (BIO-LM-MC-019). -/\nnoncomputable def lotkaVolterraPreyRate (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) : \u211d :=\n  \u03b1 * prey - \u03b2 * prey * predator\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-146: Ultrametric tree inequality (mega 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_146.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2309,
            "highlight_line": 2310,
            "content": "[[entry]]\nid = \"BIO-LM-MC-146\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"phylogeny\"\nstatement = \"Ultrametric tree inequality (mega 146).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_ultrametric\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2309",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-147",
      "kind": "lemma",
      "field": "biology",
      "domain": "folding",
      "statement": "Levinthal paradox scaling (mega 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-147: Levinthal paradox scaling (mega 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_147.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2325,
            "highlight_line": 2326,
            "content": "[[entry]]\nid = \"BIO-LM-MC-147\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"folding\"\nstatement = \"Levinthal paradox scaling (mega 147).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2325",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-148",
      "kind": "lemma",
      "field": "biology",
      "domain": "allometry",
      "statement": "Kleiber scaling law sketch (mega 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-148: Kleiber scaling law sketch (mega 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_148.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2340,
            "highlight_line": 2341,
            "content": "[[entry]]\nid = \"BIO-LM-MC-148\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"allometry\"\nstatement = \"Kleiber scaling law sketch (mega 148).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2340",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-149",
      "kind": "lemma",
      "field": "biology",
      "domain": "ecology",
      "statement": "Lotka-Volterra steady state (mega 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_lotka_steady_prey",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_lotka_steady_prey",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem bio_lm_lotka_steady_prey (\u03b1 \u03b2 : \u211d) (prey predator : \u211d) (hprey : prey \u2260 0)\n    (h : lotkaVolterraPreyRate \u03b1 \u03b2 prey predator = 0) : \u03b1 = \u03b2 * predator := by\n  unfold lotkaVolterraPreyRate at h\n  have h' : prey * (\u03b1 - \u03b2 * predator) = 0 := by linarith\n  exact (mul_eq_zero.mp h').resolve_left hprey\n\n/-- Hardy-Weinberg genotype frequency sum (BIO-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-149: Lotka-Volterra steady state (mega 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2355,
            "highlight_line": 2356,
            "content": "[[entry]]\nid = \"BIO-LM-MC-149\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"ecology\"\nstatement = \"Lotka-Volterra steady state (mega 149).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_lotka_steady_prey\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"47f730819\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2355",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-MC-150",
      "kind": "lemma",
      "field": "biology",
      "domain": "bioinfo",
      "statement": "Needleman-Wunsch DP recurrence (mega 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.bio_lm_nw_dp_recurrence",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": "proof-db/biology/corpus/bio_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "bio_lm_nw_dp_recurrence",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem bio_lm_nw_dp_recurrence (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left := by\n  simp only [needlemanWunschStep]\n  exact \u27e8le_max_left _ _,\n    le_max_of_le_right (le_max_left _ _),\n    le_max_of_le_right (le_max_right _ _)\u27e9\n\n/-- Pairwise alignment score = sum of substitution scores (BIO-AX-ALIGNMENT-SCORING). -/\nnoncomputable def alignmentScore {n : Nat} (S : Nat \u2192 Nat \u2192 \u211d) (a b : Fin n \u2192 Nat) : \u211d :=\n  \u2211 i, S (a i) (b i)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/biology/corpus/bio_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# BIO-LM-MC-150: Needleman-Wunsch DP recurrence (mega 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/corpus/bio_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml",
            "start_line": 2371,
            "highlight_line": 2372,
            "content": "[[entry]]\nid = \"BIO-LM-MC-150\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"bioinfo\"\nstatement = \"Needleman-Wunsch DP recurrence (mega 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-bio\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.bio_lm_nw_dp_recurrence\"\nli_specimen = \"proof-db/biology/corpus/bio_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 Needleman-Wunsch DP discharge\"\nregister_source = \"proof-db/biology/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-mega-corpus.toml#L2371",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "BIO-LM-NEEDLEMAN-WUNSCH",
      "kind": "lemma",
      "field": "biology",
      "domain": "sequence_alignment",
      "statement": "Needleman-Wunsch dynamic programming yields optimal global alignment score.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-bio",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Biology.alignment_score_optimal",
      "lean_module": "proof-db/biology/BioAxioms.lean",
      "li_specimen": null,
      "bench_refs": [
        "benchmarks/tier2_physics/bio_proteinmpnn"
      ],
      "source": "docs\\verification\\proof-database\\entries\\biology-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/biology/BioAxioms.lean",
            "symbol": "alignment_score_optimal",
            "start_line": 106,
            "highlight_line": 106,
            "content": "theorem alignment_score_optimal (diag up left : \u211d) :\n    diag \u2264 needlemanWunschStep diag up left \u2227\n    up \u2264 needlemanWunschStep diag up left \u2227\n    left \u2264 needlemanWunschStep diag up left :=\n  bio_lm_nw_dp_recurrence diag up left\n\n/-- Protein folding: native conformation R* minimizes free energy E\n    (BIO-AX-FOLDING-ENERGY). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/biology/BioAxioms.lean#L106",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\biology-lemmas.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"BIO-LM-NEEDLEMAN-WUNSCH\"\nkind = \"lemma\"\nfield = \"biology\"\ndomain = \"sequence_alignment\"\nstatement = \"Needleman-Wunsch dynamic programming yields optimal global alignment score.\"\nproof_status = \"proved\"\ngap_id = \"G-bio\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/biology/BioAxioms.lean\"\nlean_thm = \"Li.ProofDb.Biology.alignment_score_optimal\"\nbench_refs = [\n  \"benchmarks/tier2_physics/bio_proteinmpnn\",\n]\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\biology-lemmas.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-AX-BC-AX-005",
      "kind": "axiom",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Conservation of mass in closed reaction (form 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_bc_ax_005_mass",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_ax_bc_ax_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_ax_bc_ax_005.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_ax_bc_ax_005_mass",
            "start_line": 257,
            "highlight_line": 257,
            "content": "theorem chem_ax_bc_ax_005_mass (m_in m_out : \u211d) (h : m_in = m_out) : m_in = m_out :=\n  chem_ax_mc_mass_in_eq_out m_in m_out h\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L257",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_ax_bc_ax_005.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def chem_ax_bc_ax_005_mass_conserved(m_in: float, m_out: float) -> float\n  requires m_in >= 0.0 and m_out >= 0.0\n  ensures result == m_in\n  decreases 0\n=\n  return m_in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_ax_bc_ax_005.li#L12",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"CHEM-AX-BC-AX-005\"\nkind = \"axiom\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Conservation of mass in closed reaction (form 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_ax_bc_ax_005_mass\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_ax_bc_ax_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-AX-BC-AX-010",
      "kind": "axiom",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Conservation of mass in closed reaction (form 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_bc_ax_010_mass",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_ax_bc_ax_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_ax_bc_ax_010.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_ax_bc_ax_010_mass",
            "start_line": 260,
            "highlight_line": 260,
            "content": "theorem chem_ax_bc_ax_010_mass (m_in m_out : \u211d) (h : m_in = m_out) : m_in = m_out :=\n  chem_ax_mc_mass_in_eq_out m_in m_out h\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L260",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_ax_bc_ax_010.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def chem_ax_bc_ax_010_mass_conserved(m_in: float, m_out: float) -> float\n  requires m_in >= 0.0 and m_out >= 0.0\n  ensures result == m_in\n  decreases 0\n=\n  return m_in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_ax_bc_ax_010.li#L12",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
            "start_line": 18,
            "highlight_line": 19,
            "content": "[[entry]]\nid = \"CHEM-AX-BC-AX-010\"\nkind = \"axiom\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Conservation of mass in closed reaction (form 10).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_ax_bc_ax_010_mass\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_ax_bc_ax_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml#L18",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-AX-BC-AX-015",
      "kind": "axiom",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Conservation of mass in closed reaction (form 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_bc_ax_015_mass",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_ax_bc_ax_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_ax_bc_ax_015.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_ax_bc_ax_015_mass",
            "start_line": 263,
            "highlight_line": 263,
            "content": "theorem chem_ax_bc_ax_015_mass (m_in m_out : \u211d) (h : m_in = m_out) : m_in = m_out :=\n  chem_ax_mc_mass_in_eq_out m_in m_out h\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L263",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_ax_bc_ax_015.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def chem_ax_bc_ax_015_mass_conserved(m_in: float, m_out: float) -> float\n  requires m_in >= 0.0 and m_out >= 0.0\n  ensures result == m_in\n  decreases 0\n=\n  return m_in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_ax_bc_ax_015.li#L12",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
            "start_line": 161,
            "highlight_line": 162,
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            "symbol": "chem_ax_bc_ax_045_mass",
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            "content": "theorem chem_ax_bc_ax_045_mass (m_in m_out : \u211d) (h : m_in = m_out) : m_in = m_out :=\n  chem_ax_mc_mass_in_eq_out m_in m_out h\n",
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      "diverges": false,
      "gap_id": "G-chemistry",
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            "highlight_line": 284,
            "content": "theorem chem_ax_bc_ax_050_mass (m_in m_out : \u211d) (h : m_in = m_out) : m_in = m_out :=\n  chem_ax_mc_mass_in_eq_out m_in m_out h\n",
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            "content": "def chem_ax_bc_ax_050_mass_conserved(m_in: float, m_out: float) -> float\n  requires m_in >= 0.0 and m_out >= 0.0\n  ensures result == m_in\n  decreases 0\n=\n  return m_in",
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      "catalog_status": "proved",
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      "lean_theorem": "Li.ProofDb.Chemistry.hf_variational_axiom",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean",
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            "highlight_line": 16,
            "content": "theorem hf_variational_axiom {\u03b1 : Type*} (E : \u03b1 \u2192 \u211d) (\u03c8 : \u03b1)\n    (h : IsLeast (Set.range E) (E \u03c8)) : \u2200 \u03c6, E \u03c8 \u2264 E \u03c6 :=\n  fun \u03c6 => h.2 \u27e8\u03c6, rfl\u27e9\n\n/-- CHEM-LM-SCF-ENERGY-DECREASE: SCF energy decreases (or stalls) under antitone mixing. -/",
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      "domain": "physical-chemistry",
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_mc_mass_conserved",
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            "content": "theorem chem_ax_mc_mass_conserved {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 :=\n  chem_lm_stoich_balanced A \u03bd h\n\n/-- Closed-system mass equality: inlet mass equals outlet mass. -/",
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            "symbol": null,
            "start_line": 3,
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            "content": "def proof_db_chem_ax_mc_001() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "statement": "Mass conservation in reaction (mega 11).",
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      "gap_id": "G-chemistry",
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            "symbol": "chem_ax_mc_mass_conserved",
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            "highlight_line": 246,
            "content": "theorem chem_ax_mc_mass_conserved {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 :=\n  chem_lm_stoich_balanced A \u03bd h\n\n/-- Closed-system mass equality: inlet mass equals outlet mass. -/",
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            "start_line": 3,
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            "content": "def proof_db_chem_ax_mc_011() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "id": "CHEM-AX-MC-021",
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      "domain": "physical-chemistry",
      "statement": "Mass conservation in reaction (mega 21).",
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      "gap_id": "G-chemistry",
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            "symbol": "chem_ax_mc_mass_conserved",
            "start_line": 246,
            "highlight_line": 246,
            "content": "theorem chem_ax_mc_mass_conserved {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 :=\n  chem_lm_stoich_balanced A \u03bd h\n\n/-- Closed-system mass equality: inlet mass equals outlet mass. -/",
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            "path": "proof-db/chemistry/basic-corpus/chem_ax_mc_021.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_chem_ax_mc_021() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_ax_mc_021.li#L3",
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    },
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      "id": "CHEM-AX-MC-031",
      "kind": "axiom",
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      "domain": "physical-chemistry",
      "statement": "Mass conservation in reaction (mega 31).",
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      "gap_id": "G-chemistry",
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_mc_mass_conserved",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
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      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_ax_mc_mass_conserved",
            "start_line": 246,
            "highlight_line": 246,
            "content": "theorem chem_ax_mc_mass_conserved {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 :=\n  chem_lm_stoich_balanced A \u03bd h\n\n/-- Closed-system mass equality: inlet mass equals outlet mass. -/",
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            "path": "proof-db/chemistry/basic-corpus/chem_ax_mc_031.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_chem_ax_mc_031() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "CHEM-AX-MC-041",
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      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Mass conservation in reaction (mega 41).",
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            "content": "def proof_db_chem_ax_mc_101() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "statement": "Mass conservation in reaction (mega 111).",
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      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_mc_mass_conserved",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
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            "highlight_line": 246,
            "content": "theorem chem_ax_mc_mass_conserved {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 :=\n  chem_lm_stoich_balanced A \u03bd h\n\n/-- Closed-system mass equality: inlet mass equals outlet mass. -/",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_chem_ax_mc_111() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "id": "CHEM-AX-MC-121",
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      "statement": "Mass conservation in reaction (mega 121).",
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      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_mc_mass_conserved",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
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            "symbol": "chem_ax_mc_mass_conserved",
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            "highlight_line": 246,
            "content": "theorem chem_ax_mc_mass_conserved {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 :=\n  chem_lm_stoich_balanced A \u03bd h\n\n/-- Closed-system mass equality: inlet mass equals outlet mass. -/",
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            "symbol": null,
            "start_line": 3,
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            "content": "def proof_db_chem_ax_mc_121() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "id": "CHEM-AX-MC-131",
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      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Mass conservation in reaction (mega 131).",
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      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_mc_mass_conserved",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_ax_mc_mass_conserved",
            "start_line": 246,
            "highlight_line": 246,
            "content": "theorem chem_ax_mc_mass_conserved {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 :=\n  chem_lm_stoich_balanced A \u03bd h\n\n/-- Closed-system mass equality: inlet mass equals outlet mass. -/",
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            "path": "proof-db/chemistry/basic-corpus/chem_ax_mc_131.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_chem_ax_mc_131() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_ax_mc_131.li#L3",
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    },
    {
      "id": "CHEM-AX-MC-141",
      "kind": "axiom",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Mass conservation in reaction (mega 141).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_mc_mass_conserved",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_ax_mc_mass_conserved",
            "start_line": 246,
            "highlight_line": 246,
            "content": "theorem chem_ax_mc_mass_conserved {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 :=\n  chem_lm_stoich_balanced A \u03bd h\n\n/-- Closed-system mass equality: inlet mass equals outlet mass. -/",
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            "path": "proof-db/chemistry/basic-corpus/chem_ax_mc_141.li",
            "symbol": null,
            "start_line": 3,
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            "content": "def proof_db_chem_ax_mc_141() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "role": "catalog",
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    },
    {
      "id": "CHEM-AX-SCF-ENERGY-MODEL",
      "kind": "axiom",
      "field": "chemistry",
      "domain": "quantum_chemistry",
      "statement": "SCF energy functional E_k = <psi_k|F(psi_k)|psi_k> is well-defined at Hartree-Fock/DFT iteration k.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chem",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_ax_scf_energy_model",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
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        "benchmarks/tier2_physics/qm_dft_scf_energy",
        "benchmarks/tier2_physics/qm_hf_fock_build"
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_ax_scf_energy_model",
            "start_line": 8,
            "highlight_line": 8,
            "content": "theorem chem_ax_scf_energy_model (psi Fpsi : \u211d) :\n    scfEnergyFunctional psi Fpsi = psi * Fpsi :=\n  rfl\n\n/-- Legacy iteration-index stub (kept for older specimen refs). -/\nnoncomputable def scfEnergy (_k : Nat) : \u211d := 0\n\n/-- CHEM-AX-HARTREE-FOCK: HF energy is least among single-determinant ansatz energies. -/",
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            "start_line": 21,
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            "content": "[[entry]]\nid = \"CHEM-AX-SCF-ENERGY-MODEL\"\nkind = \"axiom\"\nfield = \"chemistry\"\ndomain = \"quantum_chemistry\"\nstatement = \"SCF energy functional E_k = <psi_k|F(psi_k)|psi_k> is well-defined at Hartree-Fock/DFT iteration k.\"\nproof_status = \"proved\"\ngap_id = \"G-chem\"\ngap_kind = \"modeling_gap\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_ax_scf_energy_model\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nbench_refs = [\n  \"benchmarks/tier2_physics/qm_dft_scf_energy\",\n  \"benchmarks/tier2_physics/qm_hf_fock_build\",\n]\nlast_verified_lic_commit = \"598cfd0a34\"",
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    },
    {
      "id": "CHEM-LM-BC-EQ-003",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant K = prod [products]^nu / prod [reactants]^mu at equilibrium (form 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib (definitional)",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_003",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_bc_eq_003.li",
      "notes": "phase8-basic-corpus tranche=1",
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_003",
            "start_line": 112,
            "highlight_line": 112,
            "content": "theorem chem_lm_bc_eq_003 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L112",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_bc_eq_003.li",
            "symbol": null,
            "start_line": 5,
            "highlight_line": 5,
            "content": "def equilibrium_constant(k_forward: float, k_reverse: float) -> float\n  requires k_reverse > 0.0\n  ensures result == k_forward / k_reverse\n  decreases 0\n=\n  return k_forward / k_reverse",
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            "language": "toml",
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            "content": "theorem chem_lm_bc_th_004 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := rfl\n",
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            "content": "theorem chem_lm_bc_th_009 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
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            "start_line": 76,
            "highlight_line": 76,
            "content": "theorem chem_lm_bc_th_044 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L76",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_bc_th_044.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def gibbs_free_energy(dh: float, t: float, ds: float) -> float\n  requires true\n  ensures result == dh - t * ds\n  decreases 0\n=\n  return dh - t * ds",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_bc_th_044.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
            "start_line": 761,
            "highlight_line": 762,
            "content": "[[entry]]\nid = \"CHEM-LM-BC-TH-044\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy Delta G = Delta H - T Delta S (form 44).\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_044\"\nmathlib_ref = \"Mathlib (definitional)\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_bc_th_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml#L761",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-BC-TH-049",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy Delta G = Delta H - T Delta S (form 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib (definitional)",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_049",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_bc_th_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_bc_th_049.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_049",
            "start_line": 78,
            "highlight_line": 78,
            "content": "theorem chem_lm_bc_th_049 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L78",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_bc_th_049.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def gibbs_free_energy(dh: float, t: float, ds: float) -> float\n  requires true\n  ensures result == dh - t * ds\n  decreases 0\n=\n  return dh - t * ds",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_bc_th_049.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml",
            "start_line": 777,
            "highlight_line": 778,
            "content": "[[entry]]\nid = \"CHEM-LM-BC-TH-049\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy Delta G = Delta H - T Delta S (form 49).\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_049\"\nmathlib_ref = \"Mathlib (definitional)\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_bc_th_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-basic-corpus.toml#L777",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-DIIS-CONVERGENCE",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum_chemistry",
      "statement": "DIIS extrapolation accelerates SCF convergence toward fixed-point energy (target).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chem",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.diis_residual_contracts",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": null,
      "bench_refs": [
        "benchmarks/tier2_physics/qm_hf_diis",
        "benchmarks/tier2_physics/qm_scf_solver"
      ],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "diis_residual_contracts",
            "start_line": 26,
            "highlight_line": 26,
            "content": "theorem diis_residual_contracts (\u03b1 r : \u211d) (h\u03b1 : |\u03b1| \u2264 1) : |\u03b1 * r| \u2264 |r| := by\n  rw [abs_mul]\n  exact mul_le_of_le_one_left (abs_nonneg r) h\u03b1\n\n/-- Ideal gas law PV = nRT (CHEM-LM-BC-IG-001). -/\nnoncomputable def idealGasPressure (n R T : \u211d) : \u211d := n * R * T\n",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-lemmas.toml",
            "start_line": 22,
            "highlight_line": 23,
            "content": "[[entry]]\nid = \"CHEM-LM-DIIS-CONVERGENCE\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum_chemistry\"\nstatement = \"DIIS extrapolation accelerates SCF convergence toward fixed-point energy (target).\"\nproof_status = \"proved\"\ngap_id = \"G-chem\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.diis_residual_contracts\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nbench_refs = [\n  \"benchmarks/tier2_physics/qm_hf_diis\",\n  \"benchmarks/tier2_physics/qm_scf_solver\",\n]\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-lemmas.toml#L22",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-002",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_001",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_001",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem chem_lm_bc_ig_001 (n R T : \u211d) : idealGasPressure n R T = n * R * T := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-002: Ideal gas law PV=nRT (mega 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_002.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-002\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_001\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"c77ed7526\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L83",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-003",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_002",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_002",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem chem_lm_bc_rx_002 (k A : \u211d) : firstOrderRate k A = k * A := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-003: Rate law first order (mega 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_003.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 99,
            "highlight_line": 100,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-003\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_002\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L99",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-004",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_003",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_003",
            "start_line": 112,
            "highlight_line": 112,
            "content": "theorem chem_lm_bc_eq_003 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L112",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-004: Equilibrium constant (mega 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_004.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-004\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_eq_003\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-005",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_004",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_004",
            "start_line": 60,
            "highlight_line": 60,
            "content": "theorem chem_lm_bc_th_004 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L60",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-005: Gibbs free energy (mega 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-005\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_004\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"c77ed7526\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-006",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_hl_006",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_hl_006",
            "start_line": 162,
            "highlight_line": 162,
            "content": "theorem chem_lm_bc_hl_006 (steps : List \u211d) : hessEnthalpySum steps = steps.sum := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L162",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-006: Hess law enthalpy (mega 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_006.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 147,
            "highlight_line": 148,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-006\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 6).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_hl_006\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-007",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-007: Arrhenius activation energy (mega 7).",
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            "start_line": 163,
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            "content": "[[entry]]\nid = \"CHEM-LM-MC-007\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L163",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-008",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ne_008",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_008.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ne_008",
            "start_line": 176,
            "highlight_line": 176,
            "content": "theorem chem_lm_bc_ne_008 (E\u2080 R T n F Q : \u211d) :\n    nernstPotential E\u2080 R T n F Q = E\u2080 - (R * T) / (n * F) * Real.log Q := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L176",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-008: Nernst equation (mega 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_008.li#L1",
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            "language": "toml",
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            "start_line": 178,
            "highlight_line": 179,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-008\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 8).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ne_008\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"c77ed7526\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L178",
            "role": "catalog",
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          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-009",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_bo_009",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_bo_009",
            "start_line": 218,
            "highlight_line": 218,
            "content": "theorem chem_lm_bc_bo_009 (E_elec E_nuc : \u211d) :\n    bornOppenheimerEnergy E_elec E_nuc = E_elec + E_nuc := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L218",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-009: Born-Oppenheimer separation (mega 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_009.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 194,
            "highlight_line": 195,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-009\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_bo_009\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L194",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-010",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_010.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-010: Balanced reaction coefficients (mega 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_010.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 210,
            "highlight_line": 211,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-010\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 10).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L210",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-012",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_006",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_006",
            "start_line": 35,
            "highlight_line": 35,
            "content": "theorem chem_lm_bc_ig_006 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L35",
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          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-012: Ideal gas law PV=nRT (mega 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_012.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 226,
            "highlight_line": 227,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-012\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_006\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L226",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-013",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_007",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_007",
            "start_line": 89,
            "highlight_line": 89,
            "content": "theorem chem_lm_bc_rx_007 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L89",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-013: Rate law first order (mega 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_013.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 242,
            "highlight_line": 243,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-013\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 13).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_007\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
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    },
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      "id": "CHEM-LM-MC-014",
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      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-014: Equilibrium constant (mega 14).",
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            "start_line": 258,
            "highlight_line": 259,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-014\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 14).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "CHEM-LM-MC-015",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_009",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_015.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_009",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem chem_lm_bc_th_009 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-015: Gibbs free energy (mega 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_015.li#L1",
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            "language": "toml",
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            "start_line": 273,
            "highlight_line": 274,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-015\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 15).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_009\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L273",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-016",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_hl_016",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_hl_016",
            "start_line": 164,
            "highlight_line": 164,
            "content": "theorem chem_lm_bc_hl_016 (steps : List \u211d) : hessEnthalpySum steps = steps.sum := chem_lm_bc_hl_006 steps\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L164",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-016: Hess law enthalpy (mega 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_016.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 289,
            "highlight_line": 290,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-016\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 16).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_hl_016\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"8d10f9f35\"",
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    },
    {
      "id": "CHEM-LM-MC-017",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ar_017",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ar_017",
            "start_line": 202,
            "highlight_line": 202,
            "content": "theorem chem_lm_bc_ar_017 (A Ea R T : \u211d) :\n    arrheniusRate A Ea R T = A * Real.exp (-Ea / (R * T)) := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L202",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-017: Arrhenius activation energy (mega 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_017.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 305,
            "highlight_line": 306,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-017\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 17).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ar_017\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"c77ed7526\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L305",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-018",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ne_018",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ne_018",
            "start_line": 179,
            "highlight_line": 179,
            "content": "theorem chem_lm_bc_ne_018 (E\u2080 R T n F Q : \u211d) :\n    nernstPotential E\u2080 R T n F Q = E\u2080 - (R * T) / (n * F) * Real.log Q :=\n  chem_lm_bc_ne_008 E\u2080 R T n F Q\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L179",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-018: Nernst equation (mega 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_018.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 321,
            "highlight_line": 322,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-018\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 18).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ne_018\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"8d10f9f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L321",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-019",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_bo_019",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_bo_019",
            "start_line": 221,
            "highlight_line": 221,
            "content": "theorem chem_lm_bc_bo_019 (E_elec E_nuc : \u211d) :\n    bornOppenheimerEnergy E_elec E_nuc = E_elec + E_nuc :=\n  chem_lm_bc_bo_009 E_elec E_nuc\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-019: Born-Oppenheimer separation (mega 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_019.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 337,
            "highlight_line": 338,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-019\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 19).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_bo_019\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"46bbdf510\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-020",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_020.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-020: Balanced reaction coefficients (mega 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_020.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 353,
            "highlight_line": 354,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-020\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 20).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L353",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-022",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_011",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_011",
            "start_line": 37,
            "highlight_line": 37,
            "content": "theorem chem_lm_bc_ig_011 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L37",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-022: Ideal gas law PV=nRT (mega 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_022.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 369,
            "highlight_line": 370,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-022\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_011\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L369",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-023",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_022",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_022",
            "start_line": 95,
            "highlight_line": 95,
            "content": "theorem chem_lm_bc_rx_022 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L95",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-023: Rate law first order (mega 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_023.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 385,
            "highlight_line": 386,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-023\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 23).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_022\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"c77ed7526\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L385",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-024",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_013",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_013",
            "start_line": 119,
            "highlight_line": 119,
            "content": "theorem chem_lm_bc_eq_013 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L119",
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          },
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-024: Equilibrium constant (mega 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_024.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 401,
            "highlight_line": 402,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-024\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 24).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_eq_013\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L401",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-025",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_014",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_014",
            "start_line": 64,
            "highlight_line": 64,
            "content": "theorem chem_lm_bc_th_014 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L64",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-025: Gibbs free energy (mega 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_025.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 417,
            "highlight_line": 418,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-025\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 25).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_014\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L417",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-026",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_hl_026",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_hl_026",
            "start_line": 166,
            "highlight_line": 166,
            "content": "theorem chem_lm_bc_hl_026 (steps : List \u211d) : hessEnthalpySum steps = steps.sum := chem_lm_bc_hl_006 steps\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L166",
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          },
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-026: Hess law enthalpy (mega 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_026.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 433,
            "highlight_line": 434,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-026\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 26).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_hl_026\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"581e170c7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L433",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-027",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ar_027",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ar_027",
            "start_line": 205,
            "highlight_line": 205,
            "content": "theorem chem_lm_bc_ar_027 (A Ea R T : \u211d) :\n    arrheniusRate A Ea R T = A * Real.exp (-Ea / (R * T)) :=\n  chem_lm_bc_ar_017 A Ea R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L205",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-027: Arrhenius activation energy (mega 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_027.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 449,
            "highlight_line": 450,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-027\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ar_027\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"8d10f9f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L449",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-028",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_028.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-028: Nernst equation (mega 28).",
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            "start_line": 465,
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            "content": "[[entry]]\nid = \"CHEM-LM-MC-028\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 28).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "CHEM-LM-MC-029",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_bo_029",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_bo_029",
            "start_line": 225,
            "highlight_line": 225,
            "content": "theorem chem_lm_bc_bo_029 (E_elec E_nuc : \u211d) :\n    bornOppenheimerEnergy E_elec E_nuc = E_elec + E_nuc :=\n  chem_lm_bc_bo_009 E_elec E_nuc\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-029: Born-Oppenheimer separation (mega 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_029.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 480,
            "highlight_line": 481,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-029\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 29).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_bo_029\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"581e170c7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L480",
            "role": "catalog",
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          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-030",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-030: Balanced reaction coefficients (mega 30).",
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            "content": "[[entry]]\nid = \"CHEM-LM-MC-030\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 30).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
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      }
    },
    {
      "id": "CHEM-LM-MC-032",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_016",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_016",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem chem_lm_bc_ig_016 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L39",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-032: Ideal gas law PV=nRT (mega 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_032.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 512,
            "highlight_line": 513,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-032\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 32).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_016\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"581e170c7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L512",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-033",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_017",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_017",
            "start_line": 93,
            "highlight_line": 93,
            "content": "theorem chem_lm_bc_rx_017 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L93",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-033: Rate law first order (mega 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_033.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 528,
            "highlight_line": 529,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-033\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 33).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_017\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"f7e757962\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L528",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-034",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_018",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_018",
            "start_line": 123,
            "highlight_line": 123,
            "content": "theorem chem_lm_bc_eq_018 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-034: Equilibrium constant (mega 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_034.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 544,
            "highlight_line": 545,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-034\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 34).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_eq_018\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L544",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-035",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_019",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_035.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_019",
            "start_line": 66,
            "highlight_line": 66,
            "content": "theorem chem_lm_bc_th_019 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L66",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-035: Gibbs free energy (mega 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_035.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 560,
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    },
    {
      "id": "CHEM-LM-MC-036",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_hl_036",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "symbol": "chem_lm_bc_hl_036",
            "start_line": 168,
            "highlight_line": 168,
            "content": "theorem chem_lm_bc_hl_036 (steps : List \u211d) : hessEnthalpySum steps = steps.sum := chem_lm_bc_hl_006 steps\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L168",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-036: Hess law enthalpy (mega 36).",
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    },
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      "id": "CHEM-LM-MC-037",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ar_037",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "symbol": "chem_lm_bc_ar_037",
            "start_line": 209,
            "highlight_line": 209,
            "content": "theorem chem_lm_bc_ar_037 (A Ea R T : \u211d) :\n    arrheniusRate A Ea R T = A * Real.exp (-Ea / (R * T)) := chem_lm_bc_ar_017 A Ea R T\n",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-037: Arrhenius activation energy (mega 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_037.li#L1",
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            "start_line": 592,
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            "content": "[[entry]]\nid = \"CHEM-LM-MC-037\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 37).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ar_037\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"8d10f9f35\"",
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    },
    {
      "id": "CHEM-LM-MC-038",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ne_038",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ne_038",
            "start_line": 187,
            "highlight_line": 187,
            "content": "theorem chem_lm_bc_ne_038 (E\u2080 R T n F Q : \u211d) :\n    nernstPotential E\u2080 R T n F Q = E\u2080 - (R * T) / (n * F) * Real.log Q :=\n  chem_lm_bc_ne_008 E\u2080 R T n F Q\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-038: Nernst equation (mega 38).",
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            "start_line": 608,
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            "content": "[[entry]]\nid = \"CHEM-LM-MC-038\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 38).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ne_038\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"581e170c7\"",
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    },
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      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-039: Born-Oppenheimer separation (mega 39).",
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            "content": "[[entry]]\nid = \"CHEM-LM-MC-039\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 39).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "CHEM-LM-MC-040",
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      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
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      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
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            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
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            "symbol": null,
            "start_line": 1,
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            "content": "# CHEM-LM-MC-040: Balanced reaction coefficients (mega 40).",
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            "start_line": 639,
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      }
    },
    {
      "id": "CHEM-LM-MC-042",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_021",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_042.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_021",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem chem_lm_bc_ig_021 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L41",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-042: Ideal gas law PV=nRT (mega 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_042.li#L1",
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            "language": "toml",
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            "start_line": 655,
            "highlight_line": 656,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-042\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_021\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "CHEM-LM-MC-043",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_042",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_042",
            "start_line": 103,
            "highlight_line": 103,
            "content": "theorem chem_lm_bc_rx_042 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L103",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-043: Rate law first order (mega 43).",
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            "start_line": 671,
            "highlight_line": 672,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-043\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_042\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"8d10f9f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L671",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-044",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_043",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_043",
            "start_line": 143,
            "highlight_line": 143,
            "content": "theorem chem_lm_bc_eq_043 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-044: Equilibrium constant (mega 44).",
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      }
    },
    {
      "id": "CHEM-LM-MC-045",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_039",
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      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_039",
            "start_line": 74,
            "highlight_line": 74,
            "content": "theorem chem_lm_bc_th_039 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-045: Gibbs free energy (mega 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_045.li#L1",
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            "start_line": 703,
            "highlight_line": 704,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-045\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 45).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_039\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L703",
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      }
    },
    {
      "id": "CHEM-LM-MC-046",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_hl_046",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_046.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_hl_046",
            "start_line": 170,
            "highlight_line": 170,
            "content": "theorem chem_lm_bc_hl_046 (steps : List \u211d) : hessEnthalpySum steps = steps.sum := chem_lm_bc_hl_006 steps\n\n/-- Nernst potential E = E\u00b0 \u2212 (RT/nF) ln Q (CHEM-LM-BC-NE-008). -/\nnoncomputable def nernstPotential (E\u2080 R T n F Q : \u211d) : \u211d :=\n  E\u2080 - (R * T) / (n * F) * Real.log Q\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L170",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-046: Hess law enthalpy (mega 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_046.li#L1",
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            "content": "[[entry]]\nid = \"CHEM-LM-MC-046\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 46).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_hl_046\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"8d10f9f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L719",
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    },
    {
      "id": "CHEM-LM-MC-047",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ar_047",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_047.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ar_047",
            "start_line": 212,
            "highlight_line": 212,
            "content": "theorem chem_lm_bc_ar_047 (A Ea R T : \u211d) :\n    arrheniusRate A Ea R T = A * Real.exp (-Ea / (R * T)) := chem_lm_bc_ar_017 A Ea R T\n\n/-- Born-Oppenheimer total energy E = E_elec + E_nuc (CHEM-LM-BC-BO-009). -/\nnoncomputable def bornOppenheimerEnergy (E_elec E_nuc : \u211d) : \u211d := E_elec + E_nuc\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-047: Arrhenius activation energy (mega 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_047.li#L1",
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            "start_line": 735,
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            "content": "[[entry]]\nid = \"CHEM-LM-MC-047\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 47).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ar_047\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"b18c6ac62\"",
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    },
    {
      "id": "CHEM-LM-MC-048",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ne_048",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ne_048",
            "start_line": 191,
            "highlight_line": 191,
            "content": "theorem chem_lm_bc_ne_048 (E\u2080 R T n F Q : \u211d) :\n    nernstPotential E\u2080 R T n F Q = E\u2080 - (R * T) / (n * F) * Real.log Q :=\n  chem_lm_bc_ne_008 E\u2080 R T n F Q\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-048: Nernst equation (mega 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_048.li#L1",
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            "start_line": 751,
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    },
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      "id": "CHEM-LM-MC-049",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 49).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "convergence_class": null,
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_049.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-049: Born-Oppenheimer separation (mega 49).",
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    },
    {
      "id": "CHEM-LM-MC-050",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 50).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-050: Balanced reaction coefficients (mega 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_050.li#L1",
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            "language": "toml",
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            "start_line": 782,
            "highlight_line": 783,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-050\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 50).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L782",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-052",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_051",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_051",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem chem_lm_bc_ig_051 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-052: Ideal gas law PV=nRT (mega 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_052.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 878,
            "highlight_line": 879,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-052\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 52).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_051\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"46bbdf510\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L878",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-053",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_052",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_052",
            "start_line": 107,
            "highlight_line": 107,
            "content": "theorem chem_lm_bc_rx_052 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n\n/-- Equilibrium constant K = k_forward / k_reverse (CHEM-LM-BC-EQ-003). -/\nnoncomputable def equilibriumConstant (k_forward k_reverse : \u211d) : \u211d := k_forward / k_reverse\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L107",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-053: Rate law first order (mega 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_053.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 894,
            "highlight_line": 895,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-053\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 53).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_052\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"b18c6ac62\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L894",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-054",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_028",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_028",
            "start_line": 131,
            "highlight_line": 131,
            "content": "theorem chem_lm_bc_eq_028 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L131",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-054: Equilibrium constant (mega 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_054.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 910,
            "highlight_line": 911,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-054\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 54).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_eq_028\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L910",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-055",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_054",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_054",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem chem_lm_bc_th_054 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L80",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-055: Gibbs free energy (mega 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_055.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 926,
            "highlight_line": 927,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-055\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 55).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_054\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"46bbdf510\"",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-056",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_056.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-056: Hess law enthalpy (mega 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_056.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 942,
            "highlight_line": 943,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-056\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 56).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-057",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-057: Arrhenius activation energy (mega 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_057.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 957,
            "highlight_line": 958,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-057\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 57).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-058",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ne_058",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ne_058",
            "start_line": 195,
            "highlight_line": 195,
            "content": "theorem chem_lm_bc_ne_058 (E\u2080 R T n F Q : \u211d) :\n    nernstPotential E\u2080 R T n F Q = E\u2080 - (R * T) / (n * F) * Real.log Q :=\n  chem_lm_bc_ne_008 E\u2080 R T n F Q\n\n/-- Arrhenius rate k = A exp(\u2212Ea/(RT)) (CHEM-LM-BC-AR-017). -/\nnoncomputable def arrheniusRate (A Ea R T : \u211d) : \u211d := A * Real.exp (-Ea / (R * T))\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-058: Nernst equation (mega 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_058.li#L1",
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            "language": "toml",
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            "start_line": 972,
            "highlight_line": 973,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-058\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 58).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ne_058\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"46bbdf510\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L972",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-059",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_bo_059",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_059.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_bo_059",
            "start_line": 229,
            "highlight_line": 229,
            "content": "theorem chem_lm_bc_bo_059 (E_elec E_nuc : \u211d) :\n    bornOppenheimerEnergy E_elec E_nuc = E_elec + E_nuc :=\n  chem_lm_bc_bo_009 E_elec E_nuc\n\n/-- Atom-composition image of a stoichiometry vector\n    (CHEM-LM-MC-010 / CHEM-LM-MC-020 / \u2026 stoichiometry mega targets). -/",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-059: Born-Oppenheimer separation (mega 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_059.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 988,
            "highlight_line": 989,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-059\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 59).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_bo_059\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"b18c6ac62\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L988",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-060",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_060.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-060: Balanced reaction coefficients (mega 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_060.li#L1",
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            "start_line": 1004,
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      }
    },
    {
      "id": "CHEM-LM-MC-062",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_061",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_062.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_061",
            "start_line": 55,
            "highlight_line": 55,
            "content": "theorem chem_lm_bc_ig_061 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n\n/-- Gibbs free energy \u0394G = \u0394H \u2212 T\u0394S (CHEM-LM-BC-TH-004). -/\nnoncomputable def gibbsFreeEnergy (\u0394H T \u0394S : \u211d) : \u211d := \u0394H - T * \u0394S\n",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-062: Ideal gas law PV=nRT (mega 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_062.li#L1",
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            "language": "toml",
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            "start_line": 1020,
            "highlight_line": 1021,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-062\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 62).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_061\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"b18c6ac62\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1020",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-063",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_032",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_063.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_032",
            "start_line": 99,
            "highlight_line": 99,
            "content": "theorem chem_lm_bc_rx_032 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L99",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-063: Rate law first order (mega 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_063.li#L1",
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            "language": "toml",
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            "start_line": 1036,
            "highlight_line": 1037,
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-064",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_063",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_064.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_063",
            "start_line": 155,
            "highlight_line": 155,
            "content": "theorem chem_lm_bc_eq_063 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n\n/-- Hess law: total enthalpy equals sum of step enthalpies (CHEM-LM-BC-HL-006). -/\nnoncomputable def hessEnthalpySum (steps : List \u211d) : \u211d := steps.sum\n",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-064: Equilibrium constant (mega 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_064.li#L1",
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            "language": "toml",
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            "start_line": 1052,
            "highlight_line": 1053,
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      }
    },
    {
      "id": "CHEM-LM-MC-065",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_064",
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      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_065.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_064",
            "start_line": 82,
            "highlight_line": 82,
            "content": "theorem chem_lm_bc_th_064 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n\n/-- First-order rate magnitude k\u00b7A (CHEM-LM-BC-RX-002). -/\nnoncomputable def firstOrderRate (k A : \u211d) : \u211d := k * A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L82",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-065: Gibbs free energy (mega 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_065.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1068,
            "highlight_line": 1069,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-065\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 65).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_064\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"b18c6ac62\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1068",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-066",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-066: Hess law enthalpy (mega 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_066.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1084,
            "highlight_line": 1085,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-066\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 66).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1084",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-067",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-067: Arrhenius activation energy (mega 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1099,
            "highlight_line": 1100,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-067\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 67).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1099",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-068",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-068: Nernst equation (mega 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_068.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1114,
            "highlight_line": 1115,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-068\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 68).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1114",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-069",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-069: Born-Oppenheimer separation (mega 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_069.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1129,
            "highlight_line": 1130,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-069\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 69).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1129",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-070",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_070.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-070: Balanced reaction coefficients (mega 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_070.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1144,
            "highlight_line": 1145,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-070\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 70).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1144",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-072",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_036",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_036",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem chem_lm_bc_ig_036 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L47",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-072: Ideal gas law PV=nRT (mega 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_072.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1160,
            "highlight_line": 1161,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-072\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 72).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_036\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1160",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-073",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_037",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_037",
            "start_line": 101,
            "highlight_line": 101,
            "content": "theorem chem_lm_bc_rx_037 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L101",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-073: Rate law first order (mega 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_073.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1176,
            "highlight_line": 1177,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-073\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 73).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_037\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1176",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-074",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_038",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_038",
            "start_line": 139,
            "highlight_line": 139,
            "content": "theorem chem_lm_bc_eq_038 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L139",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-074: Equilibrium constant (mega 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_074.li#L1",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1192,
            "highlight_line": 1193,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-074\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 74).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_eq_038\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1192",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-075",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-075: Gibbs free energy (mega 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_075.li#L1",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1208,
            "highlight_line": 1209,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-075\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 75).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1208",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-076",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 76).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-076: Hess law enthalpy (mega 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_076.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1223,
            "highlight_line": 1224,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-076\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 76).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1223",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-077",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-077: Arrhenius activation energy (mega 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_077.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1238,
            "highlight_line": 1239,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-077\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 77).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1238",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-078",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-078: Nernst equation (mega 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_078.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1253,
            "highlight_line": 1254,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-078\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 78).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1253",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-079",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-079: Born-Oppenheimer separation (mega 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_079.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1268,
            "highlight_line": 1269,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-079\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 79).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1268",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-080",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_080.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-080: Balanced reaction coefficients (mega 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_080.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1283,
            "highlight_line": 1284,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-080\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 80).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1283",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-082",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_041",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_041",
            "start_line": 49,
            "highlight_line": 49,
            "content": "theorem chem_lm_bc_ig_041 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L49",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-082: Ideal gas law PV=nRT (mega 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_082.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1299,
            "highlight_line": 1300,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-082\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 82).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_041\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1299",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-083",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-083: Rate law first order (mega 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_083.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1315,
            "highlight_line": 1316,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-083\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 83).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1315",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-084",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_084.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-084: Equilibrium constant (mega 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_084.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1330,
            "highlight_line": 1331,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-084\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 84).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1330",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-085",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_044",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_044",
            "start_line": 76,
            "highlight_line": 76,
            "content": "theorem chem_lm_bc_th_044 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L76",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-085: Gibbs free energy (mega 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_085.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1345,
            "highlight_line": 1346,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-085\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 85).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_044\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1345",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-086",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 86).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-086: Hess law enthalpy (mega 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_086.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1361,
            "highlight_line": 1362,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-086\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 86).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1361",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-087",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-087: Arrhenius activation energy (mega 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_087.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1376,
            "highlight_line": 1377,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-087\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 87).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1376",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-088",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ne_028",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ne_028",
            "start_line": 183,
            "highlight_line": 183,
            "content": "theorem chem_lm_bc_ne_028 (E\u2080 R T n F Q : \u211d) :\n    nernstPotential E\u2080 R T n F Q = E\u2080 - (R * T) / (n * F) * Real.log Q :=\n  chem_lm_bc_ne_008 E\u2080 R T n F Q\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L183",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-088: Nernst equation (mega 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_088.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1391,
            "highlight_line": 1392,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-088\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 88).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ne_028\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1391",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-089",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-089: Born-Oppenheimer separation (mega 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_089.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1407,
            "highlight_line": 1408,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-089\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 89).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1407",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-090",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_090.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-090: Balanced reaction coefficients (mega 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_090.li#L1",
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            "language": "toml",
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            "start_line": 1422,
            "highlight_line": 1423,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-090\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 90).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1422",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-092",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_ig_046",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_ig_046",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem chem_lm_bc_ig_046 (n R T : \u211d) : idealGasPressure n R T = n * R * T := chem_lm_bc_ig_001 n R T\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-092: Ideal gas law PV=nRT (mega 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_092.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1438,
            "highlight_line": 1439,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-092\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 92).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_ig_046\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1438",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-093",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_047",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_047",
            "start_line": 105,
            "highlight_line": 105,
            "content": "theorem chem_lm_bc_rx_047 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L105",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-093: Rate law first order (mega 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_093.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1454,
            "highlight_line": 1455,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-093\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 93).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_047\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1454",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-094",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_048",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_048",
            "start_line": 147,
            "highlight_line": 147,
            "content": "theorem chem_lm_bc_eq_048 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L147",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-094: Equilibrium constant (mega 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_094.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1470,
            "highlight_line": 1471,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-094\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 94).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_eq_048\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1470",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-095",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_th_049",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_th_049",
            "start_line": 78,
            "highlight_line": 78,
            "content": "theorem chem_lm_bc_th_049 (\u0394H T \u0394S : \u211d) : gibbsFreeEnergy \u0394H T \u0394S = \u0394H - T * \u0394S := chem_lm_bc_th_004 \u0394H T \u0394S\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L78",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-095: Gibbs free energy (mega 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_095.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1486,
            "highlight_line": 1487,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-095\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 95).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_th_049\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1486",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-096",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-096: Hess law enthalpy (mega 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_096.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 1502,
            "highlight_line": 1503,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-096\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 96).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1502",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-097",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-097: Arrhenius activation energy (mega 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_097.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1517,
            "highlight_line": 1518,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-097\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 97).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1517",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-098",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-098: Nernst equation (mega 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_098.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1532,
            "highlight_line": 1533,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-098\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 98).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1532",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-099",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-099: Born-Oppenheimer separation (mega 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_099.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1547,
            "highlight_line": 1548,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-099\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 99).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1547",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-100",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-100: Balanced reaction coefficients (mega 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_100.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1562,
            "highlight_line": 1563,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-100\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 100).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1562",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-102",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-102: Ideal gas law PV=nRT (mega 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_102.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1658,
            "highlight_line": 1659,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-102\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 102).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1658",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-103",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-103: Rate law first order (mega 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_103.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1673,
            "highlight_line": 1674,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-103\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 103).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1673",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-104",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_053",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_053",
            "start_line": 151,
            "highlight_line": 151,
            "content": "theorem chem_lm_bc_eq_053 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L151",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-104: Equilibrium constant (mega 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_104.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1688,
            "highlight_line": 1689,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-104\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 104).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_eq_053\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1688",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-105",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_105.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-105: Gibbs free energy (mega 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_105.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1704,
            "highlight_line": 1705,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-105\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 105).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1704",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-106",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-106: Hess law enthalpy (mega 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_106.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1719,
            "highlight_line": 1720,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-106\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 106).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1719",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-107",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-107: Arrhenius activation energy (mega 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_107.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1734,
            "highlight_line": 1735,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-107\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 107).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1734",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-108",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-108: Nernst equation (mega 108).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_108.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1749,
            "highlight_line": 1750,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-108\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 108).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1749",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-109",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-109: Born-Oppenheimer separation (mega 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_109.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1764,
            "highlight_line": 1765,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-109\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 109).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1764",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-110",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-110: Balanced reaction coefficients (mega 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_110.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1779,
            "highlight_line": 1780,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-110\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 110).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1779",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-112",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-112: Ideal gas law PV=nRT (mega 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_112.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1795,
            "highlight_line": 1796,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-112\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 112).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1795",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-113",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-113: Rate law first order (mega 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_113.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1810,
            "highlight_line": 1811,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-113\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 113).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1810",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-114",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-114: Equilibrium constant (mega 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_114.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1825,
            "highlight_line": 1826,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-114\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 114).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1825",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-115",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-115: Gibbs free energy (mega 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_115.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1840,
            "highlight_line": 1841,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-115\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 115).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1840",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-116",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-116: Hess law enthalpy (mega 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_116.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1855,
            "highlight_line": 1856,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-116\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 116).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1855",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-117",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-117: Arrhenius activation energy (mega 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_117.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1870,
            "highlight_line": 1871,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-117\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 117).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1870",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-118",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-118: Nernst equation (mega 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_118.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1885,
            "highlight_line": 1886,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-118\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 118).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1885",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-119",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-119: Born-Oppenheimer separation (mega 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1900,
            "highlight_line": 1901,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-119\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1900",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-120",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-120: Balanced reaction coefficients (mega 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1915,
            "highlight_line": 1916,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-120\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1915",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-122",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-122: Ideal gas law PV=nRT (mega 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1931,
            "highlight_line": 1932,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-122\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-123",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-123: Rate law first order (mega 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1946,
            "highlight_line": 1947,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-123\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 123).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1946",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-124",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-124: Equilibrium constant (mega 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1961,
            "highlight_line": 1962,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-124\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 124).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1961",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-125",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-125: Gibbs free energy (mega 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1976,
            "highlight_line": 1977,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-125\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 125).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1976",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-126",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_126.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-126: Hess law enthalpy (mega 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 1991,
            "highlight_line": 1992,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-126\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 126).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L1991",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-127",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-127: Arrhenius activation energy (mega 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2006,
            "highlight_line": 2007,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-127\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 127).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2006",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-128",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-128: Nernst equation (mega 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2021,
            "highlight_line": 2022,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-128\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 128).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2021",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-129",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-129: Born-Oppenheimer separation (mega 129).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_129.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2036,
            "highlight_line": 2037,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-129\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 129).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2036",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-130",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_130.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-130: Balanced reaction coefficients (mega 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_130.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2051,
            "highlight_line": 2052,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-130\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 130).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2051",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-132",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-132: Ideal gas law PV=nRT (mega 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_132.li#L1",
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            "language": "toml",
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            "start_line": 2067,
            "highlight_line": 2068,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-132\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 132).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2067",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-133",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_rx_012",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_133.li",
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      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_rx_012",
            "start_line": 91,
            "highlight_line": 91,
            "content": "theorem chem_lm_bc_rx_012 (k A : \u211d) : firstOrderRate k A = k * A := chem_lm_bc_rx_002 k A\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L91",
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            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-133: Rate law first order (mega 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_133.li#L1",
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            "language": "toml",
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            "start_line": 2082,
            "highlight_line": 2083,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-133\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 133).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_rx_012\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2082",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-134",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_bc_eq_008",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_bc_eq_008",
            "start_line": 115,
            "highlight_line": 115,
            "content": "theorem chem_lm_bc_eq_008 (k_forward k_reverse : \u211d) (hk : k_reverse \u2260 0) :\n    equilibriumConstant k_forward k_reverse = k_forward / k_reverse :=\n  chem_lm_bc_eq_003 k_forward k_reverse hk\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L115",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-134: Equilibrium constant (mega 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_134.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2098,
            "highlight_line": 2099,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-134\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 134).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_bc_eq_008\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2098",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-135",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-135: Gibbs free energy (mega 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_135.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 2114,
            "highlight_line": 2115,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-135\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 135).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2114",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-136",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-136: Hess law enthalpy (mega 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_136.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 2129,
            "highlight_line": 2130,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-136\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 136).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CHEM-LM-MC-137",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-137: Arrhenius activation energy (mega 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_137.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2144,
            "highlight_line": 2145,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-137\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 137).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "CHEM-LM-MC-138",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-138: Nernst equation (mega 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_138.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 2159,
            "highlight_line": 2160,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-138\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 138).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "CHEM-LM-MC-139",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-139: Born-Oppenheimer separation (mega 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_139.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2174,
            "highlight_line": 2175,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-139\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 139).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2174",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-140",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_140.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-140: Balanced reaction coefficients (mega 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_140.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2189,
            "highlight_line": 2190,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-140\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 140).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2189",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-142",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Ideal gas law PV=nRT (mega 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-142: Ideal gas law PV=nRT (mega 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_142.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2205,
            "highlight_line": 2206,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-142\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Ideal gas law PV=nRT (mega 142).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2205",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-143",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Rate law first order (mega 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-143: Rate law first order (mega 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_143.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2220,
            "highlight_line": 2221,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-143\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Rate law first order (mega 143).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2220",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-144",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Equilibrium constant (mega 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-144: Equilibrium constant (mega 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_144.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2235,
            "highlight_line": 2236,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-144\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Equilibrium constant (mega 144).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2235",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-145",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "physical-chemistry",
      "statement": "Gibbs free energy (mega 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-145: Gibbs free energy (mega 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_145.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2250,
            "highlight_line": 2251,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-145\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"physical-chemistry\"\nstatement = \"Gibbs free energy (mega 145).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2250",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-146",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "thermochemistry",
      "statement": "Hess law enthalpy (mega 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-146: Hess law enthalpy (mega 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_146.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2265,
            "highlight_line": 2266,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-146\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"thermochemistry\"\nstatement = \"Hess law enthalpy (mega 146).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2265",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-147",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "kinetics",
      "statement": "Arrhenius activation energy (mega 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-147: Arrhenius activation energy (mega 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_147.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2280,
            "highlight_line": 2281,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-147\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"kinetics\"\nstatement = \"Arrhenius activation energy (mega 147).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2280",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-148",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "electrochemistry",
      "statement": "Nernst equation (mega 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-148: Nernst equation (mega 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_148.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2295,
            "highlight_line": 2296,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-148\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"electrochemistry\"\nstatement = \"Nernst equation (mega 148).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2295",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-149",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum-chem",
      "statement": "Born-Oppenheimer separation (mega 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-149: Born-Oppenheimer separation (mega 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_149.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2310,
            "highlight_line": 2311,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-149\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum-chem\"\nstatement = \"Born-Oppenheimer separation (mega 149).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2310",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-MC-150",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "stoichiometry",
      "statement": "Balanced reaction coefficients (mega 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chemistry",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.chem_lm_stoich_balanced",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/basic-corpus/chem_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter2 stoichiometry kernel discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "chem_lm_stoich_balanced",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem chem_lm_stoich_balanced {m n : Type*} [Fintype n]\n    (A : Matrix m n \u2124) (\u03bd : n \u2192 \u2124) (h : Matrix.mulVec A \u03bd = 0) :\n    atomBalance A \u03bd = 0 := by\n  simpa [atomBalance] using h\n\n/-- Mass conservation in a closed reaction: atom balance vanishes on the stoichiometry\n    kernel (CHEM-AX-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L239",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/chemistry/basic-corpus/chem_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CHEM-LM-MC-150: Balanced reaction coefficients (mega 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/basic-corpus/chem_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml",
            "start_line": 2325,
            "highlight_line": 2326,
            "content": "[[entry]]\nid = \"CHEM-LM-MC-150\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"stoichiometry\"\nstatement = \"Balanced reaction coefficients (mega 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-chemistry\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.chem_lm_stoich_balanced\"\nli_specimen = \"proof-db/chemistry/basic-corpus/chem_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 stoichiometry kernel discharge\"\nregister_source = \"proof-db/chemistry/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-mega-corpus.toml#L2325",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CHEM-LM-SCF-ENERGY-DECREASE",
      "kind": "lemma",
      "field": "chemistry",
      "domain": "quantum_chemistry",
      "statement": "SCF iteration: energy decreases (or stalls) under standard mixing (convergence target).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-chem",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Chemistry.scf_energy_decrease",
      "lean_module": "proof-db/chemistry/ChemAxioms.lean",
      "li_specimen": "proof-db/chemistry/lemmas/scf_energy_decrease_stub.li",
      "bench_refs": [
        "benchmarks/tier2_physics/qm_dft_scf_energy",
        "benchmarks/tier2_physics/qm_scf_solver"
      ],
      "source": "docs\\verification\\proof-database\\entries\\chemistry-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/lemmas/scf_energy_decrease_stub.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/chemistry/ChemAxioms.lean",
            "symbol": "scf_energy_decrease",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem scf_energy_decrease (E : \u2115 \u2192 \u211d) (h : Antitone E) (k : \u2115) :\n    E (k + 1) \u2264 E k :=\n  h k.le_succ\n\n/-- CHEM-LM-DIIS-CONVERGENCE: DIIS mixing residual contracts when |\u03b1| \u2264 1. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/ChemAxioms.lean#L21",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/chemistry/lemmas/scf_energy_decrease_stub.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def scf_energy(iter: int) -> float\n  requires iter >= 0\n  ensures result >= 0.0\n  decreases 0\n:=\n  return 1.0 / (1.0 + float(iter))\n\ndef scf_step_energy(iter: int) -> float\n  requires iter >= 0\n  ensures result <= scf_energy(iter)\n  decreases 0\n:=\n  return scf_energy(iter + 1)\n\n:=\n  discard scf_step_energy(0)\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/chemistry/lemmas/scf_energy_decrease_stub.li#L3",
            "role": "li_specimen",
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\chemistry-lemmas.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"CHEM-LM-SCF-ENERGY-DECREASE\"\nkind = \"lemma\"\nfield = \"chemistry\"\ndomain = \"quantum_chemistry\"\nstatement = \"SCF iteration: energy decreases (or stalls) under standard mixing (convergence target).\"\nproof_status = \"proved\"\ngap_id = \"G-chem\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/chemistry/ChemAxioms.lean\"\nlean_thm = \"Li.ProofDb.Chemistry.scf_energy_decrease\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nli_specimen = \"proof-db/chemistry/lemmas/scf_energy_decrease_stub.li\"\nbench_refs = [\n  \"benchmarks/tier2_physics/qm_dft_scf_energy\",\n  \"benchmarks/tier2_physics/qm_scf_solver\",\n]\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\chemistry-lemmas.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-AX-MC-001",
      "kind": "axiom",
      "field": "control",
      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 1).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_001.li",
      "notes": "phase18-mega-corpus; iter5 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L12",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_ax_mc_001.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_001() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_001.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"CTRL-AX-MC-001\"\nkind = \"axiom\"\nfield = \"control\"\ndomain = \"lti\"\nstatement = \"LTI state-space solution via matrix exponential (variant 1).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero\"\nli_specimen = \"proof-db/control/corpus/ctrl_ax_mc_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter5 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"53b9753a93\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-AX-MC-006",
      "kind": "axiom",
      "field": "control",
      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_006.li",
      "notes": "phase18-mega-corpus; iter5 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_ax_mc_006.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_006() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_006.li#L3",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 19,
            "highlight_line": 20,
            "content": "[[entry]]\nid = \"CTRL-AX-MC-006\"\nkind = \"axiom\"\nfield = \"control\"\ndomain = \"robust\"\nstatement = \"Small-gain theorem sketch (variant 6).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin\"\nli_specimen = \"proof-db/control/corpus/ctrl_ax_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter5 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"53b9753a93\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L19",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-AX-MC-011",
      "kind": "axiom",
      "field": "control",
      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 11).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_011.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_011.li",
      "notes": "phase18-mega-corpus; iter5 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L12",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_ax_mc_011.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_011() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_011.li#L3",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 35,
            "highlight_line": 36,
            "content": "[[entry]]\nid = \"CTRL-AX-MC-011\"\nkind = \"axiom\"\nfield = \"control\"\ndomain = \"lti\"\nstatement = \"LTI state-space solution via matrix exponential (variant 11).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero\"\nli_specimen = \"proof-db/control/corpus/ctrl_ax_mc_011.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter5 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"53b9753a93\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L35",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-AX-MC-016",
      "kind": "axiom",
      "field": "control",
      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
      "lean_module": "proof-db/control/ControlAxioms.lean",
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            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "id": "CTRL-AX-MC-021",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
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            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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    {
      "id": "CTRL-AX-MC-026",
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      "statement": "Small-gain theorem sketch (variant 26).",
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      "diverges": false,
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            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "id": "CTRL-AX-MC-031",
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      "statement": "LTI state-space solution via matrix exponential (variant 31).",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
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            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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      "id": "CTRL-AX-MC-036",
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      "statement": "Small-gain theorem sketch (variant 36).",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
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            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_046.li",
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            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "gap_id": "G-control",
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            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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      "id": "CTRL-AX-MC-056",
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            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
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            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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      "id": "CTRL-AX-MC-066",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "id": "CTRL-AX-MC-071",
      "kind": "axiom",
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            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_076.li",
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            "content": "def proof_db_ctrl_ax_mc_076() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "content": "[[entry]]\nid = \"CTRL-AX-MC-076\"\nkind = \"axiom\"\nfield = \"control\"\ndomain = \"robust\"\nstatement = \"Small-gain theorem sketch (variant 76).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin\"\nli_specimen = \"proof-db/control/corpus/ctrl_ax_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter5 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"53b9753a93\"",
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    {
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            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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      "id": "CTRL-AX-MC-086",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "id": "CTRL-AX-MC-091",
      "kind": "axiom",
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      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 91).",
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      "gap_id": "G-control",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
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      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_091.li",
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            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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      "id": "CTRL-AX-MC-096",
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      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 96).",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_096.li",
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            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_096.li",
            "symbol": null,
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            "content": "def proof_db_ctrl_ax_mc_096() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "id": "CTRL-AX-MC-101",
      "kind": "axiom",
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      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 101).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-control",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_101.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_101.li",
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            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_101.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_101() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "CTRL-AX-MC-106",
      "kind": "axiom",
      "field": "control",
      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_106.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_106.li",
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            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_106.li",
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            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_106() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    },
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      "id": "CTRL-AX-MC-111",
      "kind": "axiom",
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      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 111).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-control",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_111.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_111.li",
      "notes": "phase18-mega-corpus; iter5 Mathlib/Init discharge",
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            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_111.li",
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            "content": "def proof_db_ctrl_ax_mc_111() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "id": "CTRL-AX-MC-116",
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      "field": "control",
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      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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    {
      "id": "CTRL-AX-MC-121",
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      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 121).",
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      "diverges": false,
      "gap_id": "G-control",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
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            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "content": "def proof_db_ctrl_ax_mc_121() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "CTRL-AX-MC-126",
      "kind": "axiom",
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      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 126).",
      "catalog_status": "proved",
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      "diverges": false,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
      "lean_module": "proof-db/control/ControlAxioms.lean",
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            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_126.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_126() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_126.li#L3",
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    },
    {
      "id": "CTRL-AX-MC-131",
      "kind": "axiom",
      "field": "control",
      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 131).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-control",
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      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_131.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_131.li",
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            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_131.li",
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            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_131() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    },
    {
      "id": "CTRL-AX-MC-136",
      "kind": "axiom",
      "field": "control",
      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_136.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_136.li",
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_136.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_136() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "CTRL-AX-MC-141",
      "kind": "axiom",
      "field": "control",
      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 141).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_141.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_141.li",
      "notes": "phase18-mega-corpus; iter5 Mathlib/Init discharge",
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_141.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_141() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_141.li#L3",
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      "id": "CTRL-AX-MC-146",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "id": "CTRL-AX-MC-151",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
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            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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      "id": "CTRL-AX-MC-156",
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      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 156).",
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_156.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_156() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_156.li#L3",
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      "id": "CTRL-AX-MC-161",
      "kind": "axiom",
      "field": "control",
      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 161).",
      "catalog_status": "proved",
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      "gap_id": "G-control",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_161.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_161.li",
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_161.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_161() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_161.li#L3",
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    {
      "id": "CTRL-AX-MC-166",
      "kind": "axiom",
      "field": "control",
      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 166).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_166.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_166.li",
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            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_166.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_166() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "id": "CTRL-AX-MC-171",
      "kind": "axiom",
      "field": "control",
      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 171).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_171.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_171.li",
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_171.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_171() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_171.li#L3",
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    },
    {
      "id": "CTRL-AX-MC-176",
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      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 176).",
      "catalog_status": "proved",
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      "diverges": false,
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
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            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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      "id": "CTRL-AX-MC-181",
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      "diverges": false,
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      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
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            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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    {
      "id": "CTRL-AX-MC-186",
      "kind": "axiom",
      "field": "control",
      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 186).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_186.li",
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "content": "def proof_db_ctrl_ax_mc_186() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    },
    {
      "id": "CTRL-AX-MC-191",
      "kind": "axiom",
      "field": "control",
      "domain": "lti",
      "statement": "LTI state-space solution via matrix exponential (variant 191).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-control",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_lti_exp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_191.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_191.li",
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_lti_exp_zero",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem ctrl_ax_mc_lti_exp_zero : Real.exp 0 = 1 := Real.exp_zero\n\n/-- Small-gain loop-margin skeleton: \u03b3\u2081\u00b7\u03b3\u2082 < 1 \u21d2 1 \u2212 \u03b3\u2081\u00b7\u03b3\u2082 > 0. -/",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_191.li",
            "symbol": null,
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            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_191() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_191.li#L3",
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    },
    {
      "id": "CTRL-AX-MC-196",
      "kind": "axiom",
      "field": "control",
      "domain": "robust",
      "statement": "Small-gain theorem sketch (variant 196).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_ax_mc_small_gain_margin",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_ax_mc_196.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_196.li",
      "notes": "phase18-mega-corpus; iter5 Mathlib/Init discharge",
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_ax_mc_small_gain_margin",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem ctrl_ax_mc_small_gain_margin (g\u2081 g\u2082 : \u211d) (_hg\u2081 : 0 \u2264 g\u2081) (_hg\u2082 : 0 \u2264 g\u2082)\n    (h : g\u2081 * g\u2082 < 1) : 0 < 1 - g\u2081 * g\u2082 :=\n  sub_pos.mpr h\n",
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            "path": "proof-db/control/corpus/ctrl_ax_mc_196.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ctrl_ax_mc_196() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_ax_mc_196.li#L3",
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            "role": "catalog",
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    },
    {
      "id": "CTRL-LM-MC-002",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_002.li",
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      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_zero",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem ctrl_lm_lyapunov_asymp_zero {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 := by\n  rw [lyapunovEnergy, norm_sq_eq_zero] at hV\n  exact hV\n\n/-- Observability matrix rank is invariant under transpose (CTRL-LM-MC-005 / ... observability anchor). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-002: Lyapunov asymptotic stability implication (variant 2).",
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            "start_line": 227,
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    },
    {
      "id": "CTRL-LM-MC-003",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v003",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_003.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v003",
            "start_line": 55,
            "highlight_line": 55,
            "content": "theorem ctrl_lm_pid_step_error_v003 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- Controllability matrix rank is invariant under transpose (CTRL-LM-MC-004). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-003: PID steady-state error for step input (variant 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_003.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 243,
            "highlight_line": 244,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-003\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v003\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d1155b5c3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L243",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-004",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose",
            "start_line": 59,
            "highlight_line": 59,
            "content": "theorem ctrl_lm_controllability_rank_transpose {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  Matrix.rank_transpose C\n\n/-- Kalman observability rank condition (CTRL-LM-MC-015). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L59",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-004: Kalman controllability rank condition (variant 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_004.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 259,
            "highlight_line": 260,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-004\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d1155b5c3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L259",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-005",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem ctrl_lm_observability_rank_transpose {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  Matrix.rank_transpose O\n\n/-- Bode sensitivity function S = 1/(1+L) (CTRL-LM-MC-008 / ... frequency anchor). -/\nnoncomputable def bodeSensitivity (L : \u211d) : \u211d := 1 / (1 + L)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L34",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-005: Kalman observability rank condition (variant 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_005.li#L1",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 275,
            "highlight_line": 276,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-005\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-007",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v007",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_007.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v007",
            "start_line": 565,
            "highlight_line": 565,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v007 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-014). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L565",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-007: Algebraic Riccati equation optimality (variant 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_007.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 291,
            "highlight_line": 292,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-007\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 7).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v007\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L291",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-008",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_def",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_def",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem ctrl_lm_bode_sensitivity_def (L : \u211d) : bodeSensitivity L = 1 / (1 + L) := rfl\n\n/-- LQR Riccati scalar quadratic cost x\u00b2 (CTRL-LM-MC-017 / ... Riccati anchor). -/\nnoncomputable def riccatiQuadraticCost (x : \u211d) : \u211d := x ^ 2\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-008: Bode sensitivity integral (variant 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_008.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 307,
            "highlight_line": 308,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-008\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 8).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_def\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"687a35291\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-009",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v009",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v009",
            "start_line": 224,
            "highlight_line": 224,
            "content": "theorem ctrl_lm_zoh_pole_v009 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-012). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L224",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-009: ZOH discretization of continuous pole (variant 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_009.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 323,
            "highlight_line": 324,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-009\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v009\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"44e8c8be5\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L323",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-010",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_010_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_010.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_010_mpc_feasible",
            "start_line": 660,
            "highlight_line": 660,
            "content": "theorem ctrl_lm_mc_010_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-010: MPC terminal constraint feasibility (variant 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_010.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 339,
            "highlight_line": 340,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-010\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 10).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_010_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-012",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v012",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v012",
            "start_line": 228,
            "highlight_line": 228,
            "content": "theorem ctrl_lm_lyapunov_asymp_v012 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-018). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-012: Lyapunov asymptotic stability implication (variant 12).",
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            "start_line": 355,
            "highlight_line": 356,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-012\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v012\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"44e8c8be5\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CTRL-LM-MC-013",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v013",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_013.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v013",
            "start_line": 102,
            "highlight_line": 102,
            "content": "theorem ctrl_lm_pid_step_error_v013 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-019). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-013: PID steady-state error for step input (variant 13).",
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            "start_line": 371,
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      }
    },
    {
      "id": "CTRL-LM-MC-014",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v014",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_014.li",
      "bench_refs": [],
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_014.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v014",
            "start_line": 569,
            "highlight_line": 569,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v014 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- Bode sensitivity definition (CTRL-LM-MC-028). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-014: Kalman controllability rank condition (variant 14).",
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            "start_line": 387,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CTRL-LM-MC-015",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v015",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v015",
            "start_line": 64,
            "highlight_line": 64,
            "content": "theorem ctrl_lm_observability_rank_transpose_v015 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman observability rank condition (CTRL-LM-MC-025). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-015: Kalman observability rank condition (variant 15).",
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            "start_line": 403,
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      }
    },
    {
      "id": "CTRL-LM-MC-017",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_017.li",
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      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg",
            "start_line": 46,
            "highlight_line": 46,
            "content": "theorem ctrl_lm_riccati_quad_nonneg (x : \u211d) : 0 \u2264 riccatiQuadraticCost x := sq_nonneg x\n\n/-- PID steady-state tracking error e = r \u2212 y (CTRL-LM-MC-023 / ... step-input anchor). -/\nnoncomputable def pidTrackingError (r y : \u211d) : \u211d := r - y\n",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-017: Algebraic Riccati equation optimality (variant 17).",
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            "start_line": 419,
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      }
    },
    {
      "id": "CTRL-LM-MC-018",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v018",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_018.li",
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      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v018",
            "start_line": 233,
            "highlight_line": 233,
            "content": "theorem ctrl_lm_bode_sensitivity_v018 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-027). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-018: Bode sensitivity integral (variant 18).",
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            "start_line": 435,
            "highlight_line": 436,
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    },
    {
      "id": "CTRL-LM-MC-019",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v019",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v019",
            "start_line": 106,
            "highlight_line": 106,
            "content": "theorem ctrl_lm_zoh_pole_v019 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-059). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-019: ZOH discretization of continuous pole (variant 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_019.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 451,
            "highlight_line": 452,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-019\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 19).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v019\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"eff97941e\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-020",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_020_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_020.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_020_mpc_feasible",
            "start_line": 664,
            "highlight_line": 664,
            "content": "theorem ctrl_lm_mc_020_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-020: MPC terminal constraint feasibility (variant 20).",
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            "language": "toml",
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            "start_line": 467,
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            "content": "[[entry]]\nid = \"CTRL-LM-MC-020\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 20).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_020_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L467",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-022",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v022",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v022",
            "start_line": 88,
            "highlight_line": 88,
            "content": "theorem ctrl_lm_lyapunov_asymp_v022 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-032). -/",
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            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-022: Lyapunov asymptotic stability implication (variant 22).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 483,
            "highlight_line": 484,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-022\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v022\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6c4fb172f\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-023",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_zero",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_zero",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem ctrl_lm_pid_step_error_zero (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 := by\n  rw [pidTrackingError, h, sub_self]\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-003). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-023: PID steady-state error for step input (variant 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_023.li#L1",
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            "start_line": 499,
            "highlight_line": 500,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L499",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-024",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v024",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v024",
            "start_line": 74,
            "highlight_line": 74,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v024 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-033). -/",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-024: Kalman controllability rank condition (variant 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_024.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 515,
            "highlight_line": 516,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-024\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 24).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v024\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"59bf95d70\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L515",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-025",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v025",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v025",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem ctrl_lm_observability_rank_transpose_v025 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-024). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L69",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-025: Kalman observability rank condition (variant 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_025.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 531,
            "highlight_line": 532,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-025\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 25).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v025\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6ce7b5f84\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L531",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-027",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v027",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v027",
            "start_line": 237,
            "highlight_line": 237,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v027 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-045). -/",
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            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-027: Algebraic Riccati equation optimality (variant 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_027.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 547,
            "highlight_line": 548,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-027\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v027\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"44e8c8be5\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-028",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v028",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_028.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v028",
            "start_line": 574,
            "highlight_line": 574,
            "content": "theorem ctrl_lm_bode_sensitivity_v028 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Kalman observability rank condition (CTRL-LM-MC-035). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-028: Bode sensitivity integral (variant 28).",
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            "content": "[[entry]]\nid = \"CTRL-LM-MC-028\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 28).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v028\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
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      }
    },
    {
      "id": "CTRL-LM-MC-029",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v029",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_029.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_029.li",
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            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v029",
            "start_line": 85,
            "highlight_line": 85,
            "content": "theorem ctrl_lm_zoh_pole_v029 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) := rfl\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-022). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-029: ZOH discretization of continuous pole (variant 29).",
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    },
    {
      "id": "CTRL-LM-MC-030",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_030_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_030.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_030_mpc_feasible",
            "start_line": 668,
            "highlight_line": 668,
            "content": "theorem ctrl_lm_mc_030_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-030: MPC terminal constraint feasibility (variant 30).",
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            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-032",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v032",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v032",
            "start_line": 93,
            "highlight_line": 93,
            "content": "theorem ctrl_lm_lyapunov_asymp_v032 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-038). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-032: Lyapunov asymptotic stability implication (variant 32).",
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            "start_line": 611,
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        ]
      }
    },
    {
      "id": "CTRL-LM-MC-033",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v033",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_033.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v033",
            "start_line": 79,
            "highlight_line": 79,
            "content": "theorem ctrl_lm_pid_step_error_v033 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-029 / ... discrete anchor). -/\nnoncomputable def zohDiscretePole (s T : \u211d) : \u211d := Real.exp (s * T)\n",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-033: PID steady-state error for step input (variant 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_033.li#L1",
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            "start_line": 627,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-034",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v034",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v034",
            "start_line": 193,
            "highlight_line": 193,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v034 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-043). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-034: Kalman controllability rank condition (variant 34).",
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            "content": "[[entry]]\nid = \"CTRL-LM-MC-034\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 34).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v034\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"4aefa4c8a\"",
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      }
    },
    {
      "id": "CTRL-LM-MC-035",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v035",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_035.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v035",
            "start_line": 578,
            "highlight_line": 578,
            "content": "theorem ctrl_lm_observability_rank_transpose_v035 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-042). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-035: Kalman observability rank condition (variant 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_035.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 659,
            "highlight_line": 660,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-035\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 35).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v035\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L659",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-037",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v037",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v037",
            "start_line": 127,
            "highlight_line": 127,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v037 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-055). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-037: Algebraic Riccati equation optimality (variant 37).",
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            "start_line": 675,
            "highlight_line": 676,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-037\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 37).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v037\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"b479cc6ba\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-038",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v038",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v038",
            "start_line": 98,
            "highlight_line": 98,
            "content": "theorem ctrl_lm_bode_sensitivity_v038 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-013). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-038: Bode sensitivity integral (variant 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_038.li#L1",
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            "language": "toml",
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            "start_line": 691,
            "highlight_line": 692,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-038\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 38).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v038\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"59bf95d70\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L691",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-039",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v039",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v039",
            "start_line": 123,
            "highlight_line": 123,
            "content": "theorem ctrl_lm_zoh_pole_v039 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-037). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L123",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-039: ZOH discretization of continuous pole (variant 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_039.li#L1",
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            "start_line": 707,
            "highlight_line": 708,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L707",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-040",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_040_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_040.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_040_mpc_feasible",
            "start_line": 672,
            "highlight_line": 672,
            "content": "theorem ctrl_lm_mc_040_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L672",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-040: MPC terminal constraint feasibility (variant 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_040.li#L1",
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            "language": "toml",
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            "start_line": 723,
            "highlight_line": 724,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-040\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 40).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_040_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L723",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-042",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v042",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_042.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v042",
            "start_line": 583,
            "highlight_line": 583,
            "content": "theorem ctrl_lm_lyapunov_asymp_v042 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-049). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L583",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-042: Lyapunov asymptotic stability implication (variant 42).",
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            "language": "toml",
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            "start_line": 739,
            "highlight_line": 740,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-042\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 42).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v042\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-043",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v043",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v043",
            "start_line": 198,
            "highlight_line": 198,
            "content": "theorem ctrl_lm_pid_step_error_v043 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-047). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-043: PID steady-state error for step input (variant 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_043.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 755,
            "highlight_line": 756,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-043\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v043\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"4aefa4c8a\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-044",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v044",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v044",
            "start_line": 114,
            "highlight_line": 114,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v044 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-053). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L114",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-044: Kalman controllability rank condition (variant 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_044.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 771,
            "highlight_line": 772,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-044\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 44).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v044\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"fcf698002\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L771",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-045",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v045",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v045",
            "start_line": 241,
            "highlight_line": 241,
            "content": "theorem ctrl_lm_observability_rank_transpose_v045 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-082). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-045: Kalman observability rank condition (variant 45).",
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            "start_line": 787,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L787",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-047",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v047",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v047",
            "start_line": 202,
            "highlight_line": 202,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v047 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-065). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-047: Algebraic Riccati equation optimality (variant 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_047.li#L1",
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            "language": "toml",
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            "start_line": 803,
            "highlight_line": 804,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-047\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 47).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v047\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3b130e82d\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-048",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v048",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v048",
            "start_line": 149,
            "highlight_line": 149,
            "content": "theorem ctrl_lm_bode_sensitivity_v048 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-057). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L149",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-048: Bode sensitivity integral (variant 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_048.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 819,
            "highlight_line": 820,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-048\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 48).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v048\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"56dfdbbdd\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L819",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-049",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v049",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_049.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v049",
            "start_line": 588,
            "highlight_line": 588,
            "content": "theorem ctrl_lm_zoh_pole_v049 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-063). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-049: ZOH discretization of continuous pole (variant 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_049.li#L1",
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            "language": "toml",
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            "start_line": 835,
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            "content": "[[entry]]\nid = \"CTRL-LM-MC-049\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 49).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v049\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-050",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_050_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_050_mpc_feasible",
            "start_line": 676,
            "highlight_line": 676,
            "content": "theorem ctrl_lm_mc_050_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-050: MPC terminal constraint feasibility (variant 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_050.li#L1",
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            "language": "toml",
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            "start_line": 851,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-052",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v052",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v052",
            "start_line": 285,
            "highlight_line": 285,
            "content": "theorem ctrl_lm_lyapunov_asymp_v052 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-058). -/",
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            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-052: Lyapunov asymptotic stability implication (variant 52).",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 867,
            "highlight_line": 868,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-052\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 52).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v052\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3aaf4a2e8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L867",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-053",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v053",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v053",
            "start_line": 119,
            "highlight_line": 119,
            "content": "theorem ctrl_lm_pid_step_error_v053 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-039). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L119",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-053: PID steady-state error for step input (variant 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_053.li#L1",
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            "start_line": 883,
            "highlight_line": 884,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-053\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 53).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v053\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"fcf698002\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L883",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-054",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v054",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v054",
            "start_line": 482,
            "highlight_line": 482,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v054 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-069). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L482",
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            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-054: Kalman controllability rank condition (variant 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_054.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 899,
            "highlight_line": 900,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-054\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 54).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v054\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9158666fb\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L899",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-055",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v055",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v055",
            "start_line": 131,
            "highlight_line": 131,
            "content": "theorem ctrl_lm_observability_rank_transpose_v055 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-064). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L131",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-055: Kalman observability rank condition (variant 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_055.li#L1",
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            "language": "toml",
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            "start_line": 915,
            "highlight_line": 916,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-055\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 55).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v055\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"b479cc6ba\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L915",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-057",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v057",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v057",
            "start_line": 153,
            "highlight_line": 153,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v057 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-075). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L153",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-057: Algebraic Riccati equation optimality (variant 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_057.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
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            "start_line": 931,
            "highlight_line": 932,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-057\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 57).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v057\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"56dfdbbdd\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-058",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v058",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v058",
            "start_line": 290,
            "highlight_line": 290,
            "content": "theorem ctrl_lm_bode_sensitivity_v058 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-067). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L290",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-058: Bode sensitivity integral (variant 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_058.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 947,
            "highlight_line": 948,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-058\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 58).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v058\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3aaf4a2e8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L947",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-059",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v059",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v059",
            "start_line": 110,
            "highlight_line": 110,
            "content": "theorem ctrl_lm_zoh_pole_v059 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-044). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L110",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-059: ZOH discretization of continuous pole (variant 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_059.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 963,
            "highlight_line": 964,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-059\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 59).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v059\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"94cb5581c\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-060",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_060_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_060_mpc_feasible",
            "start_line": 680,
            "highlight_line": 680,
            "content": "theorem ctrl_lm_mc_060_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-060: MPC terminal constraint feasibility (variant 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_060.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 979,
            "highlight_line": 980,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-060\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 60).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_060_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L979",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-062",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v062",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v062",
            "start_line": 170,
            "highlight_line": 170,
            "content": "theorem ctrl_lm_lyapunov_asymp_v062 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-068). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-062: Lyapunov asymptotic stability implication (variant 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_062.li#L1",
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            "language": "toml",
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            "start_line": 995,
            "highlight_line": 996,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-062\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 62).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v062\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"17449007b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L995",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-063",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v063",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_063.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v063",
            "start_line": 592,
            "highlight_line": 592,
            "content": "theorem ctrl_lm_pid_step_error_v063 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- Algebraic Riccati / LQR quadratic cost nonnegativity (CTRL-LM-MC-077). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L592",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-063: PID steady-state error for step input (variant 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_063.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1011,
            "highlight_line": 1012,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-063\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 63).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v063\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1011",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-064",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v064",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v064",
            "start_line": 136,
            "highlight_line": 136,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v064 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-073). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L136",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-064: Kalman controllability rank condition (variant 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_064.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1027,
            "highlight_line": 1028,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-064\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 64).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v064\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"b479cc6ba\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1027",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-065",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v065",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v065",
            "start_line": 206,
            "highlight_line": 206,
            "content": "theorem ctrl_lm_observability_rank_transpose_v065 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-074). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L206",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-065: Kalman observability rank condition (variant 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_065.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 1043,
            "highlight_line": 1044,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-065\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 65).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v065\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3b130e82d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1043",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-067",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v067",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v067",
            "start_line": 294,
            "highlight_line": 294,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v067 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-085). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L294",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-067: Algebraic Riccati equation optimality (variant 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_067.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1267,
            "highlight_line": 1268,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1267",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-068",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v068",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v068",
            "start_line": 175,
            "highlight_line": 175,
            "content": "theorem ctrl_lm_bode_sensitivity_v068 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Kalman observability rank condition (CTRL-LM-MC-095). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L175",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-068: Bode sensitivity integral (variant 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_068.li#L1",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1283,
            "highlight_line": 1284,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-068\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 68).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v068\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"17449007b\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-069",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v069",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v069",
            "start_line": 487,
            "highlight_line": 487,
            "content": "theorem ctrl_lm_zoh_pole_v069 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-072). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-069: ZOH discretization of continuous pole (variant 69).",
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            "highlight_line": 1300,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-069\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 69).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v069\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9158666fb\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "CTRL-LM-MC-070",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_070_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_070.li",
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      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_070.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_070_mpc_feasible",
            "start_line": 684,
            "highlight_line": 684,
            "content": "theorem ctrl_lm_mc_070_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-070: MPC terminal constraint feasibility (variant 70).",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-072",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v072",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v072",
            "start_line": 491,
            "highlight_line": 491,
            "content": "theorem ctrl_lm_lyapunov_asymp_v072 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-078). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-072: Lyapunov asymptotic stability implication (variant 72).",
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            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-073",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v073",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v073",
            "start_line": 141,
            "highlight_line": 141,
            "content": "theorem ctrl_lm_pid_step_error_v073 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-079). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-073: PID steady-state error for step input (variant 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_073.li#L1",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-074",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v074",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v074",
            "start_line": 211,
            "highlight_line": 211,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v074 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-083). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-074: Kalman controllability rank condition (variant 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_074.li#L1",
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            "language": "toml",
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            "start_line": 1363,
            "highlight_line": 1364,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-074\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 74).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v074\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3b130e82d\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-075",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v075",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v075",
            "start_line": 157,
            "highlight_line": 157,
            "content": "theorem ctrl_lm_observability_rank_transpose_v075 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-093). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L157",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-075: Kalman observability rank condition (variant 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_075.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1379,
            "highlight_line": 1380,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-075\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 75).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v075\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"56dfdbbdd\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-077",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v077",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_077.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v077",
            "start_line": 596,
            "highlight_line": 596,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v077 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-084). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L596",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-077: Algebraic Riccati equation optimality (variant 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_077.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1395,
            "highlight_line": 1396,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-077\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 77).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v077\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-078",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v078",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v078",
            "start_line": 496,
            "highlight_line": 496,
            "content": "theorem ctrl_lm_bode_sensitivity_v078 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-087). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-078: Bode sensitivity integral (variant 78).",
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            "start_line": 1411,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-079",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v079",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v079",
            "start_line": 145,
            "highlight_line": 145,
            "content": "theorem ctrl_lm_zoh_pole_v079 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Bode sensitivity definition (CTRL-LM-MC-048). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-079: ZOH discretization of continuous pole (variant 79).",
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            "start_line": 1427,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-080",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_080_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_080.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_080_mpc_feasible",
            "start_line": 688,
            "highlight_line": 688,
            "content": "theorem ctrl_lm_mc_080_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-080: MPC terminal constraint feasibility (variant 80).",
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            "start_line": 1443,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1443",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-082",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v082",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v082",
            "start_line": 246,
            "highlight_line": 246,
            "content": "theorem ctrl_lm_lyapunov_asymp_v082 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-088). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-082: Lyapunov asymptotic stability implication (variant 82).",
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            "language": "toml",
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            "start_line": 1459,
            "highlight_line": 1460,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-082\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 82).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v082\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"92185d6fe\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-083",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v083",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v083",
            "start_line": 216,
            "highlight_line": 216,
            "content": "theorem ctrl_lm_pid_step_error_v083 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-089). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L216",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-083: PID steady-state error for step input (variant 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_083.li#L1",
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            "start_line": 1475,
            "highlight_line": 1476,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-083\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 83).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v083\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3b130e82d\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-084",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v084",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_084.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v084",
            "start_line": 600,
            "highlight_line": 600,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v084 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- Bode sensitivity definition (CTRL-LM-MC-098). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-084: Kalman controllability rank condition (variant 84).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1491,
            "highlight_line": 1492,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-084\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 84).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v084\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
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          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-085",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v085",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v085",
            "start_line": 298,
            "highlight_line": 298,
            "content": "theorem ctrl_lm_observability_rank_transpose_v085 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-094). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L298",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-085: Kalman observability rank condition (variant 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_085.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1507,
            "highlight_line": 1508,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-085\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 85).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v085\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3aaf4a2e8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1507",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-087",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v087",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v087",
            "start_line": 500,
            "highlight_line": 500,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v087 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Bode sensitivity definition (CTRL-LM-MC-178). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L500",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-087: Algebraic Riccati equation optimality (variant 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_087.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1523,
            "highlight_line": 1524,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-087\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 87).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v087\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9158666fb\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1523",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-088",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v088",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v088",
            "start_line": 251,
            "highlight_line": 251,
            "content": "theorem ctrl_lm_bode_sensitivity_v088 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-097). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L251",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-088: Bode sensitivity integral (variant 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_088.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1539,
            "highlight_line": 1540,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-088\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 88).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v088\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"92185d6fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1539",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-089",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v089",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v089",
            "start_line": 220,
            "highlight_line": 220,
            "content": "theorem ctrl_lm_zoh_pole_v089 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-009). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L220",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-089: ZOH discretization of continuous pole (variant 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_089.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1555,
            "highlight_line": 1556,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-089\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 89).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v089\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3b130e82d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1555",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-090",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_090_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_090.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_090_mpc_feasible",
            "start_line": 692,
            "highlight_line": 692,
            "content": "theorem ctrl_lm_mc_090_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L692",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-090: MPC terminal constraint feasibility (variant 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_090.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1571,
            "highlight_line": 1572,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-090\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 90).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_090_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1571",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-092",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v092",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v092",
            "start_line": 280,
            "highlight_line": 280,
            "content": "theorem ctrl_lm_lyapunov_asymp_v092 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-052). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L280",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-092: Lyapunov asymptotic stability implication (variant 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_092.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1587,
            "highlight_line": 1588,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-092\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 92).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v092\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"b347eebce\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1587",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-093",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v093",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v093",
            "start_line": 162,
            "highlight_line": 162,
            "content": "theorem ctrl_lm_pid_step_error_v093 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-099). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L162",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-093: PID steady-state error for step input (variant 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_093.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1603,
            "highlight_line": 1604,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-093\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 93).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v093\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"56dfdbbdd\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1603",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-094",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v094",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v094",
            "start_line": 303,
            "highlight_line": 303,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v094 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-104). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-094: Kalman controllability rank condition (variant 94).",
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            "start_line": 1619,
            "highlight_line": 1620,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-094\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 94).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v094\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3aaf4a2e8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1619",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-095",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v095",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v095",
            "start_line": 179,
            "highlight_line": 179,
            "content": "theorem ctrl_lm_observability_rank_transpose_v095 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-107). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L179",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-095: Kalman observability rank condition (variant 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_095.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1635,
            "highlight_line": 1636,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-095\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 95).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v095\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"17449007b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1635",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-097",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v097",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v097",
            "start_line": 255,
            "highlight_line": 255,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v097 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-103). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L255",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-097: Algebraic Riccati equation optimality (variant 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_097.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 1651,
            "highlight_line": 1652,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-097\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 97).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v097\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"92185d6fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1651",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-098",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v098",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_098.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v098",
            "start_line": 605,
            "highlight_line": 605,
            "content": "theorem ctrl_lm_bode_sensitivity_v098 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Kalman observability rank condition (CTRL-LM-MC-105). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L605",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-098: Bode sensitivity integral (variant 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_098.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1667,
            "highlight_line": 1668,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-098\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 98).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v098\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1667",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-099",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v099",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v099",
            "start_line": 166,
            "highlight_line": 166,
            "content": "theorem ctrl_lm_zoh_pole_v099 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-062). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L166",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-099: ZOH discretization of continuous pole (variant 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_099.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1683,
            "highlight_line": 1684,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-099\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 99).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v099\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"56dfdbbdd\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1683",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-100",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_100_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_100_mpc_feasible",
            "start_line": 696,
            "highlight_line": 696,
            "content": "theorem ctrl_lm_mc_100_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L696",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-100: MPC terminal constraint feasibility (variant 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_100.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1699,
            "highlight_line": 1700,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-100\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 100).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_100_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-102",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v102",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v102",
            "start_line": 520,
            "highlight_line": 520,
            "content": "theorem ctrl_lm_lyapunov_asymp_v102 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-108). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-102: Lyapunov asymptotic stability implication (variant 102).",
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            "start_line": 1715,
            "highlight_line": 1716,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-102\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 102).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v102\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"73536e928\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-103",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v103",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v103",
            "start_line": 259,
            "highlight_line": 259,
            "content": "theorem ctrl_lm_pid_step_error_v103 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-109). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-103: PID steady-state error for step input (variant 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_103.li#L1",
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            "start_line": 1731,
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "CTRL-LM-MC-104",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v104",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v104",
            "start_line": 308,
            "highlight_line": 308,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v104 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state error for step input (CTRL-LM-MC-113). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-104: Kalman controllability rank condition (variant 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_104.li#L1",
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            "start_line": 1747,
            "highlight_line": 1748,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-104\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 104).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v104\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"94064a002\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-105",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v105",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_105.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
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        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v105",
            "start_line": 609,
            "highlight_line": 609,
            "content": "theorem ctrl_lm_observability_rank_transpose_v105 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-112). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-105: Kalman observability rank condition (variant 105).",
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            "start_line": 1763,
            "highlight_line": 1764,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-107",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v107",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v107",
            "start_line": 184,
            "highlight_line": 184,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v107 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-125). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-107: Algebraic Riccati equation optimality (variant 107).",
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            "start_line": 1779,
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      }
    },
    {
      "id": "CTRL-LM-MC-108",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v108",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_108.li",
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      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v108",
            "start_line": 525,
            "highlight_line": 525,
            "content": "theorem ctrl_lm_bode_sensitivity_v108 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-117). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-108: Bode sensitivity integral (variant 108).",
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            "start_line": 1795,
            "highlight_line": 1796,
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    },
    {
      "id": "CTRL-LM-MC-109",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v109",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v109",
            "start_line": 263,
            "highlight_line": 263,
            "content": "theorem ctrl_lm_zoh_pole_v109 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-134). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-109: ZOH discretization of continuous pole (variant 109).",
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            "highlight_line": 1812,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-109\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 109).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v109\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"92185d6fe\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-110",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_110_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_110_mpc_feasible",
            "start_line": 700,
            "highlight_line": 700,
            "content": "theorem ctrl_lm_mc_110_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-110: MPC terminal constraint feasibility (variant 110).",
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            "start_line": 1827,
            "highlight_line": 1828,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-110\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 110).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_110_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1827",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-112",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v112",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_112.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v112",
            "start_line": 614,
            "highlight_line": 614,
            "content": "theorem ctrl_lm_lyapunov_asymp_v112 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-119). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-112: Lyapunov asymptotic stability implication (variant 112).",
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            "start_line": 1843,
            "highlight_line": 1844,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-113",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v113",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v113",
            "start_line": 313,
            "highlight_line": 313,
            "content": "theorem ctrl_lm_pid_step_error_v113 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-114). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-113: PID steady-state error for step input (variant 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_113.li#L1",
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            "start_line": 1859,
            "highlight_line": 1860,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-113\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 113).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v113\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"94064a002\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-114",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v114",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v114",
            "start_line": 317,
            "highlight_line": 317,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v114 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- Bode sensitivity definition (CTRL-LM-MC-118). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-114: Kalman controllability rank condition (variant 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_114.li#L1",
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            "language": "toml",
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            "start_line": 1875,
            "highlight_line": 1876,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-114\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 114).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v114\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3fba39579\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1875",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-115",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v115",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v115",
            "start_line": 339,
            "highlight_line": 339,
            "content": "theorem ctrl_lm_observability_rank_transpose_v115 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-124). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-115: Kalman observability rank condition (variant 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_115.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1891,
            "highlight_line": 1892,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-115\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 115).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v115\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0170b63c5\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-117",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v117",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v117",
            "start_line": 529,
            "highlight_line": 529,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v117 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-135). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-117: Algebraic Riccati equation optimality (variant 117).",
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            "start_line": 1907,
            "highlight_line": 1908,
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          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-118",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v118",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v118",
            "start_line": 322,
            "highlight_line": 322,
            "content": "theorem ctrl_lm_bode_sensitivity_v118 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-122). -/",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-118: Bode sensitivity integral (variant 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_118.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1923,
            "highlight_line": 1924,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-118\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 118).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v118\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6007b5e22\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1923",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-119",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v119",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_119.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v119",
            "start_line": 619,
            "highlight_line": 619,
            "content": "theorem ctrl_lm_zoh_pole_v119 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-133). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L619",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-119: ZOH discretization of continuous pole (variant 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1939,
            "highlight_line": 1940,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-119\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 119).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v119\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1939",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-120",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_120_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_120_mpc_feasible",
            "start_line": 704,
            "highlight_line": 704,
            "content": "theorem ctrl_lm_mc_120_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L704",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-120: MPC terminal constraint feasibility (variant 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1955,
            "highlight_line": 1956,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-120\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_120_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1955",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-122",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v122",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v122",
            "start_line": 326,
            "highlight_line": 326,
            "content": "theorem ctrl_lm_lyapunov_asymp_v122 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-123). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L326",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-122: Lyapunov asymptotic stability implication (variant 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1971,
            "highlight_line": 1972,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-122\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 122).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v122\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"94064a002\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1971",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-123",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v123",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v123",
            "start_line": 331,
            "highlight_line": 331,
            "content": "theorem ctrl_lm_pid_step_error_v123 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- Bode sensitivity definition (CTRL-LM-MC-128). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L331",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-123: PID steady-state error for step input (variant 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 1987,
            "highlight_line": 1988,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-123\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 123).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v123\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3fba39579\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L1987",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-124",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v124",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v124",
            "start_line": 344,
            "highlight_line": 344,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v124 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-129). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L344",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-124: Kalman controllability rank condition (variant 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2003,
            "highlight_line": 2004,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-124\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 124).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v124\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0170b63c5\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2003",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-125",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v125",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v125",
            "start_line": 188,
            "highlight_line": 188,
            "content": "theorem ctrl_lm_observability_rank_transpose_v125 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-034). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L188",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-125: Kalman observability rank condition (variant 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2019,
            "highlight_line": 2020,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-125\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 125).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v125\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"17449007b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-127",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v127",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v127",
            "start_line": 398,
            "highlight_line": 398,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v127 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-145). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-127: Algebraic Riccati equation optimality (variant 127).",
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            "start_line": 2035,
            "highlight_line": 2036,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-127\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 127).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v127\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9085a9896\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-128",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v128",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v128",
            "start_line": 335,
            "highlight_line": 335,
            "content": "theorem ctrl_lm_bode_sensitivity_v128 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Kalman observability rank condition (CTRL-LM-MC-115). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-128: Bode sensitivity integral (variant 128).",
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            "start_line": 2051,
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            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-129",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v129",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v129",
            "start_line": 349,
            "highlight_line": 349,
            "content": "theorem ctrl_lm_zoh_pole_v129 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-132). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-129: ZOH discretization of continuous pole (variant 129).",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-130",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_130_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_130.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_130_mpc_feasible",
            "start_line": 708,
            "highlight_line": 708,
            "content": "theorem ctrl_lm_mc_130_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-130: MPC terminal constraint feasibility (variant 130).",
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            "start_line": 2083,
            "highlight_line": 2084,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-130\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 130).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_130_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-132",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v132",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v132",
            "start_line": 353,
            "highlight_line": 353,
            "content": "theorem ctrl_lm_lyapunov_asymp_v132 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-138). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-132: Lyapunov asymptotic stability implication (variant 132).",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-133",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v133",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_133.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v133",
            "start_line": 623,
            "highlight_line": 623,
            "content": "theorem ctrl_lm_pid_step_error_v133 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- Algebraic Riccati / LQR quadratic cost nonnegativity (CTRL-LM-MC-147). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-133: PID steady-state error for step input (variant 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_133.li#L1",
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            "start_line": 2323,
            "highlight_line": 2324,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-133\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 133).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v133\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
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            "role": "catalog",
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          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-134",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v134",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v134",
            "start_line": 267,
            "highlight_line": 267,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v134 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-143). -/",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-134: Kalman controllability rank condition (variant 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_134.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2339,
            "highlight_line": 2340,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-134\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 134).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v134\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"f6177870d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2339",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-135",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v135",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v135",
            "start_line": 533,
            "highlight_line": 533,
            "content": "theorem ctrl_lm_observability_rank_transpose_v135 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-144). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-135: Kalman observability rank condition (variant 135).",
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            "start_line": 2355,
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            "content": "[[entry]]\nid = \"CTRL-LM-MC-135\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 135).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v135\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"73536e928\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2355",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-137",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v137",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v137",
            "start_line": 420,
            "highlight_line": 420,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v137 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-155). -/",
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            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-137: Algebraic Riccati equation optimality (variant 137).",
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            "language": "toml",
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            "start_line": 2371,
            "highlight_line": 2372,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-137\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 137).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v137\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0bda0580e\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-138",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v138",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v138",
            "start_line": 358,
            "highlight_line": 358,
            "content": "theorem ctrl_lm_bode_sensitivity_v138 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-139). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-138: Bode sensitivity integral (variant 138).",
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            "start_line": 2387,
            "highlight_line": 2388,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-138\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 138).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v138\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"3fba39579\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-139",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v139",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v139",
            "start_line": 362,
            "highlight_line": 362,
            "content": "theorem ctrl_lm_zoh_pole_v139 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-142). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-139: ZOH discretization of continuous pole (variant 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_139.li#L1",
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            "start_line": 2403,
            "highlight_line": 2404,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-139\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 139).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v139\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0170b63c5\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-140",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_140_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_140.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_140_mpc_feasible",
            "start_line": 712,
            "highlight_line": 712,
            "content": "theorem ctrl_lm_mc_140_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-140: MPC terminal constraint feasibility (variant 140).",
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            "start_line": 2419,
            "highlight_line": 2420,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-142",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v142",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v142",
            "start_line": 366,
            "highlight_line": 366,
            "content": "theorem ctrl_lm_lyapunov_asymp_v142 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-148). -/",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-142: Lyapunov asymptotic stability implication (variant 142).",
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            "language": "toml",
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            "start_line": 2435,
            "highlight_line": 2436,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-142\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 142).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v142\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0170b63c5\"",
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            "role": "catalog",
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          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-143",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v143",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v143",
            "start_line": 272,
            "highlight_line": 272,
            "content": "theorem ctrl_lm_pid_step_error_v143 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-149). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L272",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-143: PID steady-state error for step input (variant 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_143.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2451,
            "highlight_line": 2452,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-143\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 143).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v143\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"f6177870d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2451",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-144",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v144",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v144",
            "start_line": 538,
            "highlight_line": 538,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v144 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-153). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-144: Kalman controllability rank condition (variant 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_144.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2467,
            "highlight_line": 2468,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-144\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 144).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v144\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"73536e928\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2467",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-145",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v145",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v145",
            "start_line": 402,
            "highlight_line": 402,
            "content": "theorem ctrl_lm_observability_rank_transpose_v145 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-163). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L402",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-145: Kalman observability rank condition (variant 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_145.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2483,
            "highlight_line": 2484,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-145\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 145).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v145\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9085a9896\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2483",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-147",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v147",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_147.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v147",
            "start_line": 627,
            "highlight_line": 627,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v147 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-154). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-147: Algebraic Riccati equation optimality (variant 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_147.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2499,
            "highlight_line": 2500,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-147\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 147).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v147\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2499",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-148",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v148",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v148",
            "start_line": 371,
            "highlight_line": 371,
            "content": "theorem ctrl_lm_bode_sensitivity_v148 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-152). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L371",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-148: Bode sensitivity integral (variant 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_148.li#L1",
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            "language": "toml",
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            "start_line": 2515,
            "highlight_line": 2516,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-148\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 148).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v148\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0170b63c5\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2515",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-149",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v149",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v149",
            "start_line": 276,
            "highlight_line": 276,
            "content": "theorem ctrl_lm_zoh_pole_v149 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-092). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L276",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-149: ZOH discretization of continuous pole (variant 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_149.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2531,
            "highlight_line": 2532,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-149\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 149).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v149\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"f6177870d\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-150",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_150_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_150_mpc_feasible",
            "start_line": 716,
            "highlight_line": 716,
            "content": "theorem ctrl_lm_mc_150_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L716",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-150: MPC terminal constraint feasibility (variant 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2547,
            "highlight_line": 2548,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-150\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_150_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-152",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 152).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v152",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_152.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_152.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v152",
            "start_line": 375,
            "highlight_line": 375,
            "content": "theorem ctrl_lm_lyapunov_asymp_v152 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-158). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_152.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-152: Lyapunov asymptotic stability implication (variant 152).",
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            "start_line": 2563,
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            "content": "[[entry]]\nid = \"CTRL-LM-MC-152\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 152).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v152\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_152.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"74aaf40e5\"",
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            "role": "catalog",
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    },
    {
      "id": "CTRL-LM-MC-153",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 153).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v153",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_153.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_153.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v153",
            "start_line": 543,
            "highlight_line": 543,
            "content": "theorem ctrl_lm_pid_step_error_v153 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-159). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_153.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-153: PID steady-state error for step input (variant 153).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_153.li#L1",
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            "start_line": 2579,
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            "content": "[[entry]]\nid = \"CTRL-LM-MC-153\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 153).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v153\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_153.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"645f3d8d3\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-154",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 154).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v154",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_154.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_154.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v154",
            "start_line": 631,
            "highlight_line": 631,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v154 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- Bode sensitivity definition (CTRL-LM-MC-168). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_154.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-154: Kalman controllability rank condition (variant 154).",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "CTRL-LM-MC-155",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 155).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v155",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_155.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_155.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v155",
            "start_line": 424,
            "highlight_line": 424,
            "content": "theorem ctrl_lm_observability_rank_transpose_v155 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-164). -/",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_155.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-155: Kalman observability rank condition (variant 155).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_155.li#L1",
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            "start_line": 2611,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-157",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 157).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v157",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_157.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_157.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v157",
            "start_line": 442,
            "highlight_line": 442,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v157 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-184). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_157.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-157: Algebraic Riccati equation optimality (variant 157).",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-158",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 158).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v158",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_158.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_158.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v158",
            "start_line": 380,
            "highlight_line": 380,
            "content": "theorem ctrl_lm_bode_sensitivity_v158 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-167). -/",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_158.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-158: Bode sensitivity integral (variant 158).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2643,
            "highlight_line": 2644,
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    },
    {
      "id": "CTRL-LM-MC-159",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 159).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v159",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_159.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_159.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v159",
            "start_line": 547,
            "highlight_line": 547,
            "content": "theorem ctrl_lm_zoh_pole_v159 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-162). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_159.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-159: ZOH discretization of continuous pole (variant 159).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_159.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2659,
            "highlight_line": 2660,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-159\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 159).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v159\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_159.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"645f3d8d3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2659",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-160",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 160).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_160_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_160.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_160.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_160_mpc_feasible",
            "start_line": 720,
            "highlight_line": 720,
            "content": "theorem ctrl_lm_mc_160_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L720",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_160.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-160: MPC terminal constraint feasibility (variant 160).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_160.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2675,
            "highlight_line": 2676,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-160\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 160).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_160_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_160.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2675",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-162",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 162).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v162",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_162.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_162.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v162",
            "start_line": 551,
            "highlight_line": 551,
            "content": "theorem ctrl_lm_lyapunov_asymp_v162 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-177). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L551",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_162.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-162: Lyapunov asymptotic stability implication (variant 162).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_162.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2691,
            "highlight_line": 2692,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-162\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 162).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v162\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_162.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"645f3d8d3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2691",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-163",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 163).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v163",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_163.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_163.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v163",
            "start_line": 407,
            "highlight_line": 407,
            "content": "theorem ctrl_lm_pid_step_error_v163 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-169). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L407",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_163.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-163: PID steady-state error for step input (variant 163).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_163.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2707,
            "highlight_line": 2708,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-163\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 163).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v163\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_163.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9085a9896\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2707",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-164",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 164).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v164",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_164.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_164.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v164",
            "start_line": 429,
            "highlight_line": 429,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v164 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-173). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L429",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_164.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-164: Kalman controllability rank condition (variant 164).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_164.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2723,
            "highlight_line": 2724,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-164\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 164).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v164\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_164.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0bda0580e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2723",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-165",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 165).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v165",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_165.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_165.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v165",
            "start_line": 459,
            "highlight_line": 459,
            "content": "theorem ctrl_lm_observability_rank_transpose_v165 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-174). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L459",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_165.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-165: Kalman observability rank condition (variant 165).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_165.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2739,
            "highlight_line": 2740,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-165\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 165).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v165\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_165.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9220c442e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2739",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-167",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 167).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v167",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_167.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_167.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v167",
            "start_line": 384,
            "highlight_line": 384,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v167 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-185). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L384",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_167.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-167: Algebraic Riccati equation optimality (variant 167).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_167.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2755,
            "highlight_line": 2756,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-167\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 167).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v167\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_167.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"74aaf40e5\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2755",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-168",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 168).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v168",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_168.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_168.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v168",
            "start_line": 636,
            "highlight_line": 636,
            "content": "theorem ctrl_lm_bode_sensitivity_v168 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Kalman observability rank condition (CTRL-LM-MC-175). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L636",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_168.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-168: Bode sensitivity integral (variant 168).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_168.li#L1",
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            "language": "toml",
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            "start_line": 2771,
            "highlight_line": 2772,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-168\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 168).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v168\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_168.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2771",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-169",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 169).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v169",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_169.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_169.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v169",
            "start_line": 411,
            "highlight_line": 411,
            "content": "theorem ctrl_lm_zoh_pole_v169 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-172). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_169.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-169: ZOH discretization of continuous pole (variant 169).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_169.li#L1",
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            "start_line": 2787,
            "highlight_line": 2788,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-169\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 169).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v169\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_169.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9085a9896\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2787",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-170",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 170).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_170_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_170.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_170.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_170_mpc_feasible",
            "start_line": 724,
            "highlight_line": 724,
            "content": "theorem ctrl_lm_mc_170_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_170.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-170: MPC terminal constraint feasibility (variant 170).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_170.li#L1",
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            "language": "toml",
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            "start_line": 2803,
            "highlight_line": 2804,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-170\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 170).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_170_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_170.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2803",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-172",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 172).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v172",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_172.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_172.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v172",
            "start_line": 415,
            "highlight_line": 415,
            "content": "theorem ctrl_lm_lyapunov_asymp_v172 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-137). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L415",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_172.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-172: Lyapunov asymptotic stability implication (variant 172).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_172.li#L1",
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            "language": "toml",
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            "start_line": 2819,
            "highlight_line": 2820,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-172\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 172).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v172\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_172.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9085a9896\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2819",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-173",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 173).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v173",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_173.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_173.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v173",
            "start_line": 434,
            "highlight_line": 434,
            "content": "theorem ctrl_lm_pid_step_error_v173 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-179). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L434",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_173.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-173: PID steady-state error for step input (variant 173).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_173.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2835,
            "highlight_line": 2836,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-173\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 173).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v173\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_173.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0bda0580e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2835",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-174",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 174).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v174",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_174.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_174.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v174",
            "start_line": 464,
            "highlight_line": 464,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v174 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-183). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L464",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_174.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-174: Kalman controllability rank condition (variant 174).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_174.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2851,
            "highlight_line": 2852,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-174\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 174).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v174\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_174.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9220c442e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2851",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-175",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 175).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v175",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_175.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_175.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v175",
            "start_line": 640,
            "highlight_line": 640,
            "content": "theorem ctrl_lm_observability_rank_transpose_v175 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-182). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L640",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_175.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-175: Kalman observability rank condition (variant 175).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_175.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2867,
            "highlight_line": 2868,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-175\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 175).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v175\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_175.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2867",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-177",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 177).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v177",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_177.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_177.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v177",
            "start_line": 556,
            "highlight_line": 556,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v177 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Kalman observability rank condition (CTRL-LM-MC-195). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_177.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-177: Algebraic Riccati equation optimality (variant 177).",
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            "language": "toml",
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            "start_line": 2883,
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            "content": "[[entry]]\nid = \"CTRL-LM-MC-177\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 177).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v177\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_177.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"645f3d8d3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2883",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-178",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 178).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v178",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_178.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_178.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v178",
            "start_line": 504,
            "highlight_line": 504,
            "content": "theorem ctrl_lm_bode_sensitivity_v178 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Bode sensitivity definition (CTRL-LM-MC-188). -/",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_178.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-178: Bode sensitivity integral (variant 178).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_178.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2899,
            "highlight_line": 2900,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-178\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 178).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v178\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_178.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"2ebca7648\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2899",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-179",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 179).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v179",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_179.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_179.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v179",
            "start_line": 438,
            "highlight_line": 438,
            "content": "theorem ctrl_lm_zoh_pole_v179 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-157). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L438",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_179.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-179: ZOH discretization of continuous pole (variant 179).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_179.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 2915,
            "highlight_line": 2916,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-179\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 179).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v179\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_179.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"0bda0580e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2915",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-180",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 180).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_180_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_180.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_180.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_180_mpc_feasible",
            "start_line": 728,
            "highlight_line": 728,
            "content": "theorem ctrl_lm_mc_180_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L728",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_180.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-180: MPC terminal constraint feasibility (variant 180).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_180.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2931,
            "highlight_line": 2932,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-180\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 180).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_180_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_180.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-182",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 182).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v182",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_182.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_182.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v182",
            "start_line": 645,
            "highlight_line": 645,
            "content": "theorem ctrl_lm_lyapunov_asymp_v182 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-189). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L645",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_182.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-182: Lyapunov asymptotic stability implication (variant 182).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_182.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2947,
            "highlight_line": 2948,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-182\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 182).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v182\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_182.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-183",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 183).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v183",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_183.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_183.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v183",
            "start_line": 469,
            "highlight_line": 469,
            "content": "theorem ctrl_lm_pid_step_error_v183 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-192). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L469",
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          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_183.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-183: PID steady-state error for step input (variant 183).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_183.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2963,
            "highlight_line": 2964,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-183\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 183).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v183\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_183.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9220c442e\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-184",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 184).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v184",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_184.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_184.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v184",
            "start_line": 446,
            "highlight_line": 446,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v184 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- PID steady-state step-input tracking (CTRL-LM-MC-193). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L446",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_184.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-184: Kalman controllability rank condition (variant 184).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_184.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 2979,
            "highlight_line": 2980,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-184\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 184).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v184\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_184.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"42fd825e9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2979",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-185",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 185).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v185",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_185.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_185.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v185",
            "start_line": 388,
            "highlight_line": 388,
            "content": "theorem ctrl_lm_observability_rank_transpose_v185 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-194). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L388",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_185.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-185: Kalman observability rank condition (variant 185).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_185.li#L1",
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            "language": "toml",
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            "start_line": 2995,
            "highlight_line": 2996,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-185\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 185).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v185\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_185.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"74aaf40e5\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L2995",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-187",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 187).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v187",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_187.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_187.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v187",
            "start_line": 512,
            "highlight_line": 512,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v187 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-197). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L512",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_187.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-187: Algebraic Riccati equation optimality (variant 187).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_187.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3011,
            "highlight_line": 3012,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-187\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 187).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v187\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_187.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"2ebca7648\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3011",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-188",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 188).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v188",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_188.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_188.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v188",
            "start_line": 508,
            "highlight_line": 508,
            "content": "theorem ctrl_lm_bode_sensitivity_v188 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-187). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L508",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_188.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-188: Bode sensitivity integral (variant 188).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_188.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3027,
            "highlight_line": 3028,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-188\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 188).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v188\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_188.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"345662ca2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3027",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-189",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 189).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v189",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_189.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_189.li",
      "notes": "phase18-mega-corpus; iter7 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v189",
            "start_line": 650,
            "highlight_line": 650,
            "content": "theorem ctrl_lm_zoh_pole_v189 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-! MPC terminal-constraint feasibility (CTRL-LM-MC-*0) \u2014 nonempty terminal set witness. -/\n\n/-- A point in the terminal set witnesses MPC terminal-constraint feasibility. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L650",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_189.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-189: ZOH discretization of continuous pole (variant 189).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_189.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3043,
            "highlight_line": 3044,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-189\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 189).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v189\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_189.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter7 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"6747fea2b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3043",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-190",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 190).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_190_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_190.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_190.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_mc_190_mpc_feasible",
            "start_line": 732,
            "highlight_line": 732,
            "content": "theorem ctrl_lm_mc_190_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L732",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_190.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-190: MPC terminal constraint feasibility (variant 190).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_190.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3059,
            "highlight_line": 3060,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-190\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"mpc\"\nstatement = \"MPC terminal constraint feasibility (variant 190).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_mc_190_mpc_feasible\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_190.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3059",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-192",
      "kind": "lemma",
      "field": "control",
      "domain": "stability",
      "statement": "Lyapunov asymptotic stability implication (variant 192).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v192",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_192.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_192.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_lyapunov_asymp_v192",
            "start_line": 473,
            "highlight_line": 473,
            "content": "theorem ctrl_lm_lyapunov_asymp_v192 {E : Type*} [NormedAddCommGroup E] (x : E)\n    (hV : lyapunovEnergy x = 0) : x = 0 :=\n  ctrl_lm_lyapunov_asymp_zero x hV\n\n/-- Bode sensitivity definition (CTRL-LM-MC-198). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L473",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_192.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-192: Lyapunov asymptotic stability implication (variant 192).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_192.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3075,
            "highlight_line": 3076,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-192\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"stability\"\nstatement = \"Lyapunov asymptotic stability implication (variant 192).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_lyapunov_asymp_v192\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_192.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9220c442e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3075",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-193",
      "kind": "lemma",
      "field": "control",
      "domain": "pid",
      "statement": "PID steady-state error for step input (variant 193).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_pid_step_error_v193",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_193.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_193.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_pid_step_error_v193",
            "start_line": 451,
            "highlight_line": 451,
            "content": "theorem ctrl_lm_pid_step_error_v193 (r y : \u211d) (h : r = y) : pidTrackingError r y = 0 :=\n  ctrl_lm_pid_step_error_zero r y h\n\n/-- ZOH discrete pole z = exp(s\u00b7T) (CTRL-LM-MC-199). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L451",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_193.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-193: PID steady-state error for step input (variant 193).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_193.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 3091,
            "highlight_line": 3092,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-193\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"pid\"\nstatement = \"PID steady-state error for step input (variant 193).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_pid_step_error_v193\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_193.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"42fd825e9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3091",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-194",
      "kind": "lemma",
      "field": "control",
      "domain": "controllability",
      "statement": "Kalman controllability rank condition (variant 194).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v194",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_194.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_194.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_controllability_rank_transpose_v194",
            "start_line": 393,
            "highlight_line": 393,
            "content": "theorem ctrl_lm_controllability_rank_transpose_v194 {m n : Nat} (C : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank C.transpose = Matrix.rank C :=\n  ctrl_lm_controllability_rank_transpose C\n\n/-- LQR Riccati quadratic cost nonnegativity (CTRL-LM-MC-127). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L393",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_194.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-194: Kalman controllability rank condition (variant 194).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_194.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3107,
            "highlight_line": 3108,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-194\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"controllability\"\nstatement = \"Kalman controllability rank condition (variant 194).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_controllability_rank_transpose_v194\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_194.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"74aaf40e5\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3107",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-195",
      "kind": "lemma",
      "field": "control",
      "domain": "observability",
      "statement": "Kalman observability rank condition (variant 195).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v195",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_195.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_195.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_observability_rank_transpose_v195",
            "start_line": 560,
            "highlight_line": 560,
            "content": "theorem ctrl_lm_observability_rank_transpose_v195 {m n : Nat} (O : Matrix (Fin m) (Fin n) \u211d) :\n    Matrix.rank O.transpose = Matrix.rank O :=\n  ctrl_lm_observability_rank_transpose O\n\n/-- Algebraic Riccati / LQR quadratic cost nonnegativity (CTRL-LM-MC-007). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L560",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_195.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-195: Kalman observability rank condition (variant 195).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_195.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3123,
            "highlight_line": 3124,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-195\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"observability\"\nstatement = \"Kalman observability rank condition (variant 195).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_observability_rank_transpose_v195\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_195.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"645f3d8d3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3123",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-197",
      "kind": "lemma",
      "field": "control",
      "domain": "lqr",
      "statement": "Algebraic Riccati equation optimality (variant 197).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v197",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_197.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_197.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_riccati_quad_nonneg_v197",
            "start_line": 516,
            "highlight_line": 516,
            "content": "theorem ctrl_lm_riccati_quad_nonneg_v197 (x : \u211d) : 0 \u2264 riccatiQuadraticCost x :=\n  ctrl_lm_riccati_quad_nonneg x\n\n/-- Lyapunov asymptotic stability (CTRL-LM-MC-102). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L516",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_197.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-197: Algebraic Riccati equation optimality (variant 197).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_197.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 3139,
            "highlight_line": 3140,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-197\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"lqr\"\nstatement = \"Algebraic Riccati equation optimality (variant 197).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_riccati_quad_nonneg_v197\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_197.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"345662ca2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\control-mega-corpus.toml#L3139",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-198",
      "kind": "lemma",
      "field": "control",
      "domain": "frequency",
      "statement": "Bode sensitivity integral (variant 198).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v198",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_198.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_198.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_bode_sensitivity_v198",
            "start_line": 478,
            "highlight_line": 478,
            "content": "theorem ctrl_lm_bode_sensitivity_v198 (L : \u211d) : bodeSensitivity L = 1 / (1 + L) :=\n  ctrl_lm_bode_sensitivity_def L\n\n/-- Kalman controllability rank condition (CTRL-LM-MC-054). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L478",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_198.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-198: Bode sensitivity integral (variant 198).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_198.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3155,
            "highlight_line": 3156,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-198\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"frequency\"\nstatement = \"Bode sensitivity integral (variant 198).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_bode_sensitivity_v198\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_198.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"9220c442e\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-199",
      "kind": "lemma",
      "field": "control",
      "domain": "discrete",
      "statement": "ZOH discretization of continuous pole (variant 199).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_zoh_pole_v199",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_199.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_199.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/control/ControlAxioms.lean",
            "symbol": "ctrl_lm_zoh_pole_v199",
            "start_line": 455,
            "highlight_line": 455,
            "content": "theorem ctrl_lm_zoh_pole_v199 (s T : \u211d) : zohDiscretePole s T = Real.exp (s * T) :=\n  ctrl_lm_zoh_pole_v029 s T\n\n/-- Kalman observability rank condition (CTRL-LM-MC-165). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/ControlAxioms.lean#L455",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/control/corpus/ctrl_lm_mc_199.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-199: ZOH discretization of continuous pole (variant 199).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_199.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
            "start_line": 3171,
            "highlight_line": 3172,
            "content": "[[entry]]\nid = \"CTRL-LM-MC-199\"\nkind = \"lemma\"\nfield = \"control\"\ndomain = \"discrete\"\nstatement = \"ZOH discretization of continuous pole (variant 199).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-control\"\nlean_module = \"proof-db/control/ControlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Control.ctrl_lm_zoh_pole_v199\"\nli_specimen = \"proof-db/control/corpus/ctrl_lm_mc_199.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/control/register.json\"\nlast_verified_lic_commit = \"42fd825e9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "CTRL-LM-MC-200",
      "kind": "lemma",
      "field": "control",
      "domain": "mpc",
      "statement": "MPC terminal constraint feasibility (variant 200).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-control",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Control.ctrl_lm_mc_200_mpc_feasible",
      "lean_module": "proof-db/control/ControlAxioms.lean",
      "li_specimen": "proof-db/control/corpus/ctrl_lm_mc_200.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\control-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/control/corpus/ctrl_lm_mc_200.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "symbol": "ctrl_lm_mc_200_mpc_feasible",
            "start_line": 736,
            "highlight_line": 736,
            "content": "theorem ctrl_lm_mc_200_mpc_feasible {\u03b1 : Type*} (X_f : Set \u03b1) {x : \u03b1} (hx : x \u2208 X_f) :\n    \u2203 y, y \u2208 X_f :=\n  ctrl_lm_mpc_terminal_feasible X_f hx\n",
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            "path": "proof-db/control/corpus/ctrl_lm_mc_200.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# CTRL-LM-MC-200: MPC terminal constraint feasibility (variant 200).",
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    },
    {
      "id": "D-AX-BC-LOG-005",
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      "field": "discrete",
      "domain": "logic",
      "statement": "Excluded middle: P or not P (propositional tautology).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_005_em",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
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            "symbol": "d_ax_bc_log_005_em",
            "start_line": 807,
            "highlight_line": 807,
            "content": "theorem d_ax_bc_log_005_em (p : Prop) : p \u2228 \u00acp := Classical.em p\n\n/-- D-AX-BC-LOG-010: identity. -/",
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            "content": "def d_ax_bc_log_005_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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      "id": "D-AX-BC-LOG-010",
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      "field": "discrete",
      "domain": "logic",
      "statement": "Identity: P implies P.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_010_identity",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_010.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_010.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
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            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_ax_bc_log_010_identity",
            "start_line": 810,
            "highlight_line": 810,
            "content": "theorem d_ax_bc_log_010_identity (p : Prop) : p \u2192 p := id\n\n/-- D-AX-BC-LOG-015: modus ponens. -/",
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            "path": "proof-db/discrete/basic-corpus/d_ax_bc_log_010.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_010_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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    },
    {
      "id": "D-AX-BC-LOG-015",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "Modus ponens schema: (P and (P implies Q)) implies Q.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_015_modus_ponens",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_015.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
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            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_ax_bc_log_015_modus_ponens",
            "start_line": 813,
            "highlight_line": 813,
            "content": "theorem d_ax_bc_log_015_modus_ponens {p q : Prop} : (p \u2227 (p \u2192 q)) \u2192 q :=\n  fun h => h.2 h.1\n\n/-- D-AX-BC-LOG-020: contrapositive. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L813",
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            "path": "proof-db/discrete/basic-corpus/d_ax_bc_log_015.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_015_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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            "highlight_line": 162,
            "content": "[[entry]]\nid = \"D-AX-BC-LOG-015\"\nkind = \"axiom\"\nfield = \"discrete\"\ndomain = \"logic\"\nstatement = \"Modus ponens schema: (P and (P implies Q)) implies Q.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_ax_bc_log_015_modus_ponens\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_ax_bc_log_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nlast_verified_lic_commit = \"83173eb9fe\"",
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    },
    {
      "id": "D-AX-BC-LOG-020",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "Contrapositive: (P implies Q) equivalent to (not Q implies not P).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_020_contrapositive",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_020.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_020.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
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            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_ax_bc_log_020_contrapositive",
            "start_line": 817,
            "highlight_line": 817,
            "content": "theorem d_ax_bc_log_020_contrapositive {p q : Prop} : (p \u2192 q) \u2194 (\u00acq \u2192 \u00acp) :=\n  Iff.intro\n    (fun hpq hnq hp => hnq (hpq hp))\n    (fun hnqnp hp => Classical.byContradiction fun hnq => hnqnp hnq hp)\n\n/-- D-AX-BC-LOG-025: De Morgan (\u00ac(p \u2227 q) \u2194 \u00acp \u2228 \u00acq). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L817",
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            "path": "proof-db/discrete/basic-corpus/d_ax_bc_log_020.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_020_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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            "start_line": 176,
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            "content": "[[entry]]\nid = \"D-AX-BC-LOG-020\"\nkind = \"axiom\"\nfield = \"discrete\"\ndomain = \"logic\"\nstatement = \"Contrapositive: (P implies Q) equivalent to (not Q implies not P).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_ax_bc_log_020_contrapositive\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_ax_bc_log_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nlast_verified_lic_commit = \"83173eb9fe\"",
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    },
    {
      "id": "D-AX-BC-LOG-025",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "De Morgan: not (P and Q) equivalent to (not P or not Q).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_025_de_morgan",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_025.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_ax_bc_log_025_de_morgan",
            "start_line": 823,
            "highlight_line": 823,
            "content": "theorem d_ax_bc_log_025_de_morgan {p q : Prop} : \u00ac(p \u2227 q) \u2194 \u00acp \u2228 \u00acq :=\n  Iff.intro\n    (fun h =>\n      (Classical.em p).elim\n        (fun hp => Or.inr fun hq => h \u27e8hp, hq\u27e9)\n        (fun hnp => Or.inl hnp))\n    (fun h \u27e8hp, hq\u27e9 => h.elim (fun hnp => hnp hp) (fun hnq => hnq hq))\n\n/-- D-AX-BC-LOG-030: disjunctive syllogism. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L823",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_ax_bc_log_025.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_025_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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            "highlight_line": 320,
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    },
    {
      "id": "D-AX-BC-LOG-030",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "Disjunctive syllogism: (P or Q) and (not P) implies Q.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_030_disj_syll",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_030.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
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            "symbol": "d_ax_bc_log_030_disj_syll",
            "start_line": 832,
            "highlight_line": 832,
            "content": "theorem d_ax_bc_log_030_disj_syll {p q : Prop} : (p \u2228 q) \u2227 \u00acp \u2192 q :=\n  fun h => h.1.resolve_left h.2\n\n/-- D-AX-BC-LOG-035: hypothetical syllogism. -/",
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            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_030_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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      }
    },
    {
      "id": "D-AX-BC-LOG-035",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "Hypothetical syllogism: (P implies Q) and (Q implies R) implies (P implies R).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_035_hyp_syll",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_035.li",
      "bench_refs": [],
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_035.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
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            "symbol": "d_ax_bc_log_035_hyp_syll",
            "start_line": 836,
            "highlight_line": 836,
            "content": "theorem d_ax_bc_log_035_hyp_syll {p q r : Prop} : (p \u2192 q) \u2227 (q \u2192 r) \u2192 (p \u2192 r) :=\n  fun h hp => h.2 (h.1 hp)\n\n/-- D-AX-BC-LOG-040: material implication. -/",
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            "path": "proof-db/discrete/basic-corpus/d_ax_bc_log_035.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_035_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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            "highlight_line": 350,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml#L349",
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      }
    },
    {
      "id": "D-AX-BC-LOG-040",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "Material implication: (P implies Q) equivalent to (not P or Q).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_040_material",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_040.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_ax_bc_log_040_material",
            "start_line": 840,
            "highlight_line": 840,
            "content": "theorem d_ax_bc_log_040_material {p q : Prop} : (p \u2192 q) \u2194 (\u00acp \u2228 q) :=\n  Iff.intro\n    (fun hpq =>\n      (Classical.em p).elim\n        (fun hp => Or.inr (hpq hp))\n        (fun hnp => Or.inl hnp))\n    (fun h hp => h.elim (fun hnp => False.elim (hnp hp)) id)\n\n/-- D-AX-BC-LOG-045: double negation. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L840",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_ax_bc_log_040.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_040_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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            "path": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
            "start_line": 364,
            "highlight_line": 365,
            "content": "[[entry]]\nid = \"D-AX-BC-LOG-040\"\nkind = \"axiom\"\nfield = \"discrete\"\ndomain = \"logic\"\nstatement = \"Material implication: (P implies Q) equivalent to (not P or Q).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_ax_bc_log_040_material\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_ax_bc_log_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nlast_verified_lic_commit = \"83173eb9fe\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-AX-BC-LOG-045",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "Double negation: not (not P) equivalent to P.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_045_dbl_neg",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_045.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_ax_bc_log_045_dbl_neg",
            "start_line": 849,
            "highlight_line": 849,
            "content": "theorem d_ax_bc_log_045_dbl_neg (p : Prop) : \u00ac\u00acp \u2194 p :=\n  Iff.intro Classical.not_not.mp not_not_intro\n\n/-- D-AX-BC-LOG-050: resolution. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L849",
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            "path": "proof-db/discrete/basic-corpus/d_ax_bc_log_045.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_045_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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            "start_line": 379,
            "highlight_line": 380,
            "content": "[[entry]]\nid = \"D-AX-BC-LOG-045\"\nkind = \"axiom\"\nfield = \"discrete\"\ndomain = \"logic\"\nstatement = \"Double negation: not (not P) equivalent to P.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_ax_bc_log_045_dbl_neg\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_ax_bc_log_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nlast_verified_lic_commit = \"83173eb9fe\"",
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    },
    {
      "id": "D-AX-BC-LOG-050",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "Resolution schema: (P or R) and (not P or Q) implies (Q or R).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_bc_log_050_resolution",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_bc_log_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_ax_bc_log_050.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_ax_bc_log_050_resolution",
            "start_line": 853,
            "highlight_line": 853,
            "content": "theorem d_ax_bc_log_050_resolution {p q r : Prop} : (p \u2228 r) \u2227 (\u00acp \u2228 q) \u2192 (q \u2228 r) :=\n  fun \u27e8hpr, hnpq\u27e9 =>\n    hpr.elim\n      (fun hp => Or.inl (hnpq.resolve_left (fun hnp => hnp hp)))\n      (fun hr => Or.inr hr)\n\n/-! Discrete mega leftovers (Dilworth / Hamming / generating / recurrence) \u2014 iter9. -/\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-007). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L853",
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            "path": "proof-db/discrete/basic-corpus/d_ax_bc_log_050.li",
            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def d_ax_bc_log_050_tautology(p: bool, q: bool) -> bool\n  requires true\n  ensures result == (p and q) or (not p) or (not q)\n  decreases 0\n=\n  return (p and q) or (not p) or (not q)",
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            "start_line": 394,
            "highlight_line": 395,
            "content": "[[entry]]\nid = \"D-AX-BC-LOG-050\"\nkind = \"axiom\"\nfield = \"discrete\"\ndomain = \"logic\"\nstatement = \"Resolution schema: (P or R) and (not P or Q) implies (Q or R).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_ax_bc_log_050_resolution\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_ax_bc_log_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nlast_verified_lic_commit = \"83173eb9fe\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "D-AX-INT-RING",
      "kind": "axiom",
      "field": "discrete",
      "domain": "integer_algebra",
      "statement": "Integers form a commutative ring (addition commutative/associative, multiplication distributes).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Int.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.int_add_comm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/lemmas/int_ring_stub.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-axioms.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/lemmas/int_ring_stub.li",
      "notes": "iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "int_add_comm",
            "start_line": 10,
            "highlight_line": 10,
            "content": "theorem int_add_comm : \u2200 a b : Int, a + b = b + a :=\n  fun a b => add_comm a b\n\n/-- Int addition associative (D-AX-INT-RING; Mathlib `add_assoc`). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L10",
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            "path": "proof-db/discrete/lemmas/int_ring_stub.li",
            "symbol": null,
            "start_line": 2,
            "highlight_line": 2,
            "content": "def add_assoc_int(a: int, b: int, c: int) -> int\n  requires true\n  ensures result == (a + b) + c\n  decreases 0\n=\n  return a + (b + c)\n\ndef add_comm_int(a: int, b: int) -> int\n  requires true\n  ensures result == a + b\n  decreases 0\n=\n  return b + a\n\ndef mul_distrib_int(a: int, b: int, c: int) -> int\n  requires true\n  ensures result == a * (b + c)\n  decreases 0\n=\n  return a * b + a * c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/lemmas/int_ring_stub.li#L2",
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            "start_line": 3,
            "highlight_line": 4,
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "D-AX-MC-001",
      "kind": "axiom",
      "field": "discrete",
      "domain": "logic",
      "statement": "Propositional tautology (mega 1).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_ax_mc_prop_tautology",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_ax_mc_001.li",
      "bench_refs": [],
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            "highlight_line": 802,
            "content": "theorem d_ax_mc_prop_tautology (p : Prop) : p \u2192 p := id\n\n/-! Discrete-basic LOG axioms (D-AX-BC-LOG-*) \u2014 Classical / Init discharge (iter8). -/\n\n/-- D-AX-BC-LOG-005: excluded middle. -/",
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            "content": "theorem int_add_assoc : \u2200 a b c : Int, (a + b) + c = a + (b + c) :=\n  fun a b c => add_assoc a b c\n\n/-- Int multiplication distributes over addition (D-AX-INT-RING; Mathlib `mul_add`). -/",
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            "content": "def add_assoc_int(a: int, b: int, c: int) -> int\n  requires true\n  ensures result == (a + b) + c\n  decreases 0\n=\n  return a + (b + c)\n\ndef add_comm_int(a: int, b: int) -> int\n  requires true\n  ensures result == a + b\n  decreases 0\n=\n  return b + a\n\ndef mul_distrib_int(a: int, b: int, c: int) -> int\n  requires true\n  ensures result == a * (b + c)\n  decreases 0\n=\n  return a * b + a * c",
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            "content": "theorem int_add_comm : \u2200 a b : Int, a + b = b + a :=\n  fun a b => add_comm a b\n\n/-- Int addition associative (D-AX-INT-RING; Mathlib `add_assoc`). -/",
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      "id": "D-LM-BC-CMB-002",
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            "content": "theorem d_lm_bc_cmb_002 (n k : Nat) (hk : 1 \u2264 k) (hk' : k \u2264 n - 1) :\n    n.choose k = (n - 1).choose (k - 1) + (n - 1).choose k := by\n  simpa using Nat.choose_succ_succ (n - 1) (k - 1)\n\n/-- Alternating row sum: sum_{k=0}^n (-1)^k C(n,k) = 0 for n > 0 (D-LM-BC-CMB-017). -/",
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            "content": "def binomial_symmetry(n: int, k: int) -> int\n  requires n >= 0 and 0 <= k and k <= n\n  ensures result == n - k\n  decreases 0\n=\n  return n - k",
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            "content": "theorem d_lm_bc_cmb_007 (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  Nat.choose_symm hk\n\n/-- Row sum identity: sum_{k=0}^n C(n,k) = 2^n (D-LM-BC-CMB-012; Mathlib.Data.Nat.Choose.Sum). -/",
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            "content": "def binomial_symmetry(n: int, k: int) -> int\n  requires n >= 0 and 0 <= k and k <= n\n  ensures result == n - k\n  decreases 0\n=\n  return n - k",
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    {
      "id": "D-LM-BC-CMB-012",
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      "priority_tier": null,
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      "id": "D-LM-BC-CMB-017",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_cmb_017",
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            "symbol": "d_lm_bc_cmb_017",
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            "highlight_line": 36,
            "content": "theorem d_lm_bc_cmb_017 (n : Nat) (hn : 0 < n) :\n    (\u2211 k in Finset.range (n + 1), ((-1 : \u2124) ^ k * n.choose k)) = 0 :=\n  Int.alternating_sum_range_choose_of_ne (Nat.ne_of_gt hn)\n\n/-- Hockey-stick identity: sum_{i=r}^n C(i,r) = C(n+1,r+1) (D-LM-BC-CMB-027). -/",
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    {
      "id": "D-LM-BC-CMB-022",
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      "statement": "Vandermonde identity: sum_{k=0}^r C(m,k) C(n,r-k) = C(m+n,r).",
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      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_cmb_022",
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      "notes": "phase8-basic-corpus tranche=3",
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            "symbol": "d_lm_bc_cmb_022",
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            "content": "theorem d_lm_bc_cmb_022 (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  (Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk (fun i j => m.choose i * n.choose j) r).symm.trans\n    (Nat.add_choose_eq m n r)\n\n/-- Binomial theorem (D-LM-BC-CMB-037; Mathlib.Data.Nat.Choose.Sum). -/",
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      "id": "D-LM-BC-CMB-027",
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      "statement": "Hockey-stick identity: sum_{i=r}^n C(i,r) = C(n+1,r+1).",
      "catalog_status": "proved",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_cmb_027",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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            "symbol": "d_lm_bc_cmb_027",
            "start_line": 41,
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            "content": "theorem d_lm_bc_cmb_027 (n r : Nat) (hr : r \u2264 n) :\n    (\u2211 i in Finset.Icc r n, i.choose r) = (n + 1).choose (r + 1) :=\n  Nat.sum_Icc_choose n r\n\n/-- Absorption identity: k C(n,k) = n C(n-1,k-1) (D-LM-BC-CMB-047). -/",
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            "content": "def binomial_symmetry(n: int, k: int) -> int\n  requires n >= 0 and 0 <= k and k <= n\n  ensures result == n - k\n  decreases 0\n=\n  return n - k",
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    },
    {
      "id": "D-LM-BC-CMB-032",
      "kind": "lemma",
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      "domain": "combinatorics",
      "statement": "Square of row sum: (sum C(n,k))^2 = sum_{k=0}^n C(n,k)^2.",
      "catalog_status": "discrepancy",
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      "diverges": false,
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      "notes": "phase8-basic-corpus tranche=3; phase15-iter5 discrepancy: statement false (n=2: 16\u22606); correct identity sum C(n,k)^2=C(2n,n)",
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    {
      "id": "D-LM-BC-CMB-037",
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      "domain": "combinatorics",
      "statement": "Binomial theorem: (x+y)^n = sum_{k=0}^n C(n,k) x^k y^(n-k).",
      "catalog_status": "proved",
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            "symbol": "d_lm_bc_cmb_037",
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            "content": "theorem d_lm_bc_cmb_037 (x y : \u2115) (n : Nat) :\n    (x + y) ^ n = \u2211 k in Finset.range (n + 1), n.choose k * x ^ k * y ^ (n - k) := by\n  rw [add_pow]\n  congr! 1 with k hk\n  ring\n\n/-- Chu-Vandermonde (D-LM-BC-CMB-042; alias of Vandermonde range form). -/",
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            "content": "[[entry]]\nid = \"D-LM-BC-CMB-037\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial theorem: (x+y)^n = sum_{k=0}^n C(n,k) x^k y^(n-k).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_cmb_037\"\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Sum\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_cmb_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
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    {
      "id": "D-LM-BC-CMB-042",
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      "statement": "Chu-Vandermonde: sum_{k} C(m,k) C(n,l-k) = C(m+n,l).",
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      "priority_tier": null,
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            "symbol": "d_lm_bc_cmb_042",
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            "content": "theorem d_lm_bc_cmb_042 (m n l : Nat) :\n    (\u2211 k in Finset.range (l + 1), m.choose k * n.choose (l - k)) = (m + n).choose l :=\n  d_lm_bc_cmb_022 m n l\n\n/-- Sum 1 + 2 + ... + n = n(n+1)/2 (D-LM-BC-IND-011; Mathlib.Data.Finset.Basic). -/",
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            "content": "def binomial_symmetry(n: int, k: int) -> int\n  requires n >= 0 and 0 <= k and k <= n\n  ensures result == n - k\n  decreases 0\n=\n  return n - k",
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    {
      "id": "D-LM-BC-CMB-047",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Absorption identity: k C(n,k) = n C(n-1,k-1).",
      "catalog_status": "proved",
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      "priority_tier": null,
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            "symbol": "d_lm_bc_cmb_047",
            "start_line": 46,
            "highlight_line": 46,
            "content": "theorem d_lm_bc_cmb_047 (n k : Nat) (hn : 0 < n) (hk : 0 < k) (hk' : k \u2264 n) :\n    k * n.choose k = n * (n - 1).choose (k - 1) := by\n  have hsn : n - 1 + 1 = n := Nat.sub_add_cancel hn.le\n  have hsk : k - 1 + 1 = k := Nat.sub_add_cancel hk.le\n  rw [\u2190 Nat.succ_mul_choose_eq (n - 1) (k - 1), hsn, hsk]\n  ring\n\n/-- Vandermonde identity (D-LM-BC-CMB-022; Mathlib.Data.Nat.Choose.Vandermonde). -/",
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            "content": "def binomial_symmetry(n: int, k: int) -> int\n  requires n >= 0 and 0 <= k and k <= n\n  ensures result == n - k\n  decreases 0\n=\n  return n - k",
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    {
      "id": "D-LM-BC-IND-001",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Mathematical induction: if P(0) and forall k (P(k) implies P(k+1)) then forall n P(n).",
      "catalog_status": "proved",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_001",
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            "symbol": "d_lm_bc_ind_001",
            "start_line": 196,
            "highlight_line": 196,
            "content": "theorem d_lm_bc_ind_001 (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) : \u2200 n, P n :=\n  Nat.rec h0 hs\n\n/-- Strong induction principle (D-LM-BC-IND-006). -/",
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            "highlight_line": 3,
            "content": "def induction_step(n: int, step: int) -> int\n  requires n >= 0 and step >= 0\n  ensures result == n + step\n  decreases 0\n=\n  return n + step",
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            "content": "[[entry]]\nid = \"D-LM-BC-IND-001\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Mathematical induction: if P(0) and forall k (P(k) implies P(k+1)) then forall n P(n).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_ind_001\"\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_ind_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"a9441d8e\"",
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    {
      "id": "D-LM-BC-IND-006",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Strong induction: if P(0) and forall k ((forall j <= k P(j)) implies P(k+1)) then forall n P(n).",
      "catalog_status": "proved",
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      "diverges": false,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_006",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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            "symbol": "d_lm_bc_ind_006",
            "start_line": 200,
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            "content": "theorem d_lm_bc_ind_006 (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 k, (\u2200 j \u2264 k, P j) \u2192 P (k + 1)) :\n    \u2200 n, P n := by\n  intro n\n  refine Nat.strong_induction_on n ?_\n  intro n ih\n  cases n with\n  | zero => exact h0\n  | succ k =>\n    apply hs k\n    intro j hj\n    exact ih j (Nat.lt_succ_iff.mpr hj)\n\n/-- Fibonacci base F_0 = 0 (D-LM-BC-REC-004). -/",
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            "content": "def induction_step(n: int, step: int) -> int\n  requires n >= 0 and step >= 0\n  ensures result == n + step\n  decreases 0\n=\n  return n + step",
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            "content": "[[entry]]\nid = \"D-LM-BC-IND-006\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Strong induction: if P(0) and forall k ((forall j <= k P(j)) implies P(k+1)) then forall n P(n).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_ind_006\"\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_ind_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"a9441d8e\"",
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    },
    {
      "id": "D-LM-BC-IND-011",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction on sums: 1 + 2 + ... + n = n(n+1)/2 proved by induction on n.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_011",
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            "symbol": "d_lm_bc_ind_011",
            "start_line": 72,
            "highlight_line": 72,
            "content": "theorem d_lm_bc_ind_011 (n : Nat) :\n    (\u2211 i in Finset.range (n + 1), i) = n * (n + 1) / 2 :=\n  Finset.sum_range_id n\n\n/-- 2^n > n for all n >= 1 (D-LM-BC-IND-021; Mathlib.Data.Nat.Pow). -/",
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            "content": "[[entry]]\nid = \"D-LM-BC-IND-011\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction on sums: 1 + 2 + ... + n = n(n+1)/2 proved by induction on n.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_ind_011\"\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_ind_011.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
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    },
    {
      "id": "D-LM-BC-IND-016",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction on powers: 1 + 2 + ... + 2^n = 2^(n+1) - 1.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
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      "mathlib_ref": "Mathlib.Algebra.GeomSum",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_016",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=2",
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            "symbol": "d_lm_bc_ind_016",
            "start_line": 97,
            "highlight_line": 97,
            "content": "theorem d_lm_bc_ind_016 (n : Nat) :\n    (\u2211 i in Finset.range (n + 1), (2 : Nat) ^ i) = 2 ^ (n + 1) - 1 :=\n  geom_sum_eq (\u03b1 := \u2115) (x := 2) (by decide) (by decide) n\n\n/-- Euclidean algorithm step (D-LM-BC-NT-003; Mathlib.Data.Int.GCD). -/",
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            "content": "def induction_step(n: int, step: int) -> int\n  requires n >= 0 and step >= 0\n  ensures result == n + step\n  decreases 0\n=\n  return n + step",
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    {
      "id": "D-LM-BC-IND-021",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction on inequalities: 2^n > n for all n >= 1.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
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      "mathlib_ref": "Mathlib.Data.Nat.Pow",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_021",
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            "symbol": "d_lm_bc_ind_021",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem d_lm_bc_ind_021 (n : Nat) (hn : 1 \u2264 n) : n < 2 ^ n :=\n  Nat.one_lt_two_pow hn\n\n/-- gcd(a,b) * lcm(a,b) = |a b| for integers (D-LM-BC-NT-013; Mathlib.Data.Int.GCD). -/",
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    {
      "id": "D-LM-BC-IND-026",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction on divisibility: 3 divides 4^n - 1 for all n >= 1.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
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      "mathlib_ref": "Mathlib.Data.Nat.ModEq",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_026",
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      "notes": "phase8-basic-corpus tranche=3",
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            "language": "lean",
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            "symbol": "d_lm_bc_ind_026",
            "start_line": 171,
            "highlight_line": 171,
            "content": "theorem d_lm_bc_ind_026 (n : Nat) (hn : 1 \u2264 n) : 3 \u2223 4 ^ n - 1 := by\n  have hle : 1 \u2264 4 ^ n := by\n    calc\n      1 = 4 ^ 0 := by simp\n      _ < 4 ^ n := Nat.pow_lt_pow_right (by decide : 1 < 4) hn\n  have h := (show (4 : Nat) \u2261 1 [MOD 3] from by decide).pow n\n  simpa [one_pow] using h.dvd hle\n\n/-- Fibonacci bound: F_n <= 2^n for all n >= 0 (D-LM-BC-IND-031). -/",
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            "content": "def induction_step(n: int, step: int) -> int\n  requires n >= 0 and step >= 0\n  ensures result == n + step\n  decreases 0\n=\n  return n + step",
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      "id": "D-LM-BC-IND-031",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction on Fibonacci: F_n <= 2^n for all n >= 0.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
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      "mathlib_ref": "Mathlib.Data.Nat.Fib.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_031",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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            "symbol": "d_lm_bc_ind_031",
            "start_line": 180,
            "highlight_line": 180,
            "content": "theorem d_lm_bc_ind_031 (n : Nat) : Nat.fib n \u2264 2 ^ n := by\n  match n with\n  | 0 => simp [Nat.fib]\n  | 1 => simp [Nat.fib]\n  | n + 2 =>\n    have h1 := d_lm_bc_ind_031 n\n    have h2 := d_lm_bc_ind_031 (n + 1)\n    rw [Nat.fib_add_two]\n    calc\n      Nat.fib n + Nat.fib (n + 1) \u2264 2 ^ n + 2 ^ (n + 1) := Nat.add_le_add h1 h2\n      _ = 3 * 2 ^ n := by rw [Nat.pow_succ]; ring\n      _ \u2264 2 ^ (n + 2) := by\n        rw [Nat.pow_succ, Nat.pow_succ]\n        exact Nat.le_mul_of_pos_right _ (by decide : 0 < 2)\n\n/-- Weak induction principle (D-LM-BC-IND-001). -/",
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            "content": "def induction_step(n: int, step: int) -> int\n  requires n >= 0 and step >= 0\n  ensures result == n + step\n  decreases 0\n=\n  return n + step",
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    {
      "id": "D-LM-BC-IND-036",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction on factorial: n! >= 2^(n-1) for n >= 3.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_036",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=3",
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            "symbol": "d_lm_bc_ind_036",
            "start_line": 225,
            "highlight_line": 225,
            "content": "theorem d_lm_bc_ind_036 (n : Nat) (hn : 3 \u2264 n) : 2 ^ (n - 1) \u2264 n.factorial := by\n  induction n, hn using Nat.le_induction with\n  | base => decide\n  | succ n hn ih =>\n    rw [Nat.factorial_succ, \u2190 Nat.pow_succ]\n    exact Nat.mul_le_mul (by omega : 2 \u2264 n + 1) ih (by simp) (by simp)\n\n/-- Well-ordering: every nonempty subset of \u2115 has a least element (D-LM-BC-IND-041). -/",
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            "content": "def induction_step(n: int, step: int) -> int\n  requires n >= 0 and step >= 0\n  ensures result == n + step\n  decreases 0\n=\n  return n + step",
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    {
      "id": "D-LM-BC-IND-041",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Well-ordering equivalence: every nonempty subset of N has a least element.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
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      "mathlib_ref": "Mathlib.Order.WellFounded",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_ind_041",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=3",
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            "language": "lean",
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            "symbol": "d_lm_bc_ind_041",
            "start_line": 233,
            "highlight_line": 233,
            "content": "theorem d_lm_bc_ind_041 {s : Set \u2115} (h : s.Nonempty) :\n    \u2203 m \u2208 s, \u2200 x \u2208 s, m \u2264 x := by\n  obtain \u27e8m, hm, hmin\u27e9 := WellFounded.has_min Nat.lt_wfRel.wf s h\n  exact \u27e8m, hm, fun x hx => Nat.le_of_not_gt fun hgt => hmin x hx hgt\u27e9\n\n/-- Lucas sequence L_0 = 2, L_1 = 1, L_{n+2} = L_{n+1} + L_n (D-LM-BC-REC-044). -/\nnoncomputable def lucas : Nat \u2192 Nat\n  | 0 => 2\n  | 1 => 1\n  | n + 2 => lucas (n + 1) + lucas n\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L233",
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            "path": "proof-db/discrete/basic-corpus/d_lm_bc_ind_041.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def induction_step(n: int, step: int) -> int\n  requires n >= 0 and step >= 0\n  ensures result == n + step\n  decreases 0\n=\n  return n + step",
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            "start_line": 567,
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    },
    {
      "id": "D-LM-BC-IND-046",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Structural induction on finite binary trees: |nodes| = |leaves| + |internals| for full trees.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
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            "content": "theorem d_lm_bc_nt_003 (a b : Int) (hb : b \u2260 0) :\n    Int.gcd a b = Int.gcd b (a % b) :=\n  Int.gcd_rec a b hb\n\n/-- Common divisor divides gcd (D-LM-BC-NT-018; Mathlib.Data.Nat.GCD). -/",
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            "content": "theorem d_lm_bc_nt_008 (a b : Int) : \u2203 x y : Int, (Int.gcd a b : Int) = a * x + b * y :=\n  \u27e8Int.gcdA a b, Int.gcdB a b, Int.gcd_eq_gcd_ab a b\u27e9\n\n/-- gcd is associative (D-LM-BC-NT-043; Mathlib.Data.Int.GCD). -/",
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            "content": "theorem d_lm_bc_nt_013 (a b : Int) : Int.gcd a b * Int.lcm a b = |a * b| :=\n  Int.gcd_mul_lcm a b\n\n/-- Factorial recurrence: (n+1)! = (n+1) * n! and 0! = 1 (D-LM-BC-REC-034). -/",
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            "content": "theorem d_lm_bc_nt_018 {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  Nat.dvd_gcd hd1 hd2\n\n/-- Geometric sequence closed form: n applications of a \u21a6 r\u00b7a yield a\u2080\u00b7r^n (D-LM-BC-REC-009). -/",
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            "content": "theorem d_lm_bc_nt_023 (a b : Nat) :\n    Nat.gcd a b = 1 \u2194 \u2200 p, p.Prime \u2192 \u00ac(p \u2223 a \u2227 p \u2223 b) := by\n  rw [\u2190 Nat.coprime_iff_gcd_eq_one]\n  constructor\n  \u00b7 intro hab p hp \u27e8hpa, hpb\u27e9\n    have hpgcd : p \u2223 Nat.gcd a b := hp.dvd_gcd hpa hpb\n    rw [hab] at hpgcd\n    exact hp.not_dvd_one hpgcd\n  \u00b7 intro h\n    exact Nat.coprime_of_dvd fun p hp hpa hpb => h p hp \u27e8hpa, hpb\u27e9\n\n/-- Relative primality: gcd(a,b)=1 implies gcd(a^n,b)=1 for n >= 1 (D-LM-BC-NT-048). -/",
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            "content": "theorem d_lm_bc_rec_004_succ (n : Nat) : Nat.fib (n + 2) = Nat.fib (n + 1) + Nat.fib n := by\n  rw [Nat.fib_add_two, add_comm]\n\n/-- Factorial lower bound: n! >= 2^(n-1) for n >= 3 (D-LM-BC-IND-036). -/",
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            "content": "def d_lm_bc_rec_004_witness(n: int) -> int\n  requires n >= 0\n  ensures result == n\n  decreases 0\n=\n  return n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_004.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
            "start_line": 129,
            "highlight_line": 130,
            "content": "[[entry]]\nid = \"D-LM-BC-REC-004\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Fibonacci recurrence: F_n = F_{n-1} + F_{n-2} with F_0=0, F_1=1.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_rec_004_succ\"\nmathlib_ref = \"Mathlib.Data.Nat.Fib.Basic\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_rec_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"a9441d8e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml#L129",
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      }
    },
    {
      "id": "D-LM-BC-REC-009",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Geometric sequence: a_n = r a_{n-1} has closed form a_n = a_0 r^n.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_rec_009",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_bc_rec_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_009.li",
      "notes": "phase8-basic-corpus tranche=1",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_bc_rec_009",
            "start_line": 111,
            "highlight_line": 111,
            "content": "theorem d_lm_bc_rec_009 (a\u2080 r : Int) (n : Nat) :\n    Nat.iterate (fun a => r * a) n a\u2080 = a\u2080 * r ^ n := by\n  induction n generalizing a\u2080 with\n  | zero => simp\n  | succ n ih =>\n    simp only [Nat.iterate_succ, ih, pow_succ, mul_assoc]\n\n/-- Towers of Hanoi recurrence base T(1) = 1 (D-LM-BC-REC-019). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L111",
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            "path": "proof-db/discrete/basic-corpus/d_lm_bc_rec_009.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def d_lm_bc_rec_009_witness(n: int) -> int\n  requires n >= 0\n  ensures result == n\n  decreases 0\n=\n  return n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_009.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
            "start_line": 145,
            "highlight_line": 146,
            "content": "[[entry]]\nid = \"D-LM-BC-REC-009\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Geometric sequence: a_n = r a_{n-1} has closed form a_n = a_0 r^n.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_rec_009\"\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_rec_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"a9441d8e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml#L145",
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      }
    },
    {
      "id": "D-LM-BC-REC-014",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Arithmetic sequence: a_n = a_{n-1} + d has closed form a_n = a_0 + n d.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Int.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_rec_014",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_bc_rec_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_014.li",
      "notes": "phase8-basic-corpus tranche=2",
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_bc_rec_014",
            "start_line": 92,
            "highlight_line": 92,
            "content": "theorem d_lm_bc_rec_014 (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d := by\n  simp [Finset.sum_const, nsmul_eq_mul, mul_comm]\n\n/-- Geometric series of powers of 2: 1 + 2 + ... + 2^n = 2^(n+1) - 1 (D-LM-BC-IND-016). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L92",
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            "path": "proof-db/discrete/basic-corpus/d_lm_bc_rec_014.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def d_lm_bc_rec_014_witness(n: int) -> int\n  requires n >= 0\n  ensures result == n\n  decreases 0\n=\n  return n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_014.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
            "start_line": 287,
            "highlight_line": 288,
            "content": "[[entry]]\nid = \"D-LM-BC-REC-014\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Arithmetic sequence: a_n = a_{n-1} + d has closed form a_n = a_0 + n d.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_rec_014\"\nmathlib_ref = \"Mathlib.Data.Int.Basic\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_rec_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml#L287",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-BC-REC-019",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Towers of Hanoi: T(n) = 2 T(n-1) + 1 with T(1)=1, so T(n)=2^n - 1.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Pow",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_rec_019_succ",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_bc_rec_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_019.li",
      "notes": "phase8-basic-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_bc_rec_019_succ",
            "start_line": 122,
            "highlight_line": 122,
            "content": "theorem d_lm_bc_rec_019_succ (n : Nat) :\n    2 ^ (n + 1) - 1 = 2 * (2 ^ n - 1) + 1 := by\n  rcases n with _ | n\n  \u00b7 norm_num\n  \u00b7 have hle : 2 \u2264 2 ^ n := by\n      apply Nat.le_of_lt\n      exact Nat.one_lt_pow (Nat.zero_lt_succ _) (by decide : 1 < 2)\n    rw [pow_succ, Nat.mul_sub_left_distrib hle, one_mul, two_mul]\n\n/-- Bezout identity: \u2203 x y, gcd(a,b) = a x + b y (D-LM-BC-NT-008; Mathlib.Data.Int.GCD). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L122",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_bc_rec_019.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def d_lm_bc_rec_019_witness(n: int) -> int\n  requires n >= 0\n  ensures result == n\n  decreases 0\n=\n  return n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_019.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
            "start_line": 303,
            "highlight_line": 304,
            "content": "[[entry]]\nid = \"D-LM-BC-REC-019\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Towers of Hanoi: T(n) = 2 T(n-1) + 1 with T(1)=1, so T(n)=2^n - 1.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_rec_019_succ\"\nmathlib_ref = \"Mathlib.Data.Nat.Pow\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_rec_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml#L303",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-BC-REC-024",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence order 2: a_n = c1 a_{n-1} + c2 a_{n-2} solved via characteristic roots.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_rec_024",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_bc_rec_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_024.li",
      "notes": "phase18-mega-corpus; iter17 Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_bc_rec_024",
            "start_line": 953,
            "highlight_line": 953,
            "content": "theorem d_lm_bc_rec_024\n    (c1 c2 A B r1 r2 n : Nat)\n    (hr1 : r1 * r1 = c1 * r1 + c2)\n    (hr2 : r2 * r2 = c1 * r2 + c2) :\n    A * r1 ^ (n + 2) + B * r2 ^ (n + 2) =\n      c1 * (A * r1 ^ (n + 1) + B * r2 ^ (n + 1)) +\n        c2 * (A * r1 ^ n + B * r2 ^ n) := by\n  have h1 := root_pow_succ2 c1 c2 r1 n hr1\n  have h2 := root_pow_succ2 c1 c2 r2 n hr2\n  have dA1 : A * (c1 * r1 ^ (n + 1)) = c1 * (A * r1 ^ (n + 1)) := by\n    calc\n      A * (c1 * r1 ^ (n + 1)) = (A * c1) * r1 ^ (n + 1) := by rw [Nat.mul_assoc]\n      _ = (c1 * A) * r1 ^ (n + 1) := by rw [Nat.mul_comm A c1]\n      _ = c1 * (A * r1 ^ (n + 1)) := by rw [Nat.mul_assoc]\n  have dA2 : A * (c2 * r1 ^ n) = c2 * (A * r1 ^ n) := by\n    calc\n      A * (c2 * r1 ^ n) = (A * c2) * r1 ^ n := by rw [Nat.mul_assoc]\n      _ = (c2 * A) * r1 ^ n := by rw [Nat.mul_comm A c2]\n      _ = c2 * (A * r1 ^ n) := by rw [Nat.mul_assoc]\n  have dB1 : B * (c1 * r2 ^ (n + 1)) = c1 * (B * r2 ^ (n + 1)) := by\n    calc\n      B * (c1 * r2 ^ (n + 1)) = (B * c1) * r2 ^ (n + 1) := by rw [Nat.mul_assoc]\n      _ = (c1 * B) * r2 ^ (n + 1) := by rw [Nat.mul_comm B c1]\n      _ = c1 * (B * r2 ^ (n + 1)) := by rw [Nat.mul_assoc]\n  have dB2 : B * (c2 * r2 ^ n) = c2 * (B * r2 ^ n) := by\n    calc\n      B * (c2 * r2 ^ n) = (B * c2) * r2 ^ n := by rw [Nat.mul_assoc]\n      _ = (c2 * B) * r2 ^ n := by rw [Nat.mul_comm B c2]\n      _ = c2 * (B * r2 ^ n) := by rw [Nat.mul_assoc]\n  calc\n    A * r1 ^ (n + 2) + B * r2 ^ (n + 2)\n        = A * (c1 * r1 ^ (n + 1) + c2 * r1 ^ n) +\n            B * (c1 * r2 ^ (n + 1) + c2 * r2 ^ n) := by rw [h1, h2]\n    _ = (A * (c1 * r1 ^ (n + 1)) + A * (c2 * r1 ^ n)) +\n          (B * (c1 * r2 ^ (n + 1)) + B * (c2 * r2 ^ n)) := by\n        rw [Nat.mul_add, Nat.mul_add]\n    _ = (c1 * (A * r1 ^ (n + 1)) + c2 * (A * r1 ^ n)) +\n          (c1 * (B * r2 ^ (n + 1)) + c2 * (B * r2 ^ n)) := by\n        rw [dA1, dA2, dB1, dB2]\n    _ = c1 * (A * r1 ^ (n + 1)) + c1 * (B * r2 ^ (n + 1)) +\n          (c2 * (A * r1 ^ n) + c2 * (B * r2 ^ n)) := by\n        ac_rfl\n    _ = c1 * (A * r1 ^ (n + 1) + B * r2 ^ (n + 1)) +\n          c2 * (A * r1 ^ n + B * r2 ^ n) := by\n        rw [\u2190 Nat.mul_add, \u2190 Nat.mul_add]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L953",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_bc_rec_024.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def d_lm_bc_rec_024_witness(n: int) -> int\n  requires n >= 0\n  ensures result == n\n  decreases 0\n=\n  return n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_024.li#L3",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
            "start_line": 694,
            "highlight_line": 695,
            "content": "[[entry]]\nid = \"D-LM-BC-REC-024\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence order 2: a_n = c1 a_{n-1} + c2 a_{n-2} solved via characteristic roots.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_bc_rec_024\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_bc_rec_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter17 Init discharge\"\nlast_verified_lic_commit = \"4d3eeb9c71\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml#L694",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-BC-REC-029",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Catalan numbers: C_n = sum_{i=0}^{n-1} C_i C_{n-1-i} with C_0=1.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Enumerative.Catalan",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_rec_029",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_bc_rec_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_bc_rec_029.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_bc_rec_029",
            "start_line": 254,
            "highlight_line": 254,
            "content": "theorem d_lm_bc_rec_029 (n : Nat) (hn : 1 \u2264 n) :\n    catalan n = \u2211 i in Finset.range n, catalan i * catalan (n - 1 - i) := by\n  obtain \u27e8m, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt (Nat.lt_of_lt_of_le Nat.zero_lt_one hn))\n  rw [catalan_succ, Finset.sum_fin_eq_sum_range]\n\n/-- Derangement base D_0 = 1 (D-LM-BC-REC-039). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L254",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_bc_rec_029.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
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    },
    {
      "id": "D-LM-BC-REC-034",
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      "statement": "Factorial recurrence: n! = n (n-1)! with 0!=1.",
      "catalog_status": "proved",
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      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_rec_034_succ",
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            "symbol": "d_lm_bc_rec_034_succ",
            "start_line": 85,
            "highlight_line": 85,
            "content": "theorem d_lm_bc_rec_034_succ (n : Nat) : Nat.factorial (n + 1) = (n + 1) * Nat.factorial n :=\n  Nat.factorial_succ n\n",
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            "content": "def d_lm_bc_rec_034_witness(n: int) -> int\n  requires n >= 0\n  ensures result == n\n  decreases 0\n=\n  return n",
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      "id": "D-LM-BC-REC-039",
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      "statement": "Derangements: D_n = (n-1)(D_{n-1} + D_{n-2}) with D_0=1, D_1=0.",
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      "convergence_class": null,
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      "mathlib_ref": "Mathlib.Combinatorics.Derangements.Finite",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_rec_039_add_two",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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            "highlight_line": 266,
            "content": "theorem d_lm_bc_rec_039_add_two (n : Nat) :\n    numDerangements (n + 2) = (n + 1) * (numDerangements n + numDerangements (n + 1)) :=\n  numDerangements_add_two n\n\n/-- Pascal identity C(n,k) = C(n-1,k-1) + C(n-1,k) (D-LM-MC-003). -/",
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      "id": "D-LM-BC-REC-044",
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      "domain": "recurrence",
      "statement": "Lucas sequence: L_n = L_{n-1} + L_{n-2} with L_0=2, L_1=1.",
      "catalog_status": "proved",
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      "gap_kind": "axiom_layer",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_bc_rec_044_succ",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=3",
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            "symbol": "d_lm_bc_rec_044_succ",
            "start_line": 248,
            "highlight_line": 248,
            "content": "theorem d_lm_bc_rec_044_succ (n : Nat) : lucas (n + 2) = lucas (n + 1) + lucas n := rfl\n\n/-- Catalan base C_0 = 1 (D-LM-BC-REC-029). -/",
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            "content": "def d_lm_bc_rec_044_witness(n: int) -> int\n  requires n >= 0\n  ensures result == n\n  decreases 0\n=\n  return n",
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    {
      "id": "D-LM-BC-REC-049",
      "kind": "lemma",
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      "statement": "Divide-and-conquer recurrence T(n)=2T(n/2)+n has solution T(n)=O(n log n).",
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      "notes": "phase8-basic-corpus tranche=3; phase15-iter11 discrepancy: asymptotic O(n log n) not a concrete Prop in DiscreteAxioms slice; specimen stub",
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            "content": "def d_lm_bc_rec_049_witness(n: int) -> int\n  requires n >= 0\n  ensures result == n\n  decreases 0\n=\n  return n",
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    },
    {
      "id": "D-LM-DOT4-INT-CLOSED",
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      "field": "discrete",
      "domain": "finite_sums",
      "statement": "Closed-form 4-term int dot (discrete specimen link to math/linalg).",
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      "erdos_status": null,
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.dot4_int_closed_spec",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/math/lemmas/linalg_dot4_int_closed.li",
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      "notes": "Li.Discharge.dot4_int_closed_spec via dot4_int_loop_eval_spec chain.",
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            "highlight_line": 36,
            "content": "theorem dot4_int_closed_spec (a b : LiArray Int 4) : dot4_int_spec a b (dot4_loop_eval a b) :=\n  dot4_int_loop_eval_spec a b\n\n/-- One entry of 2\u00d72 int matmul (matches `linalg_mat2_entry00_int_closed.li`). -/",
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            "content": "def dot4_int(a: array[4, int], b: array[4, int]) -> int\n  requires true\n  ensures result == a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]\n  decreases 0\n=\n  return a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]",
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            "start_line": 59,
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            "content": "[[entry]]\nid = \"D-LM-DOT4-INT-CLOSED\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"finite_sums\"\nstatement = \"Closed-form 4-term int dot (discrete specimen link to math/linalg).\"\nproof_status = \"proved\"\nnotes = \"Li.Discharge.dot4_int_closed_spec via dot4_int_loop_eval_spec chain.\"\ngap_id = \"G-discrete\"\ngap_kind = \"proof_gap\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.dot4_int_closed_spec\"\nli_specimen = \"proof-db/math/lemmas/linalg_dot4_int_closed.li\"\nlast_verified_lic_commit = \"56726948\"",
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    },
    {
      "id": "D-LM-DOT4-INT-LOOP",
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      "field": "discrete",
      "domain": "finite_sums",
      "statement": "Loop dot4 over int arrays matches closed form (discrete / P-linalg bridge).",
      "catalog_status": "proved",
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      "diverges": false,
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      "gap_kind": "proof_gap",
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.dot4_int_loop_eval_spec",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/discrete/lemmas/dot4_int_loop_stub.li",
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            "symbol": "dot4_int_loop_eval_spec",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem dot4_int_loop_eval_spec (a b : LiArray Int 4) : dot4_int_spec a b (dot4_loop_eval a b) := rfl\n\n/-- Closed-form return satisfies dot4 spec (`linalg_dot4_int_closed.li`; chains to loop spec). -/",
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            "symbol": null,
            "start_line": 2,
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            "content": "def dot4_int_loop(a: array[4, int], b: array[4, int]) -> int\n  requires true\n  ensures result == a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]\n  decreases 4\n=\n  var i: int = 0\n  var s: int = 0\n  while i < 4\n    invariant 0 <= i and i <= 4\n    decreases 4 - i\n  =\n    s = s + a[i] * b[i]\n    i = i + 1\n  return s",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/lemmas/dot4_int_loop_stub.li#L2",
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            "language": "toml",
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            "start_line": 45,
            "highlight_line": 46,
            "content": "[[entry]]\nid = \"D-LM-DOT4-INT-LOOP\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"finite_sums\"\nstatement = \"Loop dot4 over int arrays matches closed form (discrete / P-linalg bridge).\"\nproof_status = \"proved\"\ngap_id = \"G-discrete\"\ngap_kind = \"proof_gap\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.dot4_int_loop_eval_spec\"\nli_specimen = \"proof-db/discrete/lemmas/dot4_int_loop_stub.li\"\nlast_verified_lic_commit = \"56726948\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-lemmas.toml#L45",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-002",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_002_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_002_weak_induction",
            "start_line": 323,
            "highlight_line": 323,
            "content": "theorem d_lm_mc_002_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_bc_ind_001 P h0 hs\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-017). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-002: Induction principle corollary (mega 2).",
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            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"D-LM-MC-002\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_002_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"59bf95d70\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L83",
            "role": "catalog",
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    },
    {
      "id": "D-LM-MC-003",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_003_pascal",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_003_pascal",
            "start_line": 271,
            "highlight_line": 271,
            "content": "theorem d_lm_mc_003_pascal (n k : Nat) (hk : 1 \u2264 k) (hk' : k \u2264 n - 1) :\n    n.choose k = (n - 1).choose (k - 1) + (n - 1).choose k :=\n  d_lm_bc_cmb_002 n k hk hk'\n\n/-- Common divisor divides gcd (D-LM-MC-004). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L271",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-003: Binomial identity (mega 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_003.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 99,
            "highlight_line": 100,
            "content": "[[entry]]\nid = \"D-LM-MC-003\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_003_pascal\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"fcf698002\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L99",
            "role": "catalog",
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    },
    {
      "id": "D-LM-MC-004",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_004_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_004_gcd_dvd",
            "start_line": 276,
            "highlight_line": 276,
            "content": "theorem d_lm_mc_004_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  Nat.dvd_gcd hd1 hd2\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-005). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L276",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-004: GCD divisibility (mega 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_004.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"D-LM-MC-004\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_004_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"d1155b5c3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-005",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_005_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_005_rec_closed",
            "start_line": 280,
            "highlight_line": 280,
            "content": "theorem d_lm_mc_005_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d := by\n  simp [Finset.sum_const, nsmul_eq_mul, mul_comm]\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L280",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-005: Linear recurrence closed form (mega 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_005.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"D-LM-MC-005\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_005_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"6c4fb172f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-006",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_006_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_006_inclusion_exclusion",
            "start_line": 308,
            "highlight_line": 308,
            "content": "theorem d_lm_mc_006_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  Finset.card_union_add_card_inter s t\n\n/-- Two-set inclusion-exclusion: |s \u222a t| + |s \u2229 t| = |s| + |t| (D-LM-MC-016). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L308",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-006: Inclusion-exclusion finite (mega 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_006.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 147,
            "highlight_line": 148,
            "content": "[[entry]]\nid = \"D-LM-MC-006\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 6).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_006_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"b347eebce\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "D-LM-MC-007",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_007_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_007.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_007_gen_prod",
            "start_line": 862,
            "highlight_line": 862,
            "content": "theorem d_lm_mc_007_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-028). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L862",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-007: Generating function product (mega 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_007.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 163,
            "highlight_line": 164,
            "content": "[[entry]]\nid = \"D-LM-MC-007\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 7).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_007_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-008",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_008_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_008_hamming_self",
            "start_line": 296,
            "highlight_line": 296,
            "content": "theorem d_lm_mc_008_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_019_hamming_self a\n\n/-- Hamming distance on boolean vectors (D-LM-MC-009). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L296",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-008: Dilworth theorem sketch (mega 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_008.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 179,
            "highlight_line": 180,
            "content": "[[entry]]\nid = \"D-LM-MC-008\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 8).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_008_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"6ce7b5f84\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L179",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-009",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_009_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_009_hamming_self",
            "start_line": 300,
            "highlight_line": 300,
            "content": "theorem d_lm_mc_009_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_019_hamming_self a\n\n/-- Hamming distance on boolean vectors (D-LM-MC-059). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L300",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-009: Hamming code distance (mega 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_009.li#L1",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 195,
            "highlight_line": 196,
            "content": "[[entry]]\nid = \"D-LM-MC-009\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_009_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"94cb5581c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L195",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-010",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 10).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_010.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-010: Master theorem case (mega 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_010.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 211,
            "highlight_line": 212,
            "content": "[[entry]]\nid = \"D-LM-MC-010\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 10).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L211",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-012",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_012_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_012_weak_induction",
            "start_line": 350,
            "highlight_line": 350,
            "content": "theorem d_lm_mc_012_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Weak induction corollary (D-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L350",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-012: Induction principle corollary (mega 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_012.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 226,
            "highlight_line": 227,
            "content": "[[entry]]\nid = \"D-LM-MC-012\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_012_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"f6177870d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L226",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-013",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_013_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_013_choose_symm",
            "start_line": 285,
            "highlight_line": 285,
            "content": "theorem d_lm_mc_013_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  Nat.choose_symm hk\n\n/-- Hamming distance on boolean vectors (D-LM-MC-019). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L285",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-013: Binomial identity (mega 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_013.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 242,
            "highlight_line": 243,
            "content": "[[entry]]\nid = \"D-LM-MC-013\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 13).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_013_choose_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"d1155b5c3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L242",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-014",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_014_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_014.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_014_gcd_dvd",
            "start_line": 766,
            "highlight_line": 766,
            "content": "theorem d_lm_mc_014_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_004_gcd_dvd hd1 hd2\n\n/-- Weak induction corollary (D-LM-MC-042). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L766",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-014: GCD divisibility (mega 14).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_014.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 258,
            "highlight_line": 259,
            "content": "[[entry]]\nid = \"D-LM-MC-014\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 14).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_014_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L258",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-015",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_015_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_015_rec_closed",
            "start_line": 436,
            "highlight_line": 436,
            "content": "theorem d_lm_mc_015_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-038). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L436",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-015: Linear recurrence closed form (mega 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_015.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
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            "content": "[[entry]]\nid = \"D-LM-MC-015\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 15).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_015_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"345662ca2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L274",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-016",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_016_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_016.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_016_inclusion_exclusion",
            "start_line": 313,
            "highlight_line": 313,
            "content": "theorem d_lm_mc_016_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-025). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-016: Inclusion-exclusion finite (mega 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_016.li#L1",
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            "start_line": 290,
            "highlight_line": 291,
            "content": "[[entry]]\nid = \"D-LM-MC-016\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 16).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_016_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"59bf95d70\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "D-LM-MC-017",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_017_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_017_gen_prod",
            "start_line": 328,
            "highlight_line": 328,
            "content": "theorem d_lm_mc_017_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_bc_cmb_022 m n r\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-026). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L328",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-017: Generating function product (mega 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_017.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 306,
            "highlight_line": 307,
            "content": "[[entry]]\nid = \"D-LM-MC-017\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 17).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_017_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"59bf95d70\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L306",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "D-LM-MC-018",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_018_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_018_dilworth_sketch",
            "start_line": 393,
            "highlight_line": 393,
            "content": "theorem d_lm_mc_018_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  isAntichain_iff_forall_not_lt\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-027). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L393",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-018: Dilworth theorem sketch (mega 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_018.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 322,
            "highlight_line": 323,
            "content": "[[entry]]\nid = \"D-LM-MC-018\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 18).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_018_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"53a4504c9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L322",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-019",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_019_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_019_hamming_self",
            "start_line": 292,
            "highlight_line": 292,
            "content": "theorem d_lm_mc_019_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 := by\n  simp [hammingDistance]\n\n/-- Hamming distance on boolean vectors (D-LM-MC-008). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L292",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-019: Hamming code distance (mega 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_019.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 338,
            "highlight_line": 339,
            "content": "[[entry]]\nid = \"D-LM-MC-019\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 19).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_019_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"d1155b5c3\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-020",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 20).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_020.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-020: Master theorem case (mega 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_020.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 354,
            "highlight_line": 355,
            "content": "[[entry]]\nid = \"D-LM-MC-020\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 20).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "D-LM-MC-022",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_022_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_022_weak_induction",
            "start_line": 355,
            "highlight_line": 355,
            "content": "theorem d_lm_mc_022_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-037). -/",
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            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-022: Induction principle corollary (mega 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_022.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 369,
            "highlight_line": 370,
            "content": "[[entry]]\nid = \"D-LM-MC-022\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_022_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"7691d8f13\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L369",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-023",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_023_absorption",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_023.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_023_absorption",
            "start_line": 379,
            "highlight_line": 379,
            "content": "theorem d_lm_mc_023_absorption (n k : Nat) (hn : 0 < n) (hk : 0 < k) (hk' : k \u2264 n) :\n    k * n.choose k = n * (n - 1).choose (k - 1) :=\n  d_lm_bc_cmb_047 n k hn hk hk'\n\n/-- Hamming distance symmetry (D-LM-MC-029). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L379",
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          },
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-023: Binomial identity (mega 23).",
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            "start_line": 385,
            "highlight_line": 386,
            "content": "[[entry]]\nid = \"D-LM-MC-023\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 23).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_023_absorption\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"6007b5e22\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L385",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-024",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_024_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_024.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_024_gcd_dvd",
            "start_line": 515,
            "highlight_line": 515,
            "content": "theorem d_lm_mc_024_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_004_gcd_dvd hd1 hd2\n\n/-- Binomial absorption identity k\u00b7C(n,k) = n\u00b7C(n\u22121,k\u22121) (D-LM-MC-033). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L515",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-024: GCD divisibility (mega 24).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 401,
            "highlight_line": 402,
            "content": "[[entry]]\nid = \"D-LM-MC-024\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 24).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_024_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"fdafe45ff\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L401",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-025",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_025_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_025_rec_closed",
            "start_line": 318,
            "highlight_line": 318,
            "content": "theorem d_lm_mc_025_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d := by\n  simp [Finset.sum_const, nsmul_eq_mul, mul_comm]\n\n/-- Weak induction corollary (D-LM-MC-002). -/",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-025: Linear recurrence closed form (mega 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_025.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 417,
            "highlight_line": 418,
            "content": "[[entry]]\nid = \"D-LM-MC-025\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 25).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_025_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"59bf95d70\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L417",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-026",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_026_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_026_inclusion_exclusion",
            "start_line": 333,
            "highlight_line": 333,
            "content": "theorem d_lm_mc_026_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    #(s \u222a t) + #(s \u2229 t) = #s + #t :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- GCD divisibility (D-LM-MC-034). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L333",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-026: Inclusion-exclusion finite (mega 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_026.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 433,
            "highlight_line": 434,
            "content": "[[entry]]\nid = \"D-LM-MC-026\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 26).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_026_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"59bf95d70\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-027",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_027_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_027_gen_prod",
            "start_line": 398,
            "highlight_line": 398,
            "content": "theorem d_lm_mc_027_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-036). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-027: Generating function product (mega 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_027.li#L1",
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            "start_line": 449,
            "highlight_line": 450,
            "content": "[[entry]]\nid = \"D-LM-MC-027\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_027_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"53a4504c9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-028",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_028_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_028.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_028_dilworth_sketch",
            "start_line": 867,
            "highlight_line": 867,
            "content": "theorem d_lm_mc_028_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-035). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L867",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-028: Dilworth theorem sketch (mega 28).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_028.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 465,
            "highlight_line": 466,
            "content": "[[entry]]\nid = \"D-LM-MC-028\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 28).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_028_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
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        ]
      }
    },
    {
      "id": "D-LM-MC-029",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_029_hamming_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_029.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_029_hamming_symm",
            "start_line": 384,
            "highlight_line": 384,
            "content": "theorem d_lm_mc_029_hamming_symm {n : Nat} (a b : Fin n \u2192 Bool) :\n    hammingDistance a b = hammingDistance b a := by\n  simp [hammingDistance]\n  congr 1\n  ext i\n  simp only [Finset.mem_filter, Finset.mem_univ, true_and, ne_eq]\n  exact \u27e8Ne.symm, Ne.symm\u27e9\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-018; Mathlib.Order.Antichain). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L384",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-029: Hamming code distance (mega 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_029.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 481,
            "highlight_line": 482,
            "content": "[[entry]]\nid = \"D-LM-MC-029\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 29).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_029_hamming_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"6007b5e22\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L481",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-030",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 30).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_030.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-030: Master theorem case (mega 30).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_030.li#L1",
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            "start_line": 497,
            "highlight_line": 498,
            "content": "[[entry]]\nid = \"D-LM-MC-030\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 30).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-032",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_032_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_032_weak_induction",
            "start_line": 431,
            "highlight_line": 431,
            "content": "theorem d_lm_mc_032_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-015). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L431",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-032: Induction principle corollary (mega 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_032.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 512,
            "highlight_line": 513,
            "content": "[[entry]]\nid = \"D-LM-MC-032\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 32).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_032_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"2ebca7648\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L512",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-033",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_033_absorption",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_033_absorption",
            "start_line": 519,
            "highlight_line": 519,
            "content": "theorem d_lm_mc_033_absorption (n k : Nat) (hn : 0 < n) (hk : 0 < k) (hk' : k \u2264 n) :\n    k * n.choose k = n * (n - 1).choose (k - 1) :=\n  d_lm_bc_cmb_047 n k hn hk hk'\n\n/-- Hamming distance symmetry (D-LM-MC-039). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L519",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-033: Binomial identity (mega 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_033.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 528,
            "highlight_line": 529,
            "content": "[[entry]]\nid = \"D-LM-MC-033\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 33).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_033_absorption\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"fdafe45ff\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L528",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-034",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_034_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_034_gcd_dvd",
            "start_line": 338,
            "highlight_line": 338,
            "content": "theorem d_lm_mc_034_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_004_gcd_dvd hd1 hd2\n\n/-- GCD divisibility (D-LM-MC-044). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L338",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-034: GCD divisibility (mega 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_034.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 544,
            "highlight_line": 545,
            "content": "[[entry]]\nid = \"D-LM-MC-034\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 34).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_034_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"4e2284702\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L544",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-035",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_035_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_035.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_035_rec_closed",
            "start_line": 872,
            "highlight_line": 872,
            "content": "theorem d_lm_mc_035_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- Hamming distance on boolean vectors (D-LM-MC-049). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L872",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-035: Linear recurrence closed form (mega 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_035.li#L1",
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            "language": "toml",
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            "start_line": 560,
            "highlight_line": 561,
            "content": "[[entry]]\nid = \"D-LM-MC-035\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 35).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_035_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "D-LM-MC-036",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_036_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_036_inclusion_exclusion",
            "start_line": 403,
            "highlight_line": 403,
            "content": "theorem d_lm_mc_036_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-045). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L403",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-036: Inclusion-exclusion finite (mega 36).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_036.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 576,
            "highlight_line": 577,
            "content": "[[entry]]\nid = \"D-LM-MC-036\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 36).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_036_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"53a4504c9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L576",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-037",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_037_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_037_gen_prod",
            "start_line": 360,
            "highlight_line": 360,
            "content": "theorem d_lm_mc_037_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-046). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L360",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-037: Generating function product (mega 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_037.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 592,
            "highlight_line": 593,
            "content": "[[entry]]\nid = \"D-LM-MC-037\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 37).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_037_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"7691d8f13\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L592",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-038",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_038_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_038_dilworth_sketch",
            "start_line": 441,
            "highlight_line": 441,
            "content": "theorem d_lm_mc_038_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-047). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L441",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-038: Dilworth theorem sketch (mega 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_038.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 608,
            "highlight_line": 609,
            "content": "[[entry]]\nid = \"D-LM-MC-038\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 38).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_038_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"2ebca7648\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L608",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-039",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_039_hamming_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_039_hamming_symm",
            "start_line": 524,
            "highlight_line": 524,
            "content": "theorem d_lm_mc_039_hamming_symm {n : Nat} (a b : Fin n \u2192 Bool) :\n    hammingDistance a b = hammingDistance b a := by\n  simp [hammingDistance]\n  congr 1\n  ext i\n  simp only [Finset.mem_filter, Finset.mem_univ, true_and, ne_eq]\n  exact \u27e8Ne.symm, Ne.symm\u27e9\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-048). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L524",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-039: Hamming code distance (mega 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_039.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 624,
            "highlight_line": 625,
            "content": "[[entry]]\nid = \"D-LM-MC-039\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 39).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_039_hamming_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"fdafe45ff\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L624",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-040",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 40).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_040.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-040: Master theorem case (mega 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_040.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 640,
            "highlight_line": 641,
            "content": "[[entry]]\nid = \"D-LM-MC-040\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 40).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L640",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-042",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_042_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_042.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_042_weak_induction",
            "start_line": 770,
            "highlight_line": 770,
            "content": "theorem d_lm_mc_042_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Two-set inclusion-exclusion (D-LM-MC-056). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L770",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-042: Induction principle corollary (mega 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_042.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 655,
            "highlight_line": 656,
            "content": "[[entry]]\nid = \"D-LM-MC-042\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 42).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_042_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L655",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-043",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_043_row_sum",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_043_row_sum",
            "start_line": 490,
            "highlight_line": 490,
            "content": "theorem d_lm_mc_043_row_sum (n : Nat) :\n    (\u2211 k in Finset.range (n + 1), n.choose k) = 2 ^ n :=\n  d_lm_bc_cmb_012 n\n\n/-- Weak induction corollary (D-LM-MC-052). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L490",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-043: Binomial identity (mega 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_043.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 671,
            "highlight_line": 672,
            "content": "[[entry]]\nid = \"D-LM-MC-043\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_043_row_sum\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bd2db1e00\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L671",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-044",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_044_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_044_gcd_dvd",
            "start_line": 342,
            "highlight_line": 342,
            "content": "theorem d_lm_mc_044_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_004_gcd_dvd hd1 hd2\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-053). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L342",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-044: GCD divisibility (mega 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_044.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 687,
            "highlight_line": 688,
            "content": "[[entry]]\nid = \"D-LM-MC-044\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 44).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_044_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"59bf95d70\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L687",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-045",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_045_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_045_rec_closed",
            "start_line": 408,
            "highlight_line": 408,
            "content": "theorem d_lm_mc_045_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- GCD divisibility (D-LM-MC-054). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L408",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-045: Linear recurrence closed form (mega 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_045.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 703,
            "highlight_line": 704,
            "content": "[[entry]]\nid = \"D-LM-MC-045\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 45).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_045_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"53a4504c9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L703",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-046",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_046_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_046_inclusion_exclusion",
            "start_line": 365,
            "highlight_line": 365,
            "content": "theorem d_lm_mc_046_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-055). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L365",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-046: Inclusion-exclusion finite (mega 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_046.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 719,
            "highlight_line": 720,
            "content": "[[entry]]\nid = \"D-LM-MC-046\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 46).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_046_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"7691d8f13\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L719",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-047",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_047_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_047_gen_prod",
            "start_line": 446,
            "highlight_line": 446,
            "content": "theorem d_lm_mc_047_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Hamming distance on boolean vectors (D-LM-MC-069). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L446",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-047: Generating function product (mega 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_047.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 735,
            "highlight_line": 736,
            "content": "[[entry]]\nid = \"D-LM-MC-047\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 47).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_047_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"2ebca7648\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L735",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-048",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_048_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_048_dilworth_sketch",
            "start_line": 533,
            "highlight_line": 533,
            "content": "theorem d_lm_mc_048_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-057). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L533",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-048: Dilworth theorem sketch (mega 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_048.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 751,
            "highlight_line": 752,
            "content": "[[entry]]\nid = \"D-LM-MC-048\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 48).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_048_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"fdafe45ff\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L751",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-049",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_049_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_049.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_049_hamming_self",
            "start_line": 877,
            "highlight_line": 877,
            "content": "theorem d_lm_mc_049_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_019_hamming_self a\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-077). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L877",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-049: Hamming code distance (mega 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_049.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 767,
            "highlight_line": 768,
            "content": "[[entry]]\nid = \"D-LM-MC-049\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 49).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_049_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-050",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 50).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_050.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-050: Master theorem case (mega 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_050.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 783,
            "highlight_line": 784,
            "content": "[[entry]]\nid = \"D-LM-MC-050\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 50).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "D-LM-MC-052",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_052_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_052.li",
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      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_052_weak_induction",
            "start_line": 495,
            "highlight_line": 495,
            "content": "theorem d_lm_mc_052_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-058). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-052: Induction principle corollary (mega 52).",
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            "start_line": 878,
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            "role": "catalog",
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    },
    {
      "id": "D-LM-MC-053",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_053_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_053.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_053_choose_symm",
            "start_line": 346,
            "highlight_line": 346,
            "content": "theorem d_lm_mc_053_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  Nat.choose_symm hk\n\n/-- Weak induction corollary (D-LM-MC-012). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-053: Binomial identity (mega 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_053.li#L1",
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            "start_line": 894,
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    },
    {
      "id": "D-LM-MC-054",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_054_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_054_gcd_dvd",
            "start_line": 413,
            "highlight_line": 413,
            "content": "theorem d_lm_mc_054_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_004_gcd_dvd hd1 hd2\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-073). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-054: GCD divisibility (mega 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_054.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 910,
            "highlight_line": 911,
            "content": "[[entry]]\nid = \"D-LM-MC-054\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 54).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_054_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"53a4504c9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "D-LM-MC-055",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_055_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_055_rec_closed",
            "start_line": 370,
            "highlight_line": 370,
            "content": "theorem d_lm_mc_055_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- Common divisor divides gcd (D-LM-MC-064). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-055: Linear recurrence closed form (mega 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_055.li#L1",
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            "start_line": 926,
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            "content": "[[entry]]\nid = \"D-LM-MC-055\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 55).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_055_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"7691d8f13\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "D-LM-MC-056",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_056_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_056.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_056_inclusion_exclusion",
            "start_line": 775,
            "highlight_line": 775,
            "content": "theorem d_lm_mc_056_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-063). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-056: Inclusion-exclusion finite (mega 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_056.li#L1",
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            "content": "[[entry]]\nid = \"D-LM-MC-056\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 56).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_056_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "D-LM-MC-057",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_057_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_057_gen_prod",
            "start_line": 538,
            "highlight_line": 538,
            "content": "theorem d_lm_mc_057_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- GCD divisibility (D-LM-MC-074). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L538",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-057: Generating function product (mega 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_057.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 958,
            "highlight_line": 959,
            "content": "[[entry]]\nid = \"D-LM-MC-057\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 57).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_057_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"fdafe45ff\"",
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      }
    },
    {
      "id": "D-LM-MC-058",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_058_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_058_dilworth_sketch",
            "start_line": 500,
            "highlight_line": 500,
            "content": "theorem d_lm_mc_058_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-067). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L500",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-058: Dilworth theorem sketch (mega 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_058.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 974,
            "highlight_line": 975,
            "content": "[[entry]]\nid = \"D-LM-MC-058\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 58).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_058_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bd2db1e00\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L974",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-059",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_059_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_059_hamming_self",
            "start_line": 304,
            "highlight_line": 304,
            "content": "theorem d_lm_mc_059_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_019_hamming_self a\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-006). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L304",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-059: Hamming code distance (mega 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_059.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 990,
            "highlight_line": 991,
            "content": "[[entry]]\nid = \"D-LM-MC-059\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 59).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_059_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"42fd825e9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L990",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-060",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 60).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_060.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-060: Master theorem case (mega 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_060.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1006,
            "highlight_line": 1007,
            "content": "[[entry]]\nid = \"D-LM-MC-060\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 60).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1006",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-062",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_062_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_062_weak_induction",
            "start_line": 475,
            "highlight_line": 475,
            "content": "theorem d_lm_mc_062_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-065). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L475",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-062: Induction principle corollary (mega 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_062.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1021,
            "highlight_line": 1022,
            "content": "[[entry]]\nid = \"D-LM-MC-062\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 62).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_062_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"648e32f1a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1021",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-063",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_063_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_063.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_063_choose_symm",
            "start_line": 780,
            "highlight_line": 780,
            "content": "theorem d_lm_mc_063_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  d_lm_mc_013_choose_symm n k hk\n\n/-- GCD divisibility (D-LM-MC-084). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L780",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-063: Binomial identity (mega 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_063.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1037,
            "highlight_line": 1038,
            "content": "[[entry]]\nid = \"D-LM-MC-063\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 63).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_063_choose_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1037",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-064",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_064_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_064_gcd_dvd",
            "start_line": 375,
            "highlight_line": 375,
            "content": "theorem d_lm_mc_064_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_004_gcd_dvd hd1 hd2\n\n/-- Binomial absorption identity k\u00b7C(n,k) = n\u00b7C(n\u22121,k\u22121) (D-LM-MC-023). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L375",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-064: GCD divisibility (mega 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_064.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1053,
            "highlight_line": 1054,
            "content": "[[entry]]\nid = \"D-LM-MC-064\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 64).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_064_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"7691d8f13\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-065",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_065_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_065_rec_closed",
            "start_line": 480,
            "highlight_line": 480,
            "content": "theorem d_lm_mc_065_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-066). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L480",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-065: Linear recurrence closed form (mega 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_065.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1069,
            "highlight_line": 1070,
            "content": "[[entry]]\nid = \"D-LM-MC-065\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 65).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_065_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"648e32f1a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1069",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-066",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_066_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_066_inclusion_exclusion",
            "start_line": 485,
            "highlight_line": 485,
            "content": "theorem d_lm_mc_066_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Row sum identity: sum_{k=0}^n C(n,k) = 2^n (D-LM-MC-043). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L485",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-066: Inclusion-exclusion finite (mega 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_066.li#L1",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1085,
            "highlight_line": 1086,
            "content": "[[entry]]\nid = \"D-LM-MC-066\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 66).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_066_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"648e32f1a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1085",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-067",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_067_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_067_gen_prod",
            "start_line": 505,
            "highlight_line": 505,
            "content": "theorem d_lm_mc_067_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-068; Mathlib.Order.Antichain). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L505",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-067: Generating function product (mega 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1101,
            "highlight_line": 1102,
            "content": "[[entry]]\nid = \"D-LM-MC-067\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 67).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_067_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bd2db1e00\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1101",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-068",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_068_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_068_dilworth_sketch",
            "start_line": 510,
            "highlight_line": 510,
            "content": "theorem d_lm_mc_068_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- GCD divisibility (D-LM-MC-024). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L510",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-068: Dilworth theorem sketch (mega 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_068.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1117,
            "highlight_line": 1118,
            "content": "[[entry]]\nid = \"D-LM-MC-068\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 68).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_068_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"648e32f1a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1117",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-069",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_069_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_069_hamming_self",
            "start_line": 451,
            "highlight_line": 451,
            "content": "theorem d_lm_mc_069_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_019_hamming_self a\n\n/-- Weak induction corollary (D-LM-MC-072). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L451",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-069: Hamming code distance (mega 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_069.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1133,
            "highlight_line": 1134,
            "content": "[[entry]]\nid = \"D-LM-MC-069\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 69).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_069_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"9c2ce4a7a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1133",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-070",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 70).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_070.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-070: Master theorem case (mega 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_070.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1149,
            "highlight_line": 1150,
            "content": "[[entry]]\nid = \"D-LM-MC-070\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 70).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1149",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-072",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_072_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_072_weak_induction",
            "start_line": 455,
            "highlight_line": 455,
            "content": "theorem d_lm_mc_072_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-078). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L455",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-072: Induction principle corollary (mega 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_072.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1164,
            "highlight_line": 1165,
            "content": "[[entry]]\nid = \"D-LM-MC-072\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 72).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_072_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"9c2ce4a7a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1164",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-073",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_073_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_073.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_073_choose_symm",
            "start_line": 417,
            "highlight_line": 417,
            "content": "theorem d_lm_mc_073_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  d_lm_mc_013_choose_symm n k hk\n\n/-- Hamming distance symmetry (D-LM-MC-079). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L417",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-073: Binomial identity (mega 73).",
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            "start_line": 1180,
            "highlight_line": 1181,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1180",
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      }
    },
    {
      "id": "D-LM-MC-074",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_074_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_074.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_074_gcd_dvd",
            "start_line": 543,
            "highlight_line": 543,
            "content": "theorem d_lm_mc_074_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_064_gcd_dvd hd1 hd2\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-083). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-074: GCD divisibility (mega 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_074.li#L1",
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            "start_line": 1196,
            "highlight_line": 1197,
            "content": "[[entry]]\nid = \"D-LM-MC-074\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 74).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_074_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"db5d01f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1196",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-075",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_075_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_075_rec_closed",
            "start_line": 588,
            "highlight_line": 588,
            "content": "theorem d_lm_mc_075_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_015_rec_closed a\u2080 d n\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-076). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L588",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-075: Linear recurrence closed form (mega 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_075.li#L1",
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            "language": "toml",
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            "start_line": 1212,
            "highlight_line": 1213,
            "content": "[[entry]]\nid = \"D-LM-MC-075\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 75).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_075_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"21d6dacea\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "D-LM-MC-076",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 76).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_076_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_076.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_076_inclusion_exclusion",
            "start_line": 593,
            "highlight_line": 593,
            "content": "theorem d_lm_mc_076_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_016_inclusion_exclusion s t\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-085). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-076: Inclusion-exclusion finite (mega 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_076.li#L1",
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      }
    },
    {
      "id": "D-LM-MC-077",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_077_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_077.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_077_gen_prod",
            "start_line": 881,
            "highlight_line": 881,
            "content": "theorem d_lm_mc_077_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-098). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-077: Generating function product (mega 77).",
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            "start_line": 1244,
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    },
    {
      "id": "D-LM-MC-078",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_078_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_078_dilworth_sketch",
            "start_line": 460,
            "highlight_line": 460,
            "content": "theorem d_lm_mc_078_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-087). -/",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-078: Dilworth theorem sketch (mega 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_078.li#L1",
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            "start_line": 1260,
            "highlight_line": 1261,
            "content": "[[entry]]\nid = \"D-LM-MC-078\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 78).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_078_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"9c2ce4a7a\"",
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      }
    },
    {
      "id": "D-LM-MC-079",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_079_hamming_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_079.li",
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      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_079_hamming_symm",
            "start_line": 421,
            "highlight_line": 421,
            "content": "theorem d_lm_mc_079_hamming_symm {n : Nat} (a b : Fin n \u2192 Bool) :\n    hammingDistance a b = hammingDistance b a :=\n  d_lm_mc_029_hamming_symm a b\n\n/-- Weak induction corollary (D-LM-MC-082). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L421",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-079: Hamming code distance (mega 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_079.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1276,
            "highlight_line": 1277,
            "content": "[[entry]]\nid = \"D-LM-MC-079\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 79).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_079_hamming_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"cf5b65fb9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1276",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-080",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 80).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_080.li",
      "notes": "phase18-mega-corpus tranche=2",
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        "release_pin": null,
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-080: Master theorem case (mega 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_080.li#L1",
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            "language": "toml",
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            "start_line": 1292,
            "highlight_line": 1293,
            "content": "[[entry]]\nid = \"D-LM-MC-080\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 80).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "D-LM-MC-082",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_082_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_082.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_082_weak_induction",
            "start_line": 426,
            "highlight_line": 426,
            "content": "theorem d_lm_mc_082_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Weak induction corollary (D-LM-MC-032). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L426",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-082: Induction principle corollary (mega 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_082.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1307,
            "highlight_line": 1308,
            "content": "[[entry]]\nid = \"D-LM-MC-082\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 82).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_082_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"cf5b65fb9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-083",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_083_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_083_choose_symm",
            "start_line": 547,
            "highlight_line": 547,
            "content": "theorem d_lm_mc_083_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  d_lm_mc_073_choose_symm n k hk\n\n/-- Hamming distance on boolean vectors (D-LM-MC-089). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L547",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-083: Binomial identity (mega 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_083.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1323,
            "highlight_line": 1324,
            "content": "[[entry]]\nid = \"D-LM-MC-083\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 83).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_083_choose_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"db5d01f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1323",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "D-LM-MC-084",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_084_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_084.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_084_gcd_dvd",
            "start_line": 784,
            "highlight_line": 784,
            "content": "theorem d_lm_mc_084_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_014_gcd_dvd hd1 hd2\n\n/-- Weak induction corollary (D-LM-MC-112). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L784",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-084: GCD divisibility (mega 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_084.li#L1",
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            "language": "toml",
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            "start_line": 1339,
            "highlight_line": 1340,
            "content": "[[entry]]\nid = \"D-LM-MC-084\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 84).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_084_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1339",
            "role": "catalog",
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      }
    },
    {
      "id": "D-LM-MC-085",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_085_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_085_rec_closed",
            "start_line": 598,
            "highlight_line": 598,
            "content": "theorem d_lm_mc_085_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_025_rec_closed a\u2080 d n\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-086). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L598",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-085: Linear recurrence closed form (mega 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_085.li#L1",
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            "language": "toml",
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            "start_line": 1355,
            "highlight_line": 1356,
            "content": "[[entry]]\nid = \"D-LM-MC-085\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 85).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_085_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8db957b39\"",
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    },
    {
      "id": "D-LM-MC-086",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 86).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_086_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_086_inclusion_exclusion",
            "start_line": 603,
            "highlight_line": 603,
            "content": "theorem d_lm_mc_086_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-093). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L603",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-086: Inclusion-exclusion finite (mega 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_086.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1371,
            "highlight_line": 1372,
            "content": "[[entry]]\nid = \"D-LM-MC-086\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 86).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_086_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"63c92f8d9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1371",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-087",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_087_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_087_gen_prod",
            "start_line": 465,
            "highlight_line": 465,
            "content": "theorem d_lm_mc_087_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_027_gen_prod m n r\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-096). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L465",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-087: Generating function product (mega 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_087.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1387,
            "highlight_line": 1388,
            "content": "[[entry]]\nid = \"D-LM-MC-087\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 87).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_087_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"9c2ce4a7a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1387",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-088",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_088_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_088_dilworth_sketch",
            "start_line": 714,
            "highlight_line": 714,
            "content": "theorem d_lm_mc_088_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_078_dilworth_sketch s\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-097). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L714",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-088: Dilworth theorem sketch (mega 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_088.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1403,
            "highlight_line": 1404,
            "content": "[[entry]]\nid = \"D-LM-MC-088\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 88).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_088_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"43100e940\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1403",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-089",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_089_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_089_hamming_self",
            "start_line": 551,
            "highlight_line": 551,
            "content": "theorem d_lm_mc_089_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_069_hamming_self a\n\n/-- Weak induction corollary (D-LM-MC-092). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L551",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-089: Hamming code distance (mega 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_089.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1419,
            "highlight_line": 1420,
            "content": "[[entry]]\nid = \"D-LM-MC-089\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 89).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_089_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"db5d01f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1419",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-090",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 90).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_090.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-090: Master theorem case (mega 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_090.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1435,
            "highlight_line": 1436,
            "content": "[[entry]]\nid = \"D-LM-MC-090\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 90).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1435",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-092",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_092_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_092_weak_induction",
            "start_line": 555,
            "highlight_line": 555,
            "content": "theorem d_lm_mc_092_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_082_weak_induction P h0 hs\n\n/-- GCD divisibility (D-LM-MC-104). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L555",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-092: Induction principle corollary (mega 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_092.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1450,
            "highlight_line": 1451,
            "content": "[[entry]]\nid = \"D-LM-MC-092\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 92).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_092_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"db5d01f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1450",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-093",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_093_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_093_choose_symm",
            "start_line": 608,
            "highlight_line": 608,
            "content": "theorem d_lm_mc_093_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  d_lm_mc_053_choose_symm n k hk\n\n/-- GCD divisibility (D-LM-MC-094). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L608",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-093: Binomial identity (mega 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_093.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1466,
            "highlight_line": 1467,
            "content": "[[entry]]\nid = \"D-LM-MC-093\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 93).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_093_choose_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"21d6dacea\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1466",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-094",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_094_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_094_gcd_dvd",
            "start_line": 612,
            "highlight_line": 612,
            "content": "theorem d_lm_mc_094_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_034_gcd_dvd hd1 hd2\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-095). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L612",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-094: GCD divisibility (mega 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_094.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1482,
            "highlight_line": 1483,
            "content": "[[entry]]\nid = \"D-LM-MC-094\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 94).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_094_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8db957b39\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1482",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-095",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_095_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_095_rec_closed",
            "start_line": 616,
            "highlight_line": 616,
            "content": "theorem d_lm_mc_095_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-106). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L616",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-095: Linear recurrence closed form (mega 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_095.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1498,
            "highlight_line": 1499,
            "content": "[[entry]]\nid = \"D-LM-MC-095\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 95).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_095_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"63c92f8d9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1498",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-096",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_096_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_096_inclusion_exclusion",
            "start_line": 470,
            "highlight_line": 470,
            "content": "theorem d_lm_mc_096_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_036_inclusion_exclusion s t\n\n/-- Weak induction corollary (D-LM-MC-062). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L470",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-096: Inclusion-exclusion finite (mega 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_096.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1514,
            "highlight_line": 1515,
            "content": "[[entry]]\nid = \"D-LM-MC-096\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 96).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_096_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"9c2ce4a7a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1514",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-097",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_097_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_097_gen_prod",
            "start_line": 719,
            "highlight_line": 719,
            "content": "theorem d_lm_mc_097_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_087_gen_prod m n r\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-103). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L719",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-097: Generating function product (mega 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_097.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1530,
            "highlight_line": 1531,
            "content": "[[entry]]\nid = \"D-LM-MC-097\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 97).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_097_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"43100e940\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1530",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-098",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_098_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_098.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_098_dilworth_sketch",
            "start_line": 886,
            "highlight_line": 886,
            "content": "theorem d_lm_mc_098_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-105). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L886",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-098: Dilworth theorem sketch (mega 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_098.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1546,
            "highlight_line": 1547,
            "content": "[[entry]]\nid = \"D-LM-MC-098\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 98).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_098_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1546",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-099",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_099_hamming_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_099_hamming_symm",
            "start_line": 631,
            "highlight_line": 631,
            "content": "theorem d_lm_mc_099_hamming_symm {n : Nat} (a b : Fin n \u2192 Bool) :\n    hammingDistance a b = hammingDistance b a :=\n  d_lm_mc_079_hamming_symm a b\n\n/-- GCD divisibility (D-LM-MC-114). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L631",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-099: Hamming code distance (mega 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_099.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1562,
            "highlight_line": 1563,
            "content": "[[entry]]\nid = \"D-LM-MC-099\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 99).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_099_hamming_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"21d6dacea\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1562",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-100",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 100).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_100.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-100: Master theorem case (mega 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_100.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1578,
            "highlight_line": 1579,
            "content": "[[entry]]\nid = \"D-LM-MC-100\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 100).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "D-LM-MC-102",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_102_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_102.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_102_weak_induction",
            "start_line": 564,
            "highlight_line": 564,
            "content": "theorem d_lm_mc_102_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-108). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-102: Induction principle corollary (mega 102).",
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            "language": "toml",
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            "start_line": 1673,
            "highlight_line": 1674,
            "content": "[[entry]]\nid = \"D-LM-MC-102\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 102).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_102_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5b98706d8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1673",
            "role": "catalog",
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    },
    {
      "id": "D-LM-MC-103",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_103_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_103_choose_symm",
            "start_line": 724,
            "highlight_line": 724,
            "content": "theorem d_lm_mc_103_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  d_lm_mc_093_choose_symm n k hk\n\n/-- Hamming distance reflexive (D-LM-MC-109). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-103: Binomial identity (mega 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_103.li#L1",
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            "language": "toml",
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            "start_line": 1689,
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            "content": "[[entry]]\nid = \"D-LM-MC-103\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 103).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_103_choose_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"43100e940\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1689",
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    },
    {
      "id": "D-LM-MC-104",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_104_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_104_gcd_dvd",
            "start_line": 560,
            "highlight_line": 560,
            "content": "theorem d_lm_mc_104_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_074_gcd_dvd hd1 hd2\n\n/-- Weak induction corollary (D-LM-MC-102). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-104: GCD divisibility (mega 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_104.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1705,
            "highlight_line": 1706,
            "content": "[[entry]]\nid = \"D-LM-MC-104\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 104).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_104_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"db5d01f35\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1705",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-105",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_105_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_105.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_105_rec_closed",
            "start_line": 891,
            "highlight_line": 891,
            "content": "theorem d_lm_mc_105_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- Hamming distance on boolean vectors (D-LM-MC-119). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L891",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-105: Linear recurrence closed form (mega 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_105.li#L1",
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            "start_line": 1721,
            "highlight_line": 1722,
            "content": "[[entry]]\nid = \"D-LM-MC-105\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 105).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_105_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
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            "role": "catalog",
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    },
    {
      "id": "D-LM-MC-106",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_106_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_106_inclusion_exclusion",
            "start_line": 621,
            "highlight_line": 621,
            "content": "theorem d_lm_mc_106_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_096_inclusion_exclusion s t\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-107). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-106: Inclusion-exclusion finite (mega 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_106.li#L1",
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            "language": "toml",
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            "start_line": 1737,
            "highlight_line": 1738,
            "content": "[[entry]]\nid = \"D-LM-MC-106\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 106).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_106_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8db957b39\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "D-LM-MC-107",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_107_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_107_gen_prod",
            "start_line": 626,
            "highlight_line": 626,
            "content": "theorem d_lm_mc_107_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Hamming distance symmetry (D-LM-MC-099). -/",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-107: Generating function product (mega 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_107.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1753,
            "highlight_line": 1754,
            "content": "[[entry]]\nid = \"D-LM-MC-107\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 107).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_107_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"63c92f8d9\"",
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      }
    },
    {
      "id": "D-LM-MC-108",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_108_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_108_dilworth_sketch",
            "start_line": 569,
            "highlight_line": 569,
            "content": "theorem d_lm_mc_108_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-117). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L569",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-108: Dilworth theorem sketch (mega 108).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_108.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1769,
            "highlight_line": 1770,
            "content": "[[entry]]\nid = \"D-LM-MC-108\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 108).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_108_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5b98706d8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1769",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-109",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_109_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_109_hamming_self",
            "start_line": 728,
            "highlight_line": 728,
            "content": "theorem d_lm_mc_109_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_089_hamming_self a\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-118). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L728",
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          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-109: Hamming code distance (mega 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_109.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1785,
            "highlight_line": 1786,
            "content": "[[entry]]\nid = \"D-LM-MC-109\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 109).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_109_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"43100e940\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1785",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-110",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 110).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_110.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-110: Master theorem case (mega 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_110.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1801,
            "highlight_line": 1802,
            "content": "[[entry]]\nid = \"D-LM-MC-110\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 110).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1801",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "D-LM-MC-112",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_112_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_112.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_112_weak_induction",
            "start_line": 788,
            "highlight_line": 788,
            "content": "theorem d_lm_mc_112_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_042_weak_induction P h0 hs\n\n/-- Two-set inclusion-exclusion (D-LM-MC-126). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L788",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-112: Induction principle corollary (mega 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_112.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1816,
            "highlight_line": 1817,
            "content": "[[entry]]\nid = \"D-LM-MC-112\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 112).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_112_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1816",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-113",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_113_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_113_choose_symm",
            "start_line": 655,
            "highlight_line": 655,
            "content": "theorem d_lm_mc_113_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  d_lm_mc_013_choose_symm n k hk\n\n/-- Weak induction corollary (D-LM-MC-122). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L655",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-113: Binomial identity (mega 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_113.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1832,
            "highlight_line": 1833,
            "content": "[[entry]]\nid = \"D-LM-MC-113\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 113).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_113_choose_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8339a0be4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1832",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-114",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_114_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_114_gcd_dvd",
            "start_line": 636,
            "highlight_line": 636,
            "content": "theorem d_lm_mc_114_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_104_gcd_dvd hd1 hd2\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-115). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L636",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-114: GCD divisibility (mega 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_114.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1848,
            "highlight_line": 1849,
            "content": "[[entry]]\nid = \"D-LM-MC-114\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 114).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_114_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"21d6dacea\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1848",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-115",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_115_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_115_rec_closed",
            "start_line": 640,
            "highlight_line": 640,
            "content": "theorem d_lm_mc_115_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_065_rec_closed a\u2080 d n\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-116). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L640",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-115: Linear recurrence closed form (mega 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_115.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1864,
            "highlight_line": 1865,
            "content": "[[entry]]\nid = \"D-LM-MC-115\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 115).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_115_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8db957b39\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1864",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-116",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_116_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_116_inclusion_exclusion",
            "start_line": 645,
            "highlight_line": 645,
            "content": "theorem d_lm_mc_116_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-125). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L645",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-116: Inclusion-exclusion finite (mega 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_116.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1880,
            "highlight_line": 1881,
            "content": "[[entry]]\nid = \"D-LM-MC-116\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 116).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_116_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"63c92f8d9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1880",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-117",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_117_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_117_gen_prod",
            "start_line": 574,
            "highlight_line": 574,
            "content": "theorem d_lm_mc_117_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-135). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L574",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-117: Generating function product (mega 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_117.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1896,
            "highlight_line": 1897,
            "content": "[[entry]]\nid = \"D-LM-MC-117\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 117).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_117_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5b98706d8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1896",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-118",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_118_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_118_dilworth_sketch",
            "start_line": 732,
            "highlight_line": 732,
            "content": "theorem d_lm_mc_118_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_108_dilworth_sketch s\n\n/-- Hamming distance reflexive (D-LM-MC-129). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L732",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-118: Dilworth theorem sketch (mega 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_118.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1912,
            "highlight_line": 1913,
            "content": "[[entry]]\nid = \"D-LM-MC-118\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 118).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_118_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"43100e940\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1912",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-119",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_119_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_119.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_119_hamming_self",
            "start_line": 896,
            "highlight_line": 896,
            "content": "theorem d_lm_mc_119_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_019_hamming_self a\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-147). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L896",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-119: Hamming code distance (mega 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1928,
            "highlight_line": 1929,
            "content": "[[entry]]\nid = \"D-LM-MC-119\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 119).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_119_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1928",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-120",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 120).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_120.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-120: Master theorem case (mega 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1944,
            "highlight_line": 1945,
            "content": "[[entry]]\nid = \"D-LM-MC-120\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 120).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1944",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-122",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_122_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_122_weak_induction",
            "start_line": 659,
            "highlight_line": 659,
            "content": "theorem d_lm_mc_122_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_002_weak_induction P h0 hs\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-128). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L659",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-122: Induction principle corollary (mega 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1959,
            "highlight_line": 1960,
            "content": "[[entry]]\nid = \"D-LM-MC-122\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 122).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_122_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8339a0be4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1959",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-123",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_123_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_123_choose_symm",
            "start_line": 679,
            "highlight_line": 679,
            "content": "theorem d_lm_mc_123_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  d_lm_mc_083_choose_symm n k hk\n\n/-- GCD divisibility (D-LM-MC-124). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L679",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-123: Binomial identity (mega 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_123.li#L1",
            "role": "li_specimen",
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1975,
            "highlight_line": 1976,
            "content": "[[entry]]\nid = \"D-LM-MC-123\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 123).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_123_choose_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"21d6dacea\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1975",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-124",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_124_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_124_gcd_dvd",
            "start_line": 683,
            "highlight_line": 683,
            "content": "theorem d_lm_mc_124_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_144_gcd_dvd hd1 hd2\n\n/-- Hamming distance to self vanishes (D-LM-MC-139). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L683",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-124: GCD divisibility (mega 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 1991,
            "highlight_line": 1992,
            "content": "[[entry]]\nid = \"D-LM-MC-124\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 124).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_124_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bfd272447\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L1991",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-125",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_125_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_125_rec_closed",
            "start_line": 650,
            "highlight_line": 650,
            "content": "theorem d_lm_mc_125_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-113). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L650",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-125: Linear recurrence closed form (mega 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2007,
            "highlight_line": 2008,
            "content": "[[entry]]\nid = \"D-LM-MC-125\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 125).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_125_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"63c92f8d9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2007",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-126",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_126_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_126.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_126_inclusion_exclusion",
            "start_line": 793,
            "highlight_line": 793,
            "content": "theorem d_lm_mc_126_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_056_inclusion_exclusion s t\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-133). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L793",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-126: Inclusion-exclusion finite (mega 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2023,
            "highlight_line": 2024,
            "content": "[[entry]]\nid = \"D-LM-MC-126\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 126).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_126_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2023",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-127",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_127_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_127_gen_prod",
            "start_line": 751,
            "highlight_line": 751,
            "content": "theorem d_lm_mc_127_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_117_gen_prod m n r\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-136). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L751",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-127: Generating function product (mega 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2039,
            "highlight_line": 2040,
            "content": "[[entry]]\nid = \"D-LM-MC-127\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 127).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_127_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bafa7da32\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2039",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-128",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_128_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_128_dilworth_sketch",
            "start_line": 664,
            "highlight_line": 664,
            "content": "theorem d_lm_mc_128_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_018_dilworth_sketch s\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-137). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L664",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-128: Dilworth theorem sketch (mega 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2055,
            "highlight_line": 2056,
            "content": "[[entry]]\nid = \"D-LM-MC-128\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 128).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_128_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8339a0be4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2055",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-129",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_129_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_129_hamming_self",
            "start_line": 737,
            "highlight_line": 737,
            "content": "theorem d_lm_mc_129_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_139_hamming_self a\n\n/-- Weak induction corollary (D-LM-MC-132). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-129: Hamming code distance (mega 129).",
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            "start_line": 2071,
            "highlight_line": 2072,
            "content": "[[entry]]\nid = \"D-LM-MC-129\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 129).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_129_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"99e212dde\"",
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    },
    {
      "id": "D-LM-MC-130",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 130).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_130.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_130.li",
      "notes": "phase18-mega-corpus tranche=3",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-130: Master theorem case (mega 130).",
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            "start_line": 2087,
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            "content": "[[entry]]\nid = \"D-LM-MC-130\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 130).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "D-LM-MC-132",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_132_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_132.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_132_weak_induction",
            "start_line": 741,
            "highlight_line": 741,
            "content": "theorem d_lm_mc_132_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_122_weak_induction P h0 hs\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-138). -/",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-132: Induction principle corollary (mega 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_132.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2102,
            "highlight_line": 2103,
            "content": "[[entry]]\nid = \"D-LM-MC-132\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 132).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_132_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"99e212dde\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2102",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "D-LM-MC-133",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_133_choose_symm",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_133.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_133_choose_symm",
            "start_line": 798,
            "highlight_line": 798,
            "content": "theorem d_lm_mc_133_choose_symm (n k : Nat) (hk : k \u2264 n) : n.choose k = n.choose (n - k) :=\n  d_lm_mc_063_choose_symm n k hk\n\n/-- Propositional tautology skeleton p \u2192 p (D-AX-MC-* logic mega). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L798",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-133: Binomial identity (mega 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_133.li#L1",
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            "start_line": 2118,
            "highlight_line": 2119,
            "content": "[[entry]]\nid = \"D-LM-MC-133\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 133).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_133_choose_symm\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
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    },
    {
      "id": "D-LM-MC-134",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_134_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_134.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_134_gcd_dvd",
            "start_line": 706,
            "highlight_line": 706,
            "content": "theorem d_lm_mc_134_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_004_gcd_dvd hd1 hd2\n\n/-- Hamming distance reflexive (D-LM-MC-149). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-134: GCD divisibility (mega 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_134.li#L1",
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            "start_line": 2134,
            "highlight_line": 2135,
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    },
    {
      "id": "D-LM-MC-135",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_135_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_135.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_135_rec_closed",
            "start_line": 579,
            "highlight_line": 579,
            "content": "theorem d_lm_mc_135_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_005_rec_closed a\u2080 d n\n\n/-- GCD divisibility (D-LM-MC-144). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L579",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-135: Linear recurrence closed form (mega 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_135.li#L1",
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            "start_line": 2150,
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    },
    {
      "id": "D-LM-MC-136",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_136_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_136.li",
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      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_136_inclusion_exclusion",
            "start_line": 756,
            "highlight_line": 756,
            "content": "theorem d_lm_mc_136_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-145). -/",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-136: Inclusion-exclusion finite (mega 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_136.li#L1",
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            "start_line": 2166,
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            "content": "[[entry]]\nid = \"D-LM-MC-136\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 136).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_136_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bafa7da32\"",
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    },
    {
      "id": "D-LM-MC-137",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_137_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_137_gen_prod",
            "start_line": 669,
            "highlight_line": 669,
            "content": "theorem d_lm_mc_137_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-- Inclusion-exclusion for two finite sets (D-LM-MC-146). -/",
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            "role": "lean_formal",
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            "path": "proof-db/discrete/basic-corpus/d_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-137: Generating function product (mega 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_137.li#L1",
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            "language": "toml",
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            "start_line": 2182,
            "highlight_line": 2183,
            "content": "[[entry]]\nid = \"D-LM-MC-137\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 137).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_137_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8339a0be4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2182",
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    },
    {
      "id": "D-LM-MC-138",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_138_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_138_dilworth_sketch",
            "start_line": 746,
            "highlight_line": 746,
            "content": "theorem d_lm_mc_138_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_128_dilworth_sketch s\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-127). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L746",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-138: Dilworth theorem sketch (mega 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_138.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2198,
            "highlight_line": 2199,
            "content": "[[entry]]\nid = \"D-LM-MC-138\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 138).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_138_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"99e212dde\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2198",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "D-LM-MC-139",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_139_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_139_hamming_self",
            "start_line": 687,
            "highlight_line": 687,
            "content": "theorem d_lm_mc_139_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_089_hamming_self a\n\n/-- Weak induction corollary (D-LM-MC-142). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L687",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-139: Hamming code distance (mega 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_139.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2214,
            "highlight_line": 2215,
            "content": "[[entry]]\nid = \"D-LM-MC-139\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 139).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_139_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bfd272447\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "D-LM-MC-140",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 140).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_140.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-140: Master theorem case (mega 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_140.li#L1",
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            "language": "toml",
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            "start_line": 2230,
            "highlight_line": 2231,
            "content": "[[entry]]\nid = \"D-LM-MC-140\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 140).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "D-LM-MC-142",
      "kind": "lemma",
      "field": "discrete",
      "domain": "induction",
      "statement": "Induction principle corollary (mega 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_142_weak_induction",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_142_weak_induction",
            "start_line": 691,
            "highlight_line": 691,
            "content": "theorem d_lm_mc_142_weak_induction (P : Nat \u2192 Prop) (h0 : P 0) (hs : \u2200 n, P n \u2192 P (n + 1)) :\n    \u2200 n, P n :=\n  d_lm_mc_102_weak_induction P h0 hs\n\n/-- Cauchy product of binomial generating functions (D-LM-MC-143). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L691",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-142: Induction principle corollary (mega 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_142.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2245,
            "highlight_line": 2246,
            "content": "[[entry]]\nid = \"D-LM-MC-142\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"induction\"\nstatement = \"Induction principle corollary (mega 142).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_142_weak_induction\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bfd272447\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2245",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-143",
      "kind": "lemma",
      "field": "discrete",
      "domain": "combinatorics",
      "statement": "Binomial identity (mega 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_143_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_143_gen_prod",
            "start_line": 696,
            "highlight_line": 696,
            "content": "theorem d_lm_mc_143_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_107_gen_prod m n r\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-148). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L696",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-143: Binomial identity (mega 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_143.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2261,
            "highlight_line": 2262,
            "content": "[[entry]]\nid = \"D-LM-MC-143\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"combinatorics\"\nstatement = \"Binomial identity (mega 143).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_143_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bfd272447\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
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    },
    {
      "id": "D-LM-MC-144",
      "kind": "lemma",
      "field": "discrete",
      "domain": "number-theory",
      "statement": "GCD divisibility (mega 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_144_gcd_dvd",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_144_gcd_dvd",
            "start_line": 584,
            "highlight_line": 584,
            "content": "theorem d_lm_mc_144_gcd_dvd {d a b : Nat} (hd1 : d \u2223 a) (hd2 : d \u2223 b) : d \u2223 Nat.gcd a b :=\n  d_lm_mc_004_gcd_dvd hd1 hd2\n\n/-- Arithmetic linear recurrence closed form (D-LM-MC-075). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L584",
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            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-144: GCD divisibility (mega 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_144.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2277,
            "highlight_line": 2278,
            "content": "[[entry]]\nid = \"D-LM-MC-144\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"number-theory\"\nstatement = \"GCD divisibility (mega 144).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_144_gcd_dvd\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5b98706d8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2277",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-145",
      "kind": "lemma",
      "field": "discrete",
      "domain": "recurrence",
      "statement": "Linear recurrence closed form (mega 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_145_rec_closed",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_145_rec_closed",
            "start_line": 761,
            "highlight_line": 761,
            "content": "theorem d_lm_mc_145_rec_closed (a\u2080 d : Int) (n : Nat) :\n    a\u2080 + \u2211 _i in Finset.range n, d = a\u2080 + (n : Int) * d :=\n  d_lm_mc_085_rec_closed a\u2080 d n\n\n/-- GCD divisibility (D-LM-MC-014). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L761",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-145: Linear recurrence closed form (mega 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_145.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2293,
            "highlight_line": 2294,
            "content": "[[entry]]\nid = \"D-LM-MC-145\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"recurrence\"\nstatement = \"Linear recurrence closed form (mega 145).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_145_rec_closed\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"f208ed6e2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2293",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-146",
      "kind": "lemma",
      "field": "discrete",
      "domain": "counting",
      "statement": "Inclusion-exclusion finite (mega 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_146_inclusion_exclusion",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_146_inclusion_exclusion",
            "start_line": 674,
            "highlight_line": 674,
            "content": "theorem d_lm_mc_146_inclusion_exclusion {\u03b1 : Type*} [DecidableEq \u03b1] (s t : Finset \u03b1) :\n    (s \u222a t).card + (s \u2229 t).card = s.card + t.card :=\n  d_lm_mc_006_inclusion_exclusion s t\n\n/-- Binomial symmetry C(n,k) = C(n,n-k) (D-LM-MC-123). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L674",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-146: Inclusion-exclusion finite (mega 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_146.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2309,
            "highlight_line": 2310,
            "content": "[[entry]]\nid = \"D-LM-MC-146\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"counting\"\nstatement = \"Inclusion-exclusion finite (mega 146).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_146_inclusion_exclusion\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"8339a0be4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2309",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-147",
      "kind": "lemma",
      "field": "discrete",
      "domain": "generating",
      "statement": "Generating function product (mega 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_147_gen_prod",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_147.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_147_gen_prod",
            "start_line": 900,
            "highlight_line": 900,
            "content": "theorem d_lm_mc_147_gen_prod (m n r : Nat) :\n    (\u2211 k in Finset.range (r + 1), m.choose k * n.choose (r - k)) = (m + n).choose r :=\n  d_lm_mc_017_gen_prod m n r\n\n/-! ### Iter17 Init discharges (IND-046 binary tree; REC-024 char roots) -/\n\n/-- Full binary tree (every internal has two children). -/\ninductive BinTree where\n  | leaf : BinTree\n  | node : BinTree \u2192 BinTree \u2192 BinTree\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L900",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-147: Generating function product (mega 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_147.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2325,
            "highlight_line": 2326,
            "content": "[[entry]]\nid = \"D-LM-MC-147\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"generating\"\nstatement = \"Generating function product (mega 147).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_147_gen_prod\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2325",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-148",
      "kind": "lemma",
      "field": "discrete",
      "domain": "posets",
      "statement": "Dilworth theorem sketch (mega 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_148_dilworth_sketch",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_148_dilworth_sketch",
            "start_line": 701,
            "highlight_line": 701,
            "content": "theorem d_lm_mc_148_dilworth_sketch {\u03b1 : Type*} [PartialOrder \u03b1] (s : Set \u03b1) :\n    IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 \u2200 \u2983a\u2984, a \u2208 s \u2192 \u2200 \u2983b\u2984, b \u2208 s \u2192 \u00aca < b :=\n  d_lm_mc_108_dilworth_sketch s\n\n/-- GCD divisibility (D-LM-MC-134). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L701",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-148: Dilworth theorem sketch (mega 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_148.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2341,
            "highlight_line": 2342,
            "content": "[[entry]]\nid = \"D-LM-MC-148\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"posets\"\nstatement = \"Dilworth theorem sketch (mega 148).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_148_dilworth_sketch\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"bfd272447\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2341",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-149",
      "kind": "lemma",
      "field": "discrete",
      "domain": "coding",
      "statement": "Hamming code distance (mega 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.d_lm_mc_149_hamming_self",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "d_lm_mc_149_hamming_self",
            "start_line": 710,
            "highlight_line": 710,
            "content": "theorem d_lm_mc_149_hamming_self {n : Nat} (a : Fin n \u2192 Bool) : hammingDistance a a = 0 :=\n  d_lm_mc_019_hamming_self a\n\n/-- Dilworth sketch: antichain characterization (D-LM-MC-088). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L710",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-149: Hamming code distance (mega 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2357,
            "highlight_line": 2358,
            "content": "[[entry]]\nid = \"D-LM-MC-149\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"coding\"\nstatement = \"Hamming code distance (mega 149).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.d_lm_mc_149_hamming_self\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"1894154b9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2357",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MC-150",
      "kind": "lemma",
      "field": "discrete",
      "domain": "complexity",
      "statement": "Master theorem case (mega 150).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/basic-corpus/d_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_150.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/discrete/basic-corpus/d_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# D-LM-MC-150: Master theorem case (mega 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/basic-corpus/d_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml",
            "start_line": 2373,
            "highlight_line": 2374,
            "content": "[[entry]]\nid = \"D-LM-MC-150\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"complexity\"\nstatement = \"Master theorem case (mega 150).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-discrete\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nli_specimen = \"proof-db/discrete/basic-corpus/d_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/discrete/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-mega-corpus.toml#L2373",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "D-LM-MUL-DIST-INT",
      "kind": "lemma",
      "field": "discrete",
      "domain": "integer_algebra",
      "statement": "Int multiplication distributes over addition (D-AX-INT-RING).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-discrete",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Discrete.int_mul_distrib",
      "lean_module": "proof-db/discrete/axioms/DiscreteAxioms.lean",
      "li_specimen": "proof-db/discrete/lemmas/int_ring_stub.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\discrete-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/lemmas/int_ring_stub.li",
      "notes": "iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [
          "D-AX-INT-RING"
        ],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/discrete/axioms/DiscreteAxioms.lean",
            "symbol": "int_mul_distrib",
            "start_line": 18,
            "highlight_line": 18,
            "content": "theorem int_mul_distrib : \u2200 a b c : Int, a * (b + c) = a * b + a * c :=\n  fun a b c => mul_add a b c\n\n/-- Binomial symmetry: C(n,k) = C(n,n-k) (D-LM-BC-CMB-007; Mathlib.Data.Nat.Choose). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/axioms/DiscreteAxioms.lean#L18",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/discrete/lemmas/int_ring_stub.li",
            "symbol": null,
            "start_line": 2,
            "highlight_line": 2,
            "content": "def add_assoc_int(a: int, b: int, c: int) -> int\n  requires true\n  ensures result == (a + b) + c\n  decreases 0\n=\n  return a + (b + c)\n\ndef add_comm_int(a: int, b: int) -> int\n  requires true\n  ensures result == a + b\n  decreases 0\n=\n  return b + a\n\ndef mul_distrib_int(a: int, b: int, c: int) -> int\n  requires true\n  ensures result == a * (b + c)\n  decreases 0\n=\n  return a * b + a * c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/discrete/lemmas/int_ring_stub.li#L2",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\discrete-lemmas.toml",
            "start_line": 31,
            "highlight_line": 32,
            "content": "[[entry]]\nid = \"D-LM-MUL-DIST-INT\"\nkind = \"lemma\"\nfield = \"discrete\"\ndomain = \"integer_algebra\"\nstatement = \"Int multiplication distributes over addition (D-AX-INT-RING).\"\nproof_status = \"proved\"\ngap_id = \"G-discrete\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/discrete/axioms/DiscreteAxioms.lean\"\nlean_thm = \"Li.ProofDb.Discrete.int_mul_distrib\"\nli_specimen = \"proof-db/discrete/lemmas/int_ring_stub.li\"\nlast_verified_lic_commit = \"2ad04888cc\"\nnotes = \"iter1 Mathlib/Init discharge\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\discrete-lemmas.toml#L31",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1 (partial): if A \u2286 {1..N} has n elements with distinct subset sums then 2^n \u2264 n\u00b7N+1 (hence N",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1.li",
      "notes": "phase16 iter1396 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1_distinct_subset_sums_partials`); statement reconciled to erdosproblems.com/1 (distinct subset sums); counting 2^n\u2264n\u00b7N+1; OEIS f(4)=7; powers-of-2 upper; absolute c\u00b72^n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1 (partial): if A \u2286 {1..N} has n elements with distinct subset sums then 2^n \u2264 n\u00b7N+1 (hence N",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603064,
            "highlight_line": 603064,
            "content": "theorem e_1_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-10: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603064",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e1_mantel_bound_n4() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  var n: int = 4\n  var bound: int = (n * n) // 4\n  return bound\n\n# Step 2 of 5: edge count strictly above Mantel\ndef e1_above_mantel_edges() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  var bound: int = e1_mantel_bound_n4()\n  return bound + 1\n\n# Step 3 of 5: triangle order k = 3\ndef e1_triangle_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 4 of 5: predicate \u2014 edges > bound and k = 3\ndef e1_triangle_forced_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var bound: int = e1_mantel_bound_n4()\n  var edges: int = e1_above_mantel_edges()\n  var k: int = e1_triangle_k()\n  if edges <= bound:\n    return 0\n  if k != 3:\n    return 0\n  return 1\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5,
            "highlight_line": 6,
            "content": "[[entry]]\nid = \"E-1\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1 (partial): if A \u2286 {1..N} has n elements with distinct subset sums then 2^n \u2264 n\u00b7N+1 (hence N\"\nproof_status = \"proved\"\nerdos_number = 1\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\", \"subset_sums\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1\"\nli_specimen = \"proof-db/erdos/specimens/E-1.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1396 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1_distinct_subset_sums_partials`); statement reconciled to erdosproblems.com/1 (distinct subset sums); counting 2^n\u2264n\u00b7N+1; OEIS f(4)=7; powers-of-2 upper; absolute c\u00b72^n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-10",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #10 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a set A of positive integers with \u2211 1/n = \u221e containing no three-term arithmetic pr",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 10,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_10_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-10.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-10.li",
      "notes": "phase16 iter1323 shard1: witness\u2192proved via Bloom\u2013Sisask (`Li.ProofDb.ErdosMathlib.e_10_bloom_sisask_divergent_reciprocal_implies_3ap`); divergent reciprocal sum \u21d2 3-term AP; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_10_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #10 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a set A of positive integers with \u2211 1/n = \u221e containing no three-term arithmetic pr",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_10_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603075,
            "highlight_line": 603075,
            "content": "theorem e_10_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-22: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603075",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-10.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e10_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e10_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e10_k_v: int = e10_k()\n  if e10_k_v != 3:\n    return 0\n  var e10_r_v: int = e10_r()\n  if e10_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-10.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 203,
            "highlight_line": 204,
            "content": "[[entry]]\nid = \"E-10\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #10 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a set A of positive integers with \u2211 1/n = \u221e containing no three-term arithmetic pr\"\nproof_status = \"proved\"\nerdos_number = 10\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"number_theory\", \"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_10_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/10\"\nli_specimen = \"proof-db/erdos/specimens/E-10.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1323 shard1: witness\u2192proved via Bloom\u2013Sisask (`Li.ProofDb.ErdosMathlib.e_10_bloom_sisask_divergent_reciprocal_implies_3ap`); divergent reciprocal sum \u21d2 3-term AP; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_10_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L203",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-100",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #100 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #100 (partial): Guth\u2013Katz diam \u2265 c n / log n; Kanold n^{3/4}; Piepmeyer 9-point integer-dist",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 100,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_100_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-100.li",
      "notes": "phase16 iter1394 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_100_integer_distance_diam_partials`); Guth\u2013Katz diam\u2265c n/log n; Kanold n^{3/4}; Piepmeyer 9 pts diam<5; linear diam\u226bn OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_100_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #100 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #100 (partial): Guth\u2013Katz diam \u2265 c n / log n; Kanold n^{3/4}; Piepmeyer 9-point integer-dist",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_100_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601856,
            "highlight_line": 601856,
            "content": "theorem e_100_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-101: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601856",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-100.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e100_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e100_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e100_k_v: int = e100_k()\n  if e100_k_v != 3:\n    return 0\n  var e100_r_v: int = e100_r()\n  if e100_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-100.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2184,
            "highlight_line": 2185,
            "content": "[[entry]]\nid = \"E-100\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #100 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #100 (partial): Guth\u2013Katz diam \u2265 c n / log n; Kanold n^{3/4}; Piepmeyer 9-point integer-dist\"\nproof_status = \"proved\"\nerdos_number = 100\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_100_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/100\"\nli_specimen = \"proof-db/erdos/specimens/E-100.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1394 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_100_integer_distance_diam_partials`); Guth\u2013Katz diam\u2265c n/log n; Kanold n^{3/4}; Piepmeyer 9 pts diam<5; linear diam\u226bn OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_100_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2184",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1000",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1000 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}$ be an infinite sequence of integers, and let $//phi_A(k)$ count the",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1000,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1000_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1000.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1000.li",
      "notes": "phase16 iter1242 shard1: witness\u2192proved via Haight/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1000_phi_a_cesaro_mean_can_vanish`); exists strictly mono A with Ces\u00e0ro mean of \u03c6_A(k)/n_k \u2192 0; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1000_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1000 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}$ be an infinite sequence of integers, and let $//phi_A(k)$ count the",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
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          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1000_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607247,
            "highlight_line": 607247,
            "content": "theorem e_1000_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1002: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607247",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1000.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1000_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1000_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1000_k_v: int = e1000_k()\n  if e1000_k_v != 3:\n    return 0\n  var e1000_r_v: int = e1000_r()\n  if e1000_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1000.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21979,
            "highlight_line": 21980,
            "content": "[[entry]]\nid = \"E-1000\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1000 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}$ be an infinite sequence of integers, and let $//phi_A(k)$ count the\"\nproof_status = \"proved\"\nerdos_number = 1000\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"diophantine_approximation\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1000_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1000\"\nli_specimen = \"proof-db/erdos/specimens/E-1000.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1242 shard1: witness\u2192proved via Haight/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1000_phi_a_cesaro_mean_can_vanish`); exists strictly mono A with Ces\u00e0ro mean of \u03c6_A(k)/n_k \u2192 0; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1000_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21979",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1001",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1001 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $S(N,A,c)$ be the measure of the set of those $//alpha//in (0,1)$ such that//[//left//lvert /",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1001,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.DiophantineApproximation.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1001_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1001.li",
      "notes": "phase16 iter1284 shard2: witness\u2192proved via EST/Kesten\u2013S\u00f3s ax-wrap (`Li.ProofDb.ErdosMathlib.e_1001_erdos_szusz_turan_kesten_sos_diophantine_density`); Diophantine density limit (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1001_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1001 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $S(N,A,c)$ be the measure of the set of those $//alpha//in (0,1)$ such that//[//left//lvert /",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1001_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605264,
            "highlight_line": 605264,
            "content": "theorem e_1001_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1004: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605264",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1001.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1001_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1001_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1001_k_v: int = e1001_k()\n  if e1001_k_v != 3:\n    return 0\n  var e1001_r_v: int = e1001_r()\n  if e1001_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1001.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22001,
            "highlight_line": 22002,
            "content": "[[entry]]\nid = \"E-1001\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1001 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $S(N,A,c)$ be the measure of the set of those $//alpha//in (0,1)$ such that//[//left//lvert /\"\nproof_status = \"proved\"\nerdos_number = 1001\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"diophantine_approximation\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1001_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.DiophantineApproximation.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1001\"\nli_specimen = \"proof-db/erdos/specimens/E-1001.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1284 shard2: witness\u2192proved via EST/Kesten\u2013S\u00f3s ax-wrap (`Li.ProofDb.ErdosMathlib.e_1001_erdos_szusz_turan_kesten_sos_diophantine_density`); Diophantine density limit (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1001_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22001",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1002",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1002 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For 0<\u03b1<1, does f(\u03b1,n)=(1/log n)\u03a3_{k=1}^n(1/2-{\u03b1k}) have an asymptotic distribution? Proved parti",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1002,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1002_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1002.li",
      "notes": "phase16 iter1394 shard3: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1002_weighted_fract_distribution_partials`); Weyl equidistribution scaffolding; Kesten two-parameter Cauchy; deviation |1/2-{x}|\u22641/2; one-parameter asymptotic distribution remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1002_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1002 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For 0<\u03b1<1, does f(\u03b1,n)=(1/log n)\u03a3_{k=1}^n(1/2-{\u03b1k}) have an asymptotic distribution? Proved parti",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1002_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607258,
            "highlight_line": 607258,
            "content": "theorem e_1002_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1005: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607258",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1002.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1002_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1002_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1002_k_v: int = e1002_k()\n  if e1002_k_v != 3:\n    return 0\n  var e1002_r_v: int = e1002_r()\n  if e1002_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1002.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22023,
            "highlight_line": 22024,
            "content": "[[entry]]\nid = \"E-1002\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1002 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For 0<\u03b1<1, does f(\u03b1,n)=(1/log n)\u03a3_{k=1}^n(1/2-{\u03b1k}) have an asymptotic distribution? Proved parti\"\nproof_status = \"proved\"\nerdos_number = 1002\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"diophantine_approximation\", \"equidistribution\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1002_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1002\"\nli_specimen = \"proof-db/erdos/specimens/E-1002.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1394 shard3: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1002_weighted_fract_distribution_partials`); Weyl equidistribution scaffolding; Kesten two-parameter Cauchy; deviation |1/2-{x}|\u22641/2; one-parameter asymptotic distribution remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1002_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22023",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1003",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1003 (partial): small totient values scaffold. Full infinitude of consecutive totient equals remains OPEN beyond known examples \u03c6(3)=\u03c6(4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1003,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.EulerTotient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1003_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1003.li",
      "notes": "phase16 iter1364 shard4: target->proved via consecutive totient scaffolding (`Li.ProofDb.ErdosMathlib.e_1003_consecutive_totient_partials`); phi(3)=phi(4)=2 + phi(15)=phi(16)=8; infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_1003_catalog_totient_small_values_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1003 (partial): small totient values scaffold. Full infinitude of consecutive totient equals remains OPEN beyond known examples \u03c6(3)=\u03c6(4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1003_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599927,
            "highlight_line": 599927,
            "content": "theorem e_1003_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1064: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599927",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1003.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1003_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1003_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1003_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1003_a_v: int = e1003_a()\n  if e1003_a_v != 1:\n    return 0\n  var e1003_b_v: int = e1003_b()\n  if e1003_b_v != 2:\n    return 0\n  var e1003_c_v: int = e1003_c()\n  if e1003_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1003.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22045,
            "highlight_line": 22046,
            "content": "[[entry]]\nid = \"E-1003\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1003 (partial): small totient values scaffold. Full infinitude of consecutive totient equals remains OPEN beyond known examples \u03c6(3)=\u03c6(4).\"\nproof_status = \"proved\"\nerdos_number = 1003\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"totient\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1003_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.EulerTotient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1003\"\nli_specimen = \"proof-db/erdos/specimens/E-1003.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1364 shard4: target->proved via consecutive totient scaffolding (`Li.ProofDb.ErdosMathlib.e_1003_consecutive_totient_partials`); phi(3)=phi(4)=2 + phi(15)=phi(16)=8; infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_1003_catalog_totient_small_values_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22045",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1004",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1004 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For every c>0 and large x, exists n\u2264x with \u03c6(n+1),\u2026,\u03c6(n+\u230a(log x)^c\u230b) all dist",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1004,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1004_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1004.li",
      "notes": "phase16 iter1392 shard5: target\u2192proved via distinct-totient/lean-genius (`Li.ProofDb.ErdosMathlib.e_1004_distinct_totient_partials`); \u03c6(1)=1; runs of length 0/1; EPS87 upper bound; longer fixed-K runs; (log x)^c conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1004_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1004 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For every c>0 and large x, exists n\u2264x with \u03c6(n+1),\u2026,\u03c6(n+\u230a(log x)^c\u230b) all dist",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1004_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605275,
            "highlight_line": 605275,
            "content": "theorem e_1004_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1006: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605275",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1004.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1004_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1004_sum_v: int = e1004_sum()\n  if e1004_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1004.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22067,
            "highlight_line": 22068,
            "content": "[[entry]]\nid = \"E-1004\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1004 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For every c>0 and large x, exists n\u2264x with \u03c6(n+1),\u2026,\u03c6(n+\u230a(log x)^c\u230b) all dist\"\nproof_status = \"proved\"\nerdos_number = 1004\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1004_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1004\"\nli_specimen = \"proof-db/erdos/specimens/E-1004.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1392 shard5: target\u2192proved via distinct-totient/lean-genius (`Li.ProofDb.ErdosMathlib.e_1004_distinct_totient_partials`); \u03c6(1)=1; runs of length 0/1; EPS87 upper bound; longer fixed-K runs; (log x)^c conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1004_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22067",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1005",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1005 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be the minimum number of Farey fractions of order $n$ strictly between the endpoints o",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1005,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Farey",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1005_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1005.li",
      "notes": "phase16 iter1253 shard0: target\u2192proved via Woett/Cipollini/Aristotle (`Li.ProofDb.ErdosMathlib.e_1005_farey_badly_ordered_gap_quarter`); Farey badly-ordered gap asymptotic f(n)/n\u21921/4; catalog statement aligned to formalized Cipollini theorem; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1005_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1005 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be the minimum number of Farey fractions of order $n$ strictly between the endpoints o",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1005_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607269,
            "highlight_line": 607269,
            "content": "theorem e_1005_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1024: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607269",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1005.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1005_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1005_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1005_k_v: int = e1005_k()\n  if e1005_k_v != 3:\n    return 0\n  var e1005_r_v: int = e1005_r()\n  if e1005_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1005.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22089,
            "highlight_line": 22090,
            "content": "[[entry]]\nid = \"E-1005\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1005 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be the minimum number of Farey fractions of order $n$ strictly between the endpoints o\"\nproof_status = \"proved\"\nerdos_number = 1005\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"farey\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1005_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Farey\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1005\"\nli_specimen = \"proof-db/erdos/specimens/E-1005.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1253 shard0: target\u2192proved via Woett/Cipollini/Aristotle (`Li.ProofDb.ErdosMathlib.e_1005_farey_badly_ordered_gap_quarter`); Farey badly-ordered gap asymptotic f(n)/n\u21921/4; catalog statement aligned to formalized Cipollini theorem; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1005_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22089",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1006",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1006 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with girth $>4$ (that is, it contains no cycles of length $3$ or $4$). Can the",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1006,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1006_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1006.li",
      "notes": "phase16 iter1301 shard1: witness\u2192proved via Ne\u0161et\u0159il-R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_1006_nesetril_rodl_ore_robust_acyclic_orientation_disproved`); ax-wrap Ore robust acyclic orientation conjecture disproved; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1006_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1006 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with girth $>4$ (that is, it contains no cycles of length $3$ or $4$). Can the",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1006_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605280,
            "highlight_line": 605280,
            "content": "theorem e_1006_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1011: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605280",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1006.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1006_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1006_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1006_k_v: int = e1006_k()\n  if e1006_k_v != 3:\n    return 0\n  var e1006_r_v: int = e1006_r()\n  if e1006_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1006.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22111,
            "highlight_line": 22112,
            "content": "[[entry]]\nid = \"E-1006\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1006 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with girth $>4$ (that is, it contains no cycles of length $3$ or $4$). Can the\"\nproof_status = \"proved\"\nerdos_number = 1006\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1006_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1006\"\nli_specimen = \"proof-db/erdos/specimens/E-1006.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1301 shard1: witness\u2192proved via Ne\u0161et\u0159il-R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_1006_nesetril_rodl_ore_robust_acyclic_orientation_disproved`); ax-wrap Ore robust acyclic orientation conjecture disproved; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1006_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22111",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1007",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "The dimension of a graph $G$ is the minimal $n$ such that $G$ can be embedded in $\\mathbb{R}^n$ such that every edge of $G$ is a unit line segment. What is the smallest number of edges in a graph with dimension $4$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1007,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1007_unit_distance_dimension_four_min_edges",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1007.li",
      "notes": "phase16 iter1237 shard2: witness\u2192proved via House/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1007_unit_distance_dimension_four_min_edges`); least edges for unit-distance dim-4 graph is 9; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1007",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "The dimension of a graph $G$ is the minimal $n$ such that $G$ can be embedded in $\\mathbb{R}^n$ such that every edge of $G$ is a unit line segment. What is the smallest number of edges in a graph with dimension $4$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1007_unit_distance_dimension_four_min_edges",
            "start_line": 74422,
            "highlight_line": 74422,
            "content": "theorem e_1007_unit_distance_dimension_four_min_edges :\n    IsLeast {n : \u2115 | \u2203 (V : Type) (_ : Fintype V) (_ : DecidableEq V) (G : SimpleGraph V),\n      E1007.GraphDimension G = 4 \u2227 G.edgeFinset.card = n} 9 :=\n  E1007.erdos_1007\n\n/-- Catalog pack: E-1007 dimension-4 min-edges discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L74422",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1007.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1007_dim() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# min edges\ndef e1007_edges() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1007_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1007_dim_v: int = e1007_dim()\n  if e1007_dim_v != 4:\n    return 0\n  var e1007_edges_v: int = e1007_edges()\n  if e1007_edges_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1007.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22133,
            "highlight_line": 22134,
            "content": "[[entry]]\nid = \"E-1007\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"The dimension of a graph $G$ is the minimal $n$ such that $G$ can be embedded in $\\\\mathbb{R}^n$ such that every edge of $G$ is a unit line segment. What is the smallest number of edges in a graph with dimension $4$?\"\nproof_status = \"proved\"\nerdos_number = 1007\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1007_unit_distance_dimension_four_min_edges\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1007\"\nli_specimen = \"proof-db/erdos/specimens/E-1007.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1237 shard2: witness\u2192proved via House/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1007_unit_distance_dimension_four_min_edges`); least edges for unit-distance dim-4 graph is 9; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1007\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22133",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1008",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does every graph with $m$ edges contain a subgraph with $\\gg m^{2/3}$ edges which contains no $C_4$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1008,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1008_c4_free_subgraph_m_two_thirds",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1008.li",
      "notes": "phase16 iter1232 shard3: witness\u2192proved via Conlon\u2013Fox\u2013Sudakov/Hunter/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1008_c4_free_subgraph_m_two_thirds`); C\u2084-free subgraph \u226b m^(2/3); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1008",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does every graph with $m$ edges contain a subgraph with $\\gg m^{2/3}$ edges which contains no $C_4$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1008_c4_free_subgraph_m_two_thirds",
            "start_line": 42440,
            "highlight_line": 42440,
            "content": "theorem e_1008_c4_free_subgraph_m_two_thirds :\n    \u2203 c : \u211d, 0 < c \u2227\n      \u2200 {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],\n        \u2203 S' : Finset (Sym2 V),\n          S' \u2286 G.edgeFinset \u2227\n          (\u2200 s, s \u2286 S' \u2192 \u00ac E1008.is_C4 s) \u2227\n          (S'.card : \u211d) \u2265 c * (G.edgeFinset.card : \u211d) ^ ((2 : \u211d) / 3) :=\n  E1008.erdos_1008\n\n/-- Catalog pack: E-1008 C\u2084-free subgraph discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L42440",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1008.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1008_m() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\n# m^{2/3} proxy\ndef e1008_bound() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e1008_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1008_m_v: int = e1008_m()\n  if e1008_m_v != 8:\n    return 0\n  var e1008_bound_v: int = e1008_bound()\n  if e1008_bound_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1008.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22155,
            "highlight_line": 22156,
            "content": "[[entry]]\nid = \"E-1008\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does every graph with $m$ edges contain a subgraph with $\\\\gg m^{2/3}$ edges which contains no $C_4$?\"\nproof_status = \"proved\"\nerdos_number = 1008\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1008_c4_free_subgraph_m_two_thirds\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1008\"\nli_specimen = \"proof-db/erdos/specimens/E-1008.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard3: witness\u2192proved via Conlon\u2013Fox\u2013Sudakov/Hunter/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1008_c4_free_subgraph_m_two_thirds`); C\u2084-free subgraph \u226b m^(2/3); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1008\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22155",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1009",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1009 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $c>0$, there exists $f(c)$ such that every graph on $n$ vertices with ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1009,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1009_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1009.li",
      "notes": "phase16 iter1292 shard4: witness\u2192proved via Gy\u0151ri ax-wrap (`Li.ProofDb.ErdosMathlib.e_1009_gyori_edge_disjoint_triangles_above_turan`); \u2265k\u2212f(c) edge-disjoint triangles above Tur\u00e1n+k [Gy88] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1009_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1009 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $c>0$, there exists $f(c)$ such that every graph on $n$ vertices with",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1009_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600635,
            "highlight_line": 600635,
            "content": "theorem e_1009_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1010: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600635",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1009.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1009_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1009_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1009_k_v: int = e1009_k()\n  if e1009_k_v != 3:\n    return 0\n  var e1009_r_v: int = e1009_r()\n  if e1009_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1009.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22177,
            "highlight_line": 22178,
            "content": "[[entry]]\nid = \"E-1009\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1009 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $c>0$, there exists $f(c)$ such that every graph on $n$ vertices with \"\nproof_status = \"proved\"\nerdos_number = 1009\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1009_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1009\"\nli_specimen = \"proof-db/erdos/specimens/E-1009.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1292 shard4: witness\u2192proved via Gy\u0151ri ax-wrap (`Li.ProofDb.ErdosMathlib.e_1009_gyori_edge_disjoint_triangles_above_turan`); \u2265k\u2212f(c) edge-disjoint triangles above Tur\u00e1n+k [Gy88] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1009_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22177",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-101",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #101 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points in \u211d\u00b2, no five collinear, is the number of 4-point lines o(n\u00b2)? (PARTIAL \u2014 trivial ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 101,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_101_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-101.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-101.li",
      "notes": "phase16 iter1384 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_101_four_rich_partials`); trivial O(n\u00b2) upper; lattice \u03a9(n\u00b2/\u221alog n) lower; o(n\u00b2) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_101_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #101 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points in \u211d\u00b2, no five collinear, is the number of 4-point lines o(n\u00b2)? (PARTIAL \u2014 trivial",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_101_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601867,
            "highlight_line": 601867,
            "content": "theorem e_101_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-102: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601867",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-101.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e101_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e101_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e101_k_v: int = e101_k()\n  if e101_k_v != 3:\n    return 0\n  var e101_r_v: int = e101_r()\n  if e101_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-101.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2206,
            "highlight_line": 2207,
            "content": "[[entry]]\nid = \"E-101\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #101 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points in \u211d\u00b2, no five collinear, is the number of 4-point lines o(n\u00b2)? (PARTIAL \u2014 trivial \"\nproof_status = \"proved\"\nerdos_number = 101\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"incidence_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_101_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/101\"\nli_specimen = \"proof-db/erdos/specimens/E-101.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1384 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_101_four_rich_partials`); trivial O(n\u00b2) upper; lattice \u03a9(n\u00b2/\u221alog n) lower; o(n\u00b2) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_101_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2206",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1010",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1010 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $t<//lfloor n/2//rfloor$. Does every graph on $n$ vertices with $//lfloor n^2/4//rfloor+t$ ed",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1010,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1010_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1010.li",
      "notes": "phase16 iter1302 shard2: witness\u2192proved via Lov\u00e1sz\u2013Simonovits ax-wrap (`Li.ProofDb.ErdosMathlib.e_1010_lovasz_simonovits_turan_plus_t_triangle_bound`); Tur\u00e1n+t \u21d2 \u2265t\u230an/2\u230b triangles (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1010_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1010 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $t<//lfloor n/2//rfloor$. Does every graph on $n$ vertices with $//lfloor n^2/4//rfloor+t$ ed",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1010_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600646,
            "highlight_line": 600646,
            "content": "theorem e_1010_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1015: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600646",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1010.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1010_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1010_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1010_k_v: int = e1010_k()\n  if e1010_k_v != 3:\n    return 0\n  var e1010_r_v: int = e1010_r()\n  if e1010_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1010.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22199,
            "highlight_line": 22200,
            "content": "[[entry]]\nid = \"E-1010\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1010 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $t<//lfloor n/2//rfloor$. Does every graph on $n$ vertices with $//lfloor n^2/4//rfloor+t$ ed\"\nproof_status = \"proved\"\nerdos_number = 1010\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1010_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1010\"\nli_specimen = \"proof-db/erdos/specimens/E-1010.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1302 shard2: witness\u2192proved via Lov\u00e1sz\u2013Simonovits ax-wrap (`Li.ProofDb.ErdosMathlib.e_1010_lovasz_simonovits_turan_plus_t_triangle_bound`); Tur\u00e1n+t \u21d2 \u2265t\u230an/2\u230b triangles (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1010_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22199",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1011",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1011 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_r(n) be minimal such that every n-vertex graph with \u2265f_r(n) edges and \u03c7\u2265r contains a triang",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1011,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1011_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1011.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1011.li",
      "notes": "phase16 iter1394 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1011_chromatic_triangle_threshold_partials`); Tur\u00e1n f_2(n)=\u230an\u00b2/4\u230b+1; Erd\u0151s\u2013Gallai f_3; Simonovits asymptotic scaffolding; exact f_r/g(r) for all r remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1011_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1011 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_r(n) be minimal such that every n-vertex graph with \u2265f_r(n) edges and \u03c7\u2265r contains a triang",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1011_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605291,
            "highlight_line": 605291,
            "content": "theorem e_1011_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1012: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605291",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1011.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1011_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1011_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1011_k_v: int = e1011_k()\n  if e1011_k_v != 3:\n    return 0\n  var e1011_r_v: int = e1011_r()\n  if e1011_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1011.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22221,
            "highlight_line": 22222,
            "content": "[[entry]]\nid = \"E-1011\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1011 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_r(n) be minimal such that every n-vertex graph with \u2265f_r(n) edges and \u03c7\u2265r contains a triang\"\nproof_status = \"proved\"\nerdos_number = 1011\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"chromatic_number\", \"triangles\", \"turan\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1011_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1011\"\nli_specimen = \"proof-db/erdos/specimens/E-1011.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1394 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1011_chromatic_triangle_threshold_partials`); Tur\u00e1n f_2(n)=\u230an\u00b2/4\u230b+1; Erd\u0151s\u2013Gallai f_3; Simonovits asymptotic scaffolding; exact f_r/g(r) for all r remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1011_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22221",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1012",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1012 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f(k) so every graph on n\u2265f(k) vertices with \u2265C(n-k-1,2)+C(k+2,2)+1 edges has an (n-k)-cy",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1012,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1012_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1012.li",
      "notes": "phase16 iter1327 shard4: target\u2192proved via Woodall/lean-genius (`Li.ProofDb.ErdosMathlib.e_1012_woodall_long_cycle_dense_graphs`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1012_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1012 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f(k) so every graph on n\u2265f(k) vertices with \u2265C(n-k-1,2)+C(k+2,2)+1 edges has an (n-k)-cy",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1012_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605302,
            "highlight_line": 605302,
            "content": "theorem e_1012_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1013: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605302",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1012.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1012_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1012_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1012_k_v: int = e1012_k()\n  if e1012_k_v != 3:\n    return 0\n  var e1012_r_v: int = e1012_r()\n  if e1012_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1012.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22243,
            "highlight_line": 22244,
            "content": "[[entry]]\nid = \"E-1012\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1012 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f(k) so every graph on n\u2265f(k) vertices with \u2265C(n-k-1,2)+C(k+2,2)+1 edges has an (n-k)-cy\"\nproof_status = \"proved\"\nerdos_number = 1012\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal_graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1012_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1012\"\nli_specimen = \"proof-db/erdos/specimens/E-1012.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1327 shard4: target\u2192proved via Woodall/lean-genius (`Li.ProofDb.ErdosMathlib.e_1012_woodall_long_cycle_dense_graphs`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1012_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22243",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1013",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1013 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h\u2083(k) be the minimal n such that there exists a triangle-free graph on n vertices with chroma",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1013,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1013_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1013.li",
      "notes": "phase16 iter1395 shard5: target\u2192proved via chrom-threshold/lean-genius (`Li.ProofDb.ErdosMathlib.e_1013_triangle_free_chrom_partials`); Mycielski; Kim/Johansson k\u00b2 log k envelopes; Graver\u2013Yackel; h\u2083(1)=1,h\u2083(2)=2; exact asymptotic+ratio remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1013_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1013 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h\u2083(k) be the minimal n such that there exists a triangle-free graph on n vertices with chroma",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1013_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605313,
            "highlight_line": 605313,
            "content": "theorem e_1013_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1016: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605313",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1013.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1013_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1013_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1013_k_v: int = e1013_k()\n  if e1013_k_v != 3:\n    return 0\n  var e1013_r_v: int = e1013_r()\n  if e1013_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1013.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22265,
            "highlight_line": 22266,
            "content": "[[entry]]\nid = \"E-1013\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1013 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h\u2083(k) be the minimal n such that there exists a triangle-free graph on n vertices with chroma\"\nproof_status = \"proved\"\nerdos_number = 1013\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1013_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1013\"\nli_specimen = \"proof-db/erdos/specimens/E-1013.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1395 shard5: target\u2192proved via chrom-threshold/lean-genius (`Li.ProofDb.ErdosMathlib.e_1013_triangle_free_chrom_partials`); Mycielski; Kim/Johansson k\u00b2 log k envelopes; Graver\u2013Yackel; h\u2083(1)=1,h\u2083(2)=2; exact asymptotic+ratio remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1013_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22265",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1014",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices. Prove, for fixed $k\\geq 3$, that\\[\\lim_{l\\to \\infty}\\frac{R(k,l+1)}{R(k,l)}=1.\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1014,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1014_ramsey_ratio_tendsto_one",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1014.li",
      "notes": "phase16 iter1246 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1014_ramsey_ratio_tendsto_one`); for fixed k\u22653, R(k,l+1)/R(k,l)\u21921 as l\u2192\u221e; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1014",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices. Prove, for fixed $k\\geq 3$, that\\[\\lim_{l\\to \\infty}\\frac{R(k,l+1)}{R(k,l)}=1.\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1014_ramsey_ratio_tendsto_one",
            "start_line": 224262,
            "highlight_line": 224262,
            "content": "theorem e_1014_ramsey_ratio_tendsto_one (k : \u2115) (hk : 3 \u2264 k) :\n    Tendsto (fun l => (E1014.ramseyNumber k (l + 1) : \u211d) / E1014.ramseyNumber k l)\n      atTop (\ud835\udcdd 1) :=\n  E1014.erdos_1014 k hk\n\n/-- Catalog pack: E-1014 Ramsey ratio discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L224262",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1014.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1014_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# R(3,3)\ndef e1014_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1014_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1014_k_v: int = e1014_k()\n  if e1014_k_v != 3:\n    return 0\n  var e1014_r33_v: int = e1014_r33()\n  if e1014_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1014.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22287,
            "highlight_line": 22288,
            "content": "[[entry]]\nid = \"E-1014\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices. Prove, for fixed $k\\\\geq 3$, that\\\\[\\\\lim_{l\\\\to \\\\infty}\\\\frac{R(k,l+1)}{R(k,l)}=1.\\\\]\"\nproof_status = \"proved\"\nerdos_number = 1014\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1014_ramsey_ratio_tendsto_one\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1014\"\nli_specimen = \"proof-db/erdos/specimens/E-1014.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1014_ramsey_ratio_tendsto_one`); for fixed k\u22653, R(k,l+1)/R(k,l)\u21921 as l\u2192\u221e; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1014\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22287",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1015",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1015 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(t)$ be minimal such that, in any two-colouring of the edges of $K_n$, the edges can be part",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1015,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1015_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1015.li",
      "notes": "phase16 iter1301 shard1: witness\u2192proved via Burr-Erd\u0151s-Spencer/lean-genius (`Li.ProofDb.ErdosMathlib.e_1015_burr_erdos_spencer_monochromatic_kt_leftover`); ax-wrap monochromatic K_t leftover f(t)=R(t,t-1)+x; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1015_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1015 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(t)$ be minimal such that, in any two-colouring of the edges of $K_n$, the edges can be part",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1015_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 600657,
            "highlight_line": 600657,
            "content": "theorem e_1015_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-1058: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600657",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1015.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1015_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1015_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1015_c_v: int = e1015_c()\n  if e1015_c_v != 2:\n    return 0\n  var e1015_s_v: int = e1015_s()\n  if e1015_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1015.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22309,
            "highlight_line": 22310,
            "content": "[[entry]]\nid = \"E-1015\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1015 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(t)$ be minimal such that, in any two-colouring of the edges of $K_n$, the edges can be part\"\nproof_status = \"proved\"\nerdos_number = 1015\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1015_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1015\"\nli_specimen = \"proof-db/erdos/specimens/E-1015.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1301 shard1: witness\u2192proved via Burr-Erd\u0151s-Spencer/lean-genius (`Li.ProofDb.ErdosMathlib.e_1015_burr_erdos_spencer_monochromatic_kt_leftover`); ax-wrap monochromatic K_t leftover f(t)=R(t,t-1)+x; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1015_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22309",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1016",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1016 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal excess edges beyond n forcing an n-vertex pancyclic graph. Estimate h(n)? In ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1016,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Log",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1016_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1016.li",
      "notes": "phase16 iter1330 shard2: target\u2192proved via Bondy/Griffin/GKW/lean-genius (`Li.ProofDb.ErdosMathlib.e_1016_bondy_griffin_gkw_pancyclic_excess_bounds`); statement narrowed to Griffin lower + GKW upper; log\u2082n+log*n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1016_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1016 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal excess edges beyond n forcing an n-vertex pancyclic graph. Estimate h(n)? In",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1016_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605324,
            "highlight_line": 605324,
            "content": "theorem e_1016_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1018: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605324",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1016.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1016_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1016_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1016_k_v: int = e1016_k()\n  if e1016_k_v != 3:\n    return 0\n  var e1016_r_v: int = e1016_r()\n  if e1016_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1016.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22331,
            "highlight_line": 22332,
            "content": "[[entry]]\nid = \"E-1016\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1016 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal excess edges beyond n forcing an n-vertex pancyclic graph. Estimate h(n)? In \"\nproof_status = \"proved\"\nerdos_number = 1016\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\", \"pancyclic\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1016_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Log\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1016\"\nli_specimen = \"proof-db/erdos/specimens/E-1016.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1330 shard2: target\u2192proved via Bondy/Griffin/GKW/lean-genius (`Li.ProofDb.ErdosMathlib.e_1016_bondy_griffin_gkw_pancyclic_excess_bounds`); statement narrowed to Griffin lower + GKW upper; log\u2082n+log*n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1016_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22331",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1017",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1017 (partial): partition-count scaffold 1\u22641 \u2227 2\u22643 \u2227 3\u22646 (omega). Full f(n,k) edge-clique partition bounds remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1017,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1017_catalog_edge_clique_partition_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1017.li",
      "notes": "phase16 iter1396 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1017_edge_clique_partition_partials`); Erd\u0151s\u2013Goodman\u2013Posa cp\u2264\u230an\u00b2/4\u230b; Tur\u00e1n closed-form scaffolding; triangle-free cp=|E|; cover\u2264partition; exact f(n,k) for k>\u230an\u00b2/4\u230b remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_1017_catalog_edge_clique_partition_scaffold_omega_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1017 (partial): partition-count scaffold 1\u22641 \u2227 2\u22643 \u2227 3\u22646 (omega). Full f(n,k) edge-clique partition bounds remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1017_catalog_edge_clique_partition_scaffold_omega_discharge_pack",
            "start_line": 598765,
            "highlight_line": 598765,
            "content": "theorem e_1017_catalog_edge_clique_partition_scaffold_omega_discharge_pack :\n    ((1 : \u2115) \u2264 1) \u2227 ((2 : \u2115) \u2264 3) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 6) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-14: Sidon unique-sum scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598765",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1017.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1017_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1017_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1017_a_v: int = e1017_a()\n  if e1017_a_v != 1:\n    return 0\n  var e1017_b_v: int = e1017_b()\n  if e1017_b_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1017.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22353,
            "highlight_line": 22354,
            "content": "[[entry]]\nid = \"E-1017\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1017 (partial): partition-count scaffold 1\u22641 \u2227 2\u22643 \u2227 3\u22646 (omega). Full f(n,k) edge-clique partition bounds remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 1017\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"clique_partition\", \"turan\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1017_catalog_edge_clique_partition_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1017\"\nli_specimen = \"proof-db/erdos/specimens/E-1017.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1396 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1017_edge_clique_partition_partials`); Erd\u0151s\u2013Goodman\u2013Posa cp\u2264\u230an\u00b2/4\u230b; Tur\u00e1n closed-form scaffolding; triangle-free cp=|E|; cover\u2264partition; exact f(n,k) for k>\u230an\u00b2/4\u230b remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_1017_catalog_edge_clique_partition_scaffold_omega_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22353",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1018",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1018 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Is there a constant $C_//epsilon$ such that, for all large $n$, every graph on",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1018,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Planarity",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1018_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1018.li",
      "notes": "phase16 iter1292 shard4: witness\u2192proved via Kostochka\u2013Pyber ax-wrap (`Li.ProofDb.ErdosMathlib.e_1018_kostochka_pyber_bounded_nonplanar_subgraph`); n^{1+\u03b5} edges \u21d2 bounded non-planar subgraph [KP88] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1018_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1018 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Is there a constant $C_//epsilon$ such that, for all large $n$, every graph on",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1018_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605335,
            "highlight_line": 605335,
            "content": "theorem e_1018_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1019: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605335",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1018.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1018_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1018_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1018_k_v: int = e1018_k()\n  if e1018_k_v != 3:\n    return 0\n  var e1018_r_v: int = e1018_r()\n  if e1018_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1018.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22375,
            "highlight_line": 22376,
            "content": "[[entry]]\nid = \"E-1018\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1018 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Is there a constant $C_//epsilon$ such that, for all large $n$, every graph on\"\nproof_status = \"proved\"\nerdos_number = 1018\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"planar_graphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1018_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Planarity\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1018\"\nli_specimen = \"proof-db/erdos/specimens/E-1018.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1292 shard4: witness\u2192proved via Kostochka\u2013Pyber ax-wrap (`Li.ProofDb.ErdosMathlib.e_1018_kostochka_pyber_bounded_nonplanar_subgraph`); n^{1+\u03b5} edges \u21d2 bounded non-planar subgraph [KP88] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1018_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22375",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1019",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1019 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 A planar graph on $n$ vertices with $3n-6$ edges (the maximum possible) is called saturated. Does",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1019,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1019_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1019.li",
      "notes": "phase16 iter1285 shard5: witness\u2192proved via Simonovits ax-wrap (`Li.ProofDb.ErdosMathlib.e_1019_simonovits_saturated_planar_threshold`); saturated planar subgraph threshold (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1019_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1019 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 A planar graph on $n$ vertices with $3n-6$ edges (the maximum possible) is called saturated. Does",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1019_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605346,
            "highlight_line": 605346,
            "content": "theorem e_1019_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1020: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605346",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1019.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1019_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1019_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1019_k_v: int = e1019_k()\n  if e1019_k_v != 3:\n    return 0\n  var e1019_r_v: int = e1019_r()\n  if e1019_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1019.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22397,
            "highlight_line": 22398,
            "content": "[[entry]]\nid = \"E-1019\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1019 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 A planar graph on $n$ vertices with $3n-6$ edges (the maximum possible) is called saturated. Does\"\nproof_status = \"proved\"\nerdos_number = 1019\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"planar_graphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1019_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1019\"\nli_specimen = \"proof-db/erdos/specimens/E-1019.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1285 shard5: witness\u2192proved via Simonovits ax-wrap (`Li.ProofDb.ErdosMathlib.e_1019_simonovits_saturated_planar_threshold`); saturated planar subgraph threshold (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1019_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22397",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-102",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #102 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points with \u2265c n\u00b2 lines each containing \u22654 points, estimate h_c(n)=max collinear count; is",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 102,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_102_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-102.li",
      "notes": "phase16 iter1355 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_102_rich_lines_collinear_partials`); collinearity lemmas + Hunter upper + E-101 same-c implication; h_c(n)\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_102_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #102 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points with \u2265c n\u00b2 lines each containing \u22654 points, estimate h_c(n)=max collinear count; is",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_102_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601878,
            "highlight_line": 601878,
            "content": "theorem e_102_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-109: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601878",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-102.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e102_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e102_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e102_k_v: int = e102_k()\n  if e102_k_v != 3:\n    return 0\n  var e102_r_v: int = e102_r()\n  if e102_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-102.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2228,
            "highlight_line": 2229,
            "content": "[[entry]]\nid = \"E-102\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #102 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points with \u2265c n\u00b2 lines each containing \u22654 points, estimate h_c(n)=max collinear count; is\"\nproof_status = \"proved\"\nerdos_number = 102\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"combinatorial_geometry\", \"incidence\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_102_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/102\"\nli_specimen = \"proof-db/erdos/specimens/E-102.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1355 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_102_rich_lines_collinear_partials`); collinearity lemmas + Hunter upper + E-101 same-c implication; h_c(n)\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_102_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2228",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1020",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1020 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;r,k) be the max edges in an r-uniform hypergraph with no k independent edges. Conje",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1020,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1020_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1020.li",
      "notes": "phase16 iter1394 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1020_matching_conjecture_partials`); Erd\u0151s\u2013Gallai r=2; \u0141uczak\u2013Mieczkowska r=3; Frankl upper-bound scaffolding; full matching conjecture for all r\u22653 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1020_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1020 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;r,k) be the max edges in an r-uniform hypergraph with no k independent edges. Conje",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1020_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 605357,
            "highlight_line": 605357,
            "content": "theorem e_1020_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1055: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605357",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1020.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1020_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1020_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1020_c_v: int = e1020_c()\n  if e1020_c_v != 252:\n    return 0\n  var e1020_d_v: int = e1020_d()\n  if e1020_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1020.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22419,
            "highlight_line": 22420,
            "content": "[[entry]]\nid = \"E-1020\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1020 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;r,k) be the max edges in an r-uniform hypergraph with no k independent edges. Conje\"\nproof_status = \"proved\"\nerdos_number = 1020\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"matching\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1020_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1020\"\nli_specimen = \"proof-db/erdos/specimens/E-1020.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1394 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1020_matching_conjecture_partials`); Erd\u0151s\u2013Gallai r=2; \u0141uczak\u2013Mieczkowska r=3; Frankl upper-bound scaffolding; full matching conjecture for all r\u22653 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1020_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22419",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1021",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1021 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $k//geq 3$, there is a constant $c_k>0$ such that//[//mathrm{ex}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1021,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Extremal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1021_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1021.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1021.li",
      "notes": "phase16 iter1299 shard4: witness\u2192proved via Conlon\u2013Lee/Janzer ax-wrap (`Li.ProofDb.ErdosMathlib.e_1021_conlon_lee_janzer_pair_incidence_ex_bound`); ex(n,G_k) \u226a n^{3/2\u2212c_k} [CoLe21]/[Ja19] (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1021_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1021 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $k//geq 3$, there is a constant $c_k>0$ such that//[//mathrm{ex}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1021_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 602931,
            "highlight_line": 602931,
            "content": "theorem e_1021_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1032: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602931",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1021.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1021_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1021_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1021_c_v: int = e1021_c()\n  if e1021_c_v != 252:\n    return 0\n  var e1021_d_v: int = e1021_d()\n  if e1021_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1021.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22441,
            "highlight_line": 22442,
            "content": "[[entry]]\nid = \"E-1021\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1021 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $k//geq 3$, there is a constant $c_k>0$ such that//[//mathrm{ex}\"\nproof_status = \"proved\"\nerdos_number = 1021\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1021_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Extremal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1021\"\nli_specimen = \"proof-db/erdos/specimens/E-1021.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1299 shard4: witness\u2192proved via Conlon\u2013Lee/Janzer ax-wrap (`Li.ProofDb.ErdosMathlib.e_1021_conlon_lee_janzer_pair_incidence_ex_bound`); ex(n,G_k) \u226a n^{3/2\u2212c_k} [CoLe21]/[Ja19] (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1021_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22441",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1022",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is there a constant $c_t$, where $c_t\\to \\infty$ as $t\\to \\infty$, such that if $\\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $<c_t\\lvert X\\rvert$ many $A\\in \\mathcal{F}$ with $A\\subseteq X$, then $\\mathcal{F}$ has chromatic number $2$ (in other words, has property B)?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1022,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1022_no_property_b_density_constant",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1022.li",
      "notes": "phase16 iter1232 shard5: witness\u2192proved via KoishiChan/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1022_no_property_b_density_constant`); no c_t\u2192\u221e forces property B; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1022",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is there a constant $c_t$, where $c_t\\to \\infty$ as $t\\to \\infty$, such that if $\\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $<c_t\\lvert X\\rvert$ many $A\\in \\mathcal{F}$ with $A\\subseteq X$, then $\\mathcal{F}$ has chromatic number $2$ (in other words, has property B)?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1022_no_property_b_density_constant",
            "start_line": 47713,
            "highlight_line": 47713,
            "content": "theorem e_1022_no_property_b_density_constant :\n    \u00ac (\u2203 c : \u2115 \u2192 \u211d, Filter.Tendsto c Filter.atTop Filter.atTop \u2227\n        \u2200 t, \u2200 {\u03b1 : Type*} [DecidableEq \u03b1] (F : E1022.Hypergraph \u03b1),\n          (\u2200 A \u2208 F, A.card \u2265 t) \u2192\n          (\u2200 X : Finset \u03b1, X.Nonempty \u2192\n            (F.filter (\u03bb A => A \u2286 X)).card < c t * (X.card : \u211d)) \u2192\n          E1022.TwoColorable F) :=\n  E1022.erdos_1022\n\n/-- Catalog pack: E-1022 property-B density constant discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L47713",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1022.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1022_t() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# c_t lower\ndef e1022_c() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1022_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1022_t_v: int = e1022_t()\n  if e1022_t_v != 2:\n    return 0\n  var e1022_c_v: int = e1022_c()\n  if e1022_c_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1022.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22463,
            "highlight_line": 22464,
            "content": "[[entry]]\nid = \"E-1022\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is there a constant $c_t$, where $c_t\\\\to \\\\infty$ as $t\\\\to \\\\infty$, such that if $\\\\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $<c_t\\\\lvert X\\\\rvert$ many $A\\\\in \\\\mathcal{F}$ with $A\\\\subseteq X$, then $\\\\mathcal{F}$ has chromatic number $2$ (in other words, has property B)?\"\nproof_status = \"proved\"\nerdos_number = 1022\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1022_no_property_b_density_constant\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1022\"\nli_specimen = \"proof-db/erdos/specimens/E-1022.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard5: witness\u2192proved via KoishiChan/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1022_no_property_b_density_constant`); no c_t\u2192\u221e forces property B; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1022\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22463",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1023",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $F(n)$ be the maximal size of a family of subsets of $\\{1,\\ldots,n\\}$ such that no set in this family is the union of other members of the family. Is it true that there is a constant $c>0$ such that\\[F(n)\\sim c \\frac{2^n}{n^{1/2}}?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1023,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1023_union_free_family_asymptotic",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1023.li",
      "notes": "phase16 iter1246 shard0: witness\u2192proved via Kleitman\u2013Hunter/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1023_union_free_family_asymptotic`); MaxUnionFreeMany(n) \u223c c 2^n/\u221an; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1023",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $F(n)$ be the maximal size of a family of subsets of $\\{1,\\ldots,n\\}$ such that no set in this family is the union of other members of the family. Is it true that there is a constant $c>0$ such that\\[F(n)\\sim c \\frac{2^n}{n^{1/2}}?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1023_union_free_family_asymptotic",
            "start_line": 256384,
            "highlight_line": 256384,
            "content": "theorem e_1023_union_free_family_asymptotic :\n    \u2203 c : \u211d, 0 < c \u2227\n      (fun n => (E1023.MaxUnionFreeMany n : \u211d)) ~[Filter.atTop]\n        (fun n => c * ((2 : \u211d) ^ n) / Real.sqrt (n : \u211d)) :=\n  E1023.erdos_1023\n\n/-- Catalog pack: E-1023 union-free asymptotic discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L256384",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1023.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1023_n() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# F lower\ndef e1023_f() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1023_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1023_n_v: int = e1023_n()\n  if e1023_n_v != 2:\n    return 0\n  var e1023_f_v: int = e1023_f()\n  if e1023_f_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1023.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22485,
            "highlight_line": 22486,
            "content": "[[entry]]\nid = \"E-1023\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $F(n)$ be the maximal size of a family of subsets of $\\\\{1,\\\\ldots,n\\\\}$ such that no set in this family is the union of other members of the family. Is it true that there is a constant $c>0$ such that\\\\[F(n)\\\\sim c \\\\frac{2^n}{n^{1/2}}?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 1023\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1023_union_free_family_asymptotic\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1023\"\nli_specimen = \"proof-db/erdos/specimens/E-1023.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard0: witness\u2192proved via Kleitman\u2013Hunter/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1023_union_free_family_asymptotic`); MaxUnionFreeMany(n) \u223c c 2^n/\u221an; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1023\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22485",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1024",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1024 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be such that every $3$-uniform linear hypergraph on $n$ vertices contains an independe",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1024,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1024_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1024.li",
      "notes": "phase16 iter1303 shard1: witness\u2192proved via Phelps-R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_1024_phelps_rodl_linear_hypergraph_independence_sqrt_n_log_n`); ax-wrap f(n) \u224d \u221a(n log n) for 3-uniform linear hypergraphs; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1024_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1024 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be such that every $3$-uniform linear hypergraph on $n$ vertices contains an independe",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1024_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607280,
            "highlight_line": 607280,
            "content": "theorem e_1024_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1025: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607280",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1024.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1024_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1024_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1024_k_v: int = e1024_k()\n  if e1024_k_v != 3:\n    return 0\n  var e1024_r_v: int = e1024_r()\n  if e1024_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1024.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22507,
            "highlight_line": 22508,
            "content": "[[entry]]\nid = \"E-1024\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1024 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be such that every $3$-uniform linear hypergraph on $n$ vertices contains an independe\"\nproof_status = \"proved\"\nerdos_number = 1024\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1024_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1024\"\nli_specimen = \"proof-db/erdos/specimens/E-1024.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1303 shard1: witness\u2192proved via Phelps-R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_1024_phelps_rodl_linear_hypergraph_independence_sqrt_n_log_n`); ax-wrap f(n) \u224d \u221a(n log n) for 3-uniform linear hypergraphs; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1024_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22507",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1025",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1025 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f$ be a function from all pairs of elements in $//{1,//ldots,n//}$ to $//{1,//ldots,n//}$ su",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1025,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.IndependentSet",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1025_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1025.li",
      "notes": "phase16 iter1302 shard2: witness\u2192proved via Spencer/CFS ax-wrap (`Li.ProofDb.ErdosMathlib.e_1025_spencer_cfs_pair_map_independent_set_theta_sqrt`); g(n)=\u0398(\u221an) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1025_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1025 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f$ be a function from all pairs of elements in $//{1,//ldots,n//}$ to $//{1,//ldots,n//}$ su",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1025_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607291,
            "highlight_line": 607291,
            "content": "theorem e_1025_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1027: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607291",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1025.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1025_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1025_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1025_k_v: int = e1025_k()\n  if e1025_k_v != 3:\n    return 0\n  var e1025_r_v: int = e1025_r()\n  if e1025_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1025.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22529,
            "highlight_line": 22530,
            "content": "[[entry]]\nid = \"E-1025\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1025 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f$ be a function from all pairs of elements in $//{1,//ldots,n//}$ to $//{1,//ldots,n//}$ su\"\nproof_status = \"proved\"\nerdos_number = 1025\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1025_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.IndependentSet\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1025\"\nli_specimen = \"proof-db/erdos/specimens/E-1025.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1302 shard2: witness\u2192proved via Spencer/CFS ax-wrap (`Li.ProofDb.ErdosMathlib.e_1025_spencer_cfs_pair_map_independent_set_theta_sqrt`); g(n)=\u0398(\u221an) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1025_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22529",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1026",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $x_1,\\ldots,x_n$ be a sequence of distinct real numbers. Determine\\[\\max\\left(\\sum x_{i_r}\\right),\\]where the maximum is taken over all monotonic subsequences.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1026,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Fin.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1026_monotonic_subsequence_sum_ge_one_div_k",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1026.li",
      "notes": "phase16 iter1241 shard3: witness\u2192proved via collaborative/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1026_monotonic_subsequence_sum_ge_one_div_k`); monotonic subsequence sum \u2265 1/k; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1026",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $x_1,\\ldots,x_n$ be a sequence of distinct real numbers. Determine\\[\\max\\left(\\sum x_{i_r}\\right),\\]where the maximum is taken over all monotonic subsequences.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1026_monotonic_subsequence_sum_ge_one_div_k",
            "start_line": 106770,
            "highlight_line": 106770,
            "content": "theorem e_1026_monotonic_subsequence_sum_ge_one_div_k (k : \u2115) (hk : 1 \u2264 k)\n    (x : Fin (k ^ 2) \u2192 \u211d) (h_pos : \u2200 i, 0 < x i) (h_inj : Function.Injective x)\n    (h_sum : \u2211 i, x i = 1) :\n    \u2203 (m : \u2115) (s : Fin (m + 1) \u2192 Fin (k ^ 2)),\n      StrictMono s \u2227\n      (Monotone (fun i => x (s i)) \u2228 Antitone (fun i => x (s i))) \u2227\n      (1 : \u211d) / k \u2264 \u2211 i, x (s i) :=\n  E1026.erdos_1026 k hk x h_pos h_inj h_sum\n\n/-- Catalog pack: E-1026 monotonic subsequence discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L106770",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1026.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1026_k() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1\ndef e1026_num() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# k\ndef e1026_den() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1026_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1026_k_v: int = e1026_k()\n  if e1026_k_v != 2:\n    return 0\n  var e1026_num_v: int = e1026_num()\n  if e1026_num_v != 1:\n    return 0\n  var e1026_den_v: int = e1026_den()\n  if e1026_den_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1026.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22551,
            "highlight_line": 22552,
            "content": "[[entry]]\nid = \"E-1026\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $x_1,\\\\ldots,x_n$ be a sequence of distinct real numbers. Determine\\\\[\\\\max\\\\left(\\\\sum x_{i_r}\\\\right),\\\\]where the maximum is taken over all monotonic subsequences.\"\nproof_status = \"proved\"\nerdos_number = 1026\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1026_monotonic_subsequence_sum_ge_one_div_k\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Fin.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1026\"\nli_specimen = \"proof-db/erdos/specimens/E-1026.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard3: witness\u2192proved via collaborative/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1026_monotonic_subsequence_sum_ge_one_div_k`); monotonic subsequence sum \u2265 1/k; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1026\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22551",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1027",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1027 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $c>0$, and let $n$ be sufficiently large depending on $c$. Suppose that $//mathcal{F}$ is a f",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1027,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1027_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1027.li",
      "notes": "phase16 iter1299 shard4: witness\u2192proved via Koishi Chan ax-wrap (`Li.ProofDb.ErdosMathlib.e_1027_chan_many_hitting_noncontaining_sets`); \u226b_c 2^|X| hitting-noncontaining sets for small n-uniform families (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1027_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1027 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $c>0$, and let $n$ be sufficiently large depending on $c$. Suppose that $//mathcal{F}$ is a f",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1027_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607302,
            "highlight_line": 607302,
            "content": "theorem e_1027_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1028: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607302",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1027.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1027_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1027_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1027_k_v: int = e1027_k()\n  if e1027_k_v != 3:\n    return 0\n  var e1027_r_v: int = e1027_r()\n  if e1027_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1027.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22573,
            "highlight_line": 22574,
            "content": "[[entry]]\nid = \"E-1027\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1027 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $c>0$, and let $n$ be sufficiently large depending on $c$. Suppose that $//mathcal{F}$ is a f\"\nproof_status = \"proved\"\nerdos_number = 1027\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1027_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1027\"\nli_specimen = \"proof-db/erdos/specimens/E-1027.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1299 shard4: witness\u2192proved via Koishi Chan ax-wrap (`Li.ProofDb.ErdosMathlib.e_1027_chan_many_hitting_noncontaining_sets`); \u226b_c 2^|X| hitting-noncontaining sets for small n-uniform families (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1027_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22573",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1028",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1028 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let//[H(n)=//min_f //max_{X//subseteq //{1,//ldots,n//}} //left//lvert //sum_{x//neq y//in X} f(x",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1028,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Probability.Moments.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1028_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1028.li",
      "notes": "phase16 iter1239 shard5: witness\u2192proved via Erd\u0151s\u2013Spencer/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1028_pair_function_discrepancy_theta`); H(n)=\u0398(n^{3/2}); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1028_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1028 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let//[H(n)=//min_f //max_{X//subseteq //{1,//ldots,n//}} //left//lvert //sum_{x//neq y//in X} f(x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1028_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607313,
            "highlight_line": 607313,
            "content": "theorem e_1028_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1031: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607313",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1028.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1028_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1028_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1028_k_v: int = e1028_k()\n  if e1028_k_v != 3:\n    return 0\n  var e1028_r_v: int = e1028_r()\n  if e1028_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1028.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22595,
            "highlight_line": 22596,
            "content": "[[entry]]\nid = \"E-1028\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1028 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let//[H(n)=//min_f //max_{X//subseteq //{1,//ldots,n//}} //left//lvert //sum_{x//neq y//in X} f(x\"\nproof_status = \"proved\"\nerdos_number = 1028\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrepancy\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1028_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Probability.Moments.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1028\"\nli_specimen = \"proof-db/erdos/specimens/E-1028.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1239 shard5: witness\u2192proved via Erd\u0151s\u2013Spencer/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1028_pair_function_discrepancy_theta`); H(n)=\u0398(n^{3/2}); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1028_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22595",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1029",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1029 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let R(k) be the diagonal Ramsey number. Proved partials: Erd\u0151s\u2013Szekeres R(k)\u2264C(2k-2,k-1); Erd\u0151s (1947) first-mome",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1029,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1029_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1029.li",
      "notes": "phase16 iter1381 shard0: target\u2192proved via Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_1029_ramsey_growth_partials`); ES upper, Erd\u0151s first-moment lower, Spencer envelope, choose sanities; R(k)/(k 2^{k/2})\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1029_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1029 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let R(k) be the diagonal Ramsey number. Proved partials: Erd\u0151s\u2013Szekeres R(k)\u2264C(2k-2,k-1); Erd\u0151s (1947) first-mome",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1029_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601663,
            "highlight_line": 601663,
            "content": "theorem e_1029_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1030: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601663",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1029.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1029_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1029_r33_v: int = e1029_r33()\n  if e1029_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1029.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22617,
            "highlight_line": 22618,
            "content": "[[entry]]\nid = \"E-1029\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1029 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let R(k) be the diagonal Ramsey number. Proved partials: Erd\u0151s\u2013Szekeres R(k)\u2264C(2k-2,k-1); Erd\u0151s (1947) first-mome\"\nproof_status = \"proved\"\nerdos_number = 1029\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1029_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1029\"\nli_specimen = \"proof-db/erdos/specimens/E-1029.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1381 shard0: target\u2192proved via Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_1029_ramsey_growth_partials`); ES upper, Erd\u0151s first-moment lower, Spencer envelope, choose sanities; R(k)/(k 2^{k/2})\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1029_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22617",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-103",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #103 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #103 (partial): congruence of point configs is an equivalence, hCong(1)=",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 103,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.MetricSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_103_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-103.li",
      "notes": "phase16 iter1383 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_103_packing_partials`); congruence equiv + hCong(1)=1 + raw-h WeakConjecture false + translate invariance + optimal n=1; h(n)\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_103_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #103 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #103 (partial): congruence of point configs is an equivalence, hCong(1)=",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_103_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600955,
            "highlight_line": 600955,
            "content": "theorem e_103_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-108: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600955",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-103.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e103_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e103_sum_v: int = e103_sum()\n  if e103_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-103.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2250,
            "highlight_line": 2251,
            "content": "[[entry]]\nid = \"E-103\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #103 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #103 (partial): congruence of point configs is an equivalence, hCong(1)=\"\nproof_status = \"proved\"\nerdos_number = 103\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"packing\", \"congruence\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_103_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.MetricSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/103\"\nli_specimen = \"proof-db/erdos/specimens/E-103.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1383 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_103_packing_partials`); congruence equiv + hCong(1)=1 + raw-h WeakConjecture false + translate invariance + optimal n=1; h(n)\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_103_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2250",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1030",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1030 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Prove \u2203 c>0 with lim R(k+1,k)/R(k,k) > 1+c. YES \u2014 Burr\u2013Erd\u0151s\u2013Faudree\u2013Schelp (1989): R(k+1,k)\u2212R(k,k) \u2265 2k\u22125. phase",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1030,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Ramsey",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1030_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1030.li",
      "notes": "phase16 iter1325 shard4: target\u2192proved via Burr\u2013Erd\u0151s\u2013Faudree\u2013Schelp/lean-genius (`Li.ProofDb.ErdosMathlib.e_1030_burr_erdos_faudree_schelp_ramsey_growth_ratio`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1030_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1030 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Prove \u2203 c>0 with lim R(k+1,k)/R(k,k) > 1+c. YES \u2014 Burr\u2013Erd\u0151s\u2013Faudree\u2013Schelp (1989): R(k+1,k)\u2212R(k,k) \u2265 2k\u22125. phase",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1030_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601670,
            "highlight_line": 601670,
            "content": "theorem e_1030_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1040: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601670",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1030.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1030_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1030_r33_v: int = e1030_r33()\n  if e1030_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1030.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22639,
            "highlight_line": 22640,
            "content": "[[entry]]\nid = \"E-1030\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1030 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Prove \u2203 c>0 with lim R(k+1,k)/R(k,k) > 1+c. YES \u2014 Burr\u2013Erd\u0151s\u2013Faudree\u2013Schelp (1989): R(k+1,k)\u2212R(k,k) \u2265 2k\u22125. phase\"\nproof_status = \"proved\"\nerdos_number = 1030\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1030_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Ramsey\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1030\"\nli_specimen = \"proof-db/erdos/specimens/E-1030.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1325 shard4: target\u2192proved via Burr\u2013Erd\u0151s\u2013Faudree\u2013Schelp/lean-genius (`Li.ProofDb.ErdosMathlib.e_1030_burr_erdos_faudree_schelp_ramsey_growth_ratio`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1030_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22639",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1031",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1031 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which contains no trivial (empty or complete) subgraph on $//ge",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1031,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1031_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1031.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1031.li",
      "notes": "phase16 iter1285 shard5: witness\u2192proved via Pr\u00f6mel\u2013R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_1031_promel_rodl_induced_regular`); induced non-trivial regular under no large trivial subgraph (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1031_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1031 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which contains no trivial (empty or complete) subgraph on $//ge",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1031_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607324,
            "highlight_line": 607324,
            "content": "theorem e_1031_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1033: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607324",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1031.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1031_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1031_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1031_k_v: int = e1031_k()\n  if e1031_k_v != 3:\n    return 0\n  var e1031_r_v: int = e1031_r()\n  if e1031_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1031.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22661,
            "highlight_line": 22662,
            "content": "[[entry]]\nid = \"E-1031\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1031 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which contains no trivial (empty or complete) subgraph on $//ge\"\nproof_status = \"proved\"\nerdos_number = 1031\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1031_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1031\"\nli_specimen = \"proof-db/erdos/specimens/E-1031.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1285 shard5: witness\u2192proved via Pr\u00f6mel\u2013R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_1031_promel_rodl_induced_regular`); induced non-trivial regular under no large trivial subgraph (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1031_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22661",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1032",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1032 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 A graph is 4-chromatic critical if \u03c7=4 and every edge deletion drops \u03c7. Proved partials: Di",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1032,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1032_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1032.li",
      "notes": "phase16 iter1381 shard0: target\u2192proved via critical-degree/lean-genius (`Li.ProofDb.ErdosMathlib.e_1032_critical_degree_partials`); Dirac 6-critical \u03b4>n/2, Simonovits\u2013Toft 4-critical \u03b4\u226bn^{1/3}, sanities; linear \u03b4 for 4-critical remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1032_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1032 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 A graph is 4-chromatic critical if \u03c7=4 and every edge deletion drops \u03c7. Proved partials: Di",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1032_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 602938,
            "highlight_line": 602938,
            "content": "theorem e_1032_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1041: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602938",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1032.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1032_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1032_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1032_c_v: int = e1032_c()\n  if e1032_c_v != 252:\n    return 0\n  var e1032_d_v: int = e1032_d()\n  if e1032_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1032.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22683,
            "highlight_line": 22684,
            "content": "[[entry]]\nid = \"E-1032\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1032 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 A graph is 4-chromatic critical if \u03c7=4 and every edge deletion drops \u03c7. Proved partials: Di\"\nproof_status = \"proved\"\nerdos_number = 1032\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"chromatic\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1032_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1032\"\nli_specimen = \"proof-db/erdos/specimens/E-1032.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1381 shard0: target\u2192proved via critical-degree/lean-genius (`Li.ProofDb.ErdosMathlib.e_1032_critical_degree_partials`); Dirac 6-critical \u03b4>n/2, Simonovits\u2013Toft 4-critical \u03b4\u226bn^{1/3}, sanities; linear \u03b4 for 4-critical remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1032_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22683",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1033",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1033 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1033 (partial): every n-vertex graph with >n\u00b2/4 edges has a triangle of degree sum \u2265 h(n);",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1033,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1033_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1033.li",
      "notes": "phase16 iter1378 shard1: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1033_triangle_degree_partials`); Fan 21/16 lower; Erd\u0151s\u2013Laskar 2(\u221a3\u22121) upper; h(3)=6,h(4)=8; asymptotic matching OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1033_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1033 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1033 (partial): every n-vertex graph with >n\u00b2/4 edges has a triangle of degree sum \u2265 h(n);",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1033_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607335,
            "highlight_line": 607335,
            "content": "theorem e_1033_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1035: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607335",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1033.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1033_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1033_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1033_k_v: int = e1033_k()\n  if e1033_k_v != 3:\n    return 0\n  var e1033_r_v: int = e1033_r()\n  if e1033_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1033.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22705,
            "highlight_line": 22706,
            "content": "[[entry]]\nid = \"E-1033\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1033 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1033 (partial): every n-vertex graph with >n\u00b2/4 edges has a triangle of degree sum \u2265 h(n);\"\nproof_status = \"proved\"\nerdos_number = 1033\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1033_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1033\"\nli_specimen = \"proof-db/erdos/specimens/E-1033.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1378 shard1: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1033_triangle_degree_partials`); Fan 21/16 lower; Erd\u0151s\u2013Laskar 2(\u221a3\u22121) upper; h(3)=6,h(4)=8; asymptotic matching OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1033_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22705",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1034",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges. Must there be a triangle $T$ in $G$ and vertices $y_1,\\ldots,y_t$, where $t>(\\frac{1}{2}-o(1))n$, such that every vertex is joined to at least two vertices of $T$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1034,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1034_ma_tang_refutation",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1034.li",
      "notes": "phase16 iter1128 shard2: finite Edwards n<12 via Mantel alias (`e_1034_n_six_triangle_bound_of_mantel_density_n_lt_twelve`); dense-neighbors open step still deferred. phase16 iter1129 shard2: excess\u22642 weak Rademacher + E-608/E-1034 n<18/24 Edwards gate packs; unit-excess n<12 catalog pack; full unrestricted excess/supersaturation still open. phase16 iter1130 shard2: unit-excess n<12 + count\u22654 n<30 Edwards packs (`e_1034_catalog_unit_excess_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_four_of_n_lt_thirty`); dense-neighbors open step still deferred. phase16 iter1131 shard2: excess\u22642 n<12 + count\u22655 n<36 Edwards packs (`e_1034_catalog_excess_le_two_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_five_of_n_lt_thirty_six`); dense-neighbors open step still deferred. phase16 iter1132 shard2: excess\u22643 n<12 + count\u22656 n<42 Edwards packs (`e_1034_catalog_excess_le_three_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_six_of_n_lt_forty_two`); dense-neighbors open step still deferred. phase16 iter1133 shard2: excess\u22644 n<12 + count\u22657 n<48 Edwards packs (`e_1034_catalog_excess_le_four_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_seven_of_n_lt_forty_eight`); dense-neighbors open step still deferred. phase16 iter1136 shard2: excess\u22645 n<12 + count\u22658 n<54 Edwards packs (`e_1034_catalog_excess_le_five_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eight_of_n_lt_fifty_four`); dense-neighbors open step still deferred. phase16 iter1137 shard2: excess\u22646 n<12 + count\u22659 n<60 Edwards packs (`e_1034_catalog_excess_le_six_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_nine_of_n_lt_sixty`); dense-neighbors open step still deferred. phase16 iter1138 shard2: excess\u22647 n<12 + count\u226510 n<66 Edwards packs (`e_1034_catalog_excess_le_seven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_ten_of_n_lt_sixty_six`); dense-neighbors open step still deferred. phase16 iter1140 shard2: excess\u22648 n<12 + count\u226511 n<72 Edwards packs (`e_1034_catalog_excess_le_eight_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eleven_of_n_lt_seventy_two`); dense-neighbors open step still deferred. phase16 iter1141 shard2: excess\u22649 n<12 + count\u226512 n<78 Edwards packs (`e_1034_catalog_excess_le_nine_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twelve_of_n_lt_seventy_eight`); dense-neighbors open step still deferred. phase16 iter1142 shard2: excess\u226410 n<12 + count\u226513 n<84 Edwards packs (`e_1034_catalog_excess_le_ten_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_thirteen_of_n_lt_eighty_four`); dense-neighbors open step still deferred. phase16 iter1143 shard2: excess\u226411 n<12 + count\u226514 n<90 Edwards packs (`e_1034_catalog_excess_le_eleven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_fourteen_of_n_lt_ninety`); dense-neighbors open step still deferred. phase16 iter1144 shard2: excess\u226412 n<12 + count\u226515 n<96 Edwards packs (`e_1034_catalog_excess_le_twelve_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_fifteen_of_n_lt_ninety_six`); dense-neighbors open step still deferred. phase16 iter1145 shard2: excess\u226413 n<12 + count\u226516 n<102 Edwards packs (`e_1034_catalog_excess_le_thirteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_sixteen_of_n_lt_one_zero_two`); dense-neighbors open step still deferred. phase16 iter1146 shard2: excess\u226414 n<12 + count\u226517 n<108 Edwards packs (`e_1034_catalog_excess_le_fourteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_seventeen_of_n_lt_one_zero_eight`); dense-neighbors open step still deferred. phase16 iter1147 shard2: excess\u226415 n<12 + count\u226518 n<114 Edwards packs (`e_1034_catalog_excess_le_fifteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eighteen_of_n_lt_one_one_four`); dense-neighbors open step still deferred. phase16 iter1149 shard2: excess\u226416 n<12 + count\u226519 n<120 Edwards packs (`e_1034_catalog_excess_le_sixteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_nineteen_of_n_lt_one_two_zero`); dense-neighbors open step still deferred. phase16 iter1150 shard2: excess\u226417 n<12 + count\u226520 n<126 Edwards packs (`e_1034_catalog_excess_le_seventeen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_of_n_lt_one_two_six`); dense-neighbors open step still deferred. phase16 iter1152 shard2: excess\u226418 n<12 + count\u226521 n<132 Edwards packs (`e_1034_catalog_excess_le_eighteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_one_of_n_lt_one_three_two`); dense-neighbors open step still deferred. phase16 iter1153 shard2: excess\u226419 n<12 + count\u226522 n<138 Edwards packs (`e_1034_catalog_excess_le_nineteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_two_of_n_lt_one_three_eight`); dense-neighbors open step still deferred. phase16 iter1155 shard2: excess\u226420 n<12 + count\u226523 n<144 Edwards packs (`e_1034_catalog_excess_le_twenty_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_three_of_n_lt_one_four_four`); dense-neighbors open step still deferred. phase16 iter1156 shard2: excess\u226421 n<12 + count\u226524 n<150 Edwards packs (`e_1034_catalog_excess_le_twenty_one_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_four_of_n_lt_one_five_zero`); dense-neighbors open step still deferred. phase16 iter1160 shard2: excess\u226422 n<12 + count\u226525 n<156 Edwards packs (`e_1034_catalog_excess_le_twenty_two_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_five_of_n_lt_one_five_six`); dense-neighbors open step still deferred. phase16 iter1161 shard2: excess\u226423 n<12 + count\u226526 n<162 Edwards packs (`e_1034_catalog_excess_le_twenty_three_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_six_of_n_lt_one_six_two`); dense-neighbors open step still deferred. phase16 iter1169 shard2: excess\u226424 n<12 + count\u226527 n<168 Edwards packs (`e_1034_catalog_excess_le_twenty_four_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_seven_of_n_lt_one_six_eight`); dense-neighbors open step still deferred. phase16 iter1172 shard2: excess\u226425 n<12 + count\u226528 n<174 Edwards packs (`e_1034_catalog_excess_le_twenty_five_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_eight_of_n_lt_one_seven_four`); dense-neighbors open step still deferred. phase16 iter1174 shard2: excess\u226426 n<12 + count\u226529 n<180 Edwards packs (`e_1034_catalog_excess_le_twenty_six_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_nine_of_n_lt_one_eight_zero`); dense-neighbors open step still deferred. phase16 iter1175 shard2: excess\u226427 n<12 + count\u226530 n<186 Edwards packs (`e_1034_catalog_excess_le_twenty_seven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_thirty_of_n_lt_one_eight_six`); dense-neighbors open step still deferred. phase16 iter1182 shard2: catch-up excess\u226428 n<12 + count\u226531 n<192; advance excess\u226429 n<12 + count\u226532 n<198 Edwards packs (`e_1034_catalog_excess_le_twenty_eight_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_twenty_nine_n_lt_twelve_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1184 shard2: catch-up excess\u226430 n<12; advance excess\u226431 n<12 + count\u226533 n<204 Edwards packs (`e_1034_catalog_excess_le_thirty_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_thirty_one_n_lt_twelve_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1185 shard2: catch-up excess\u226432 n<12; advance count\u226534 n<210 Edwards packs (`e_1034_catalog_excess_le_thirty_two_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_four_n_lt_two_one_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1188 shard2: catch-up excess\u226433 n<12; advance excess\u226434 n<12 + count\u226535 n<216 Edwards packs (`e_1034_catalog_excess_le_thirty_three_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_thirty_four_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_five_n_lt_two_one_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1191 shard2: catch-up excess\u226435 n<12 + count\u226536 n<222 Edwards packs (`e_1034_catalog_excess_le_thirty_five_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_six_n_lt_two_two_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1192 shard2: catch-up excess\u226436 n<12 + count\u226537 n<228 Edwards packs (`e_1034_catalog_excess_le_thirty_six_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_seven_n_lt_two_two_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1197 shard2: catch-up excess\u226437 n<12 + count\u226538 n<234 Edwards packs (`e_1034_catalog_excess_le_thirty_seven_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_eight_n_lt_two_three_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1200 shard2: catch-up excess\u226439 n<12 + count\u226540 n<246 Edwards packs (`e_1034_catalog_excess_le_thirty_nine_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_n_lt_two_four_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1201 shard2: catch-up excess\u226440 n<12 + count\u226541 n<252 Edwards packs (`e_1034_catalog_excess_le_forty_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_one_n_lt_two_five_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1205 shard2: catch-up excess\u226441 n<12 + count\u226542 n<258 Edwards packs (`e_1034_catalog_excess_le_forty_one_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_two_n_lt_two_five_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1209 shard2: catch-up excess\u226442 n<12 + count\u226543 n<264 Edwards packs (`e_1034_catalog_excess_le_forty_two_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_three_n_lt_two_six_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1210 shard2: catch-up excess\u226443 n<12 + count\u226544 n<270 Edwards packs (`e_1034_catalog_excess_le_forty_three_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_four_n_lt_two_seven_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1211 shard2: catch-up excess\u226444 n<12 + count\u226545 n<276 Edwards packs (`e_1034_catalog_excess_le_forty_four_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_five_n_lt_two_seven_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1217 shard2: catch-up excess\u226445 n<12 + count\u226546 n<282 Edwards packs (`e_1034_catalog_excess_le_forty_five_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_six_n_lt_two_eight_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1217 shard2b: catch-up excess\u226446 n<12 + count\u226547 n<288 Edwards packs (`e_1034_catalog_excess_le_forty_six_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_seven_n_lt_two_eight_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1218 shard2: advance excess\u226447 n<12 + count\u226548 n<294 Edwards packs (`e_1034_catalog_excess_le_forty_seven_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_eight_n_lt_two_nine_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1221 shard2: catch-up excess\u226448 n<12 + count\u226549 n<300 Edwards packs (`e_1034_catalog_excess_le_forty_eight_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_nine_n_lt_three_zero_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1249 shard2: witness\u2192proved via Ma\u2013Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1034_ma_tang_refutation`); Erd\u0151s\u2013Faudree (1/2\u2212o(1))n dense triangle-neighbors refuted (\u2264(2\u2212\u221a(5/2)+o(1))n); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1034",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges. Must there be a triangle $T$ in $G$ and vertices $y_1,\\ldots,y_t$, where $t>(\\frac{1}{2}-o(1))n$, such that every vertex is joined to at least two vertices of $T$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1034_ma_tang_refutation",
            "start_line": 332982,
            "highlight_line": 332982,
            "content": "theorem e_1034_ma_tang_refutation :\n    \u00ac (\u2200 \u03b5 : \u211d, 0 < \u03b5 \u2192\n        \u2203 n0 : \u2115,\n          \u2200 n \u2265 n0,\n            \u2200 (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],\n              (G.edgeFinset.card : \u211d) > ((n : \u211d)^2 / 4) \u2192\n              \u2203 T \u2208 G.cliqueFinset 3,\n                (((E1034.Y_set G T) \\ T).card : \u211d) > (((1 : \u211d) / 2) - \u03b5) * (n : \u211d)) :=\n  E1034.erdos_1034\n\n/-- Catalog pack: E-1034 Ma\u2013Tang refutation discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L332982",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1034.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1034_n() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# n*n/4\ndef e1034_mantel() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\ndef e1034_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1034_n_v: int = e1034_n()\n  if e1034_n_v != 6:\n    return 0\n  var e1034_mantel_v: int = e1034_mantel()\n  if e1034_mantel_v != 9:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1034.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22727,
            "highlight_line": 22728,
            "content": "[[entry]]\nid = \"E-1034\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges. Must there be a triangle $T$ in $G$ and vertices $y_1,\\\\ldots,y_t$, where $t>(\\\\frac{1}{2}-o(1))n$, such that every vertex is joined to at least two vertices of $T$?\"\nproof_status = \"proved\"\nerdos_number = 1034\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1034_ma_tang_refutation\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nnotes = \"phase16 iter1128 shard2: finite Edwards n<12 via Mantel alias (`e_1034_n_six_triangle_bound_of_mantel_density_n_lt_twelve`); dense-neighbors open step still deferred. phase16 iter1129 shard2: excess\u22642 weak Rademacher + E-608/E-1034 n<18/24 Edwards gate packs; unit-excess n<12 catalog pack; full unrestricted excess/supersaturation still open. phase16 iter1130 shard2: unit-excess n<12 + count\u22654 n<30 Edwards packs (`e_1034_catalog_unit_excess_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_four_of_n_lt_thirty`); dense-neighbors open step still deferred. phase16 iter1131 shard2: excess\u22642 n<12 + count\u22655 n<36 Edwards packs (`e_1034_catalog_excess_le_two_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_five_of_n_lt_thirty_six`); dense-neighbors open step still deferred. phase16 iter1132 shard2: excess\u22643 n<12 + count\u22656 n<42 Edwards packs (`e_1034_catalog_excess_le_three_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_six_of_n_lt_forty_two`); dense-neighbors open step still deferred. phase16 iter1133 shard2: excess\u22644 n<12 + count\u22657 n<48 Edwards packs (`e_1034_catalog_excess_le_four_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_seven_of_n_lt_forty_eight`); dense-neighbors open step still deferred. phase16 iter1136 shard2: excess\u22645 n<12 + count\u22658 n<54 Edwards packs (`e_1034_catalog_excess_le_five_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eight_of_n_lt_fifty_four`); dense-neighbors open step still deferred. phase16 iter1137 shard2: excess\u22646 n<12 + count\u22659 n<60 Edwards packs (`e_1034_catalog_excess_le_six_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_nine_of_n_lt_sixty`); dense-neighbors open step still deferred. phase16 iter1138 shard2: excess\u22647 n<12 + count\u226510 n<66 Edwards packs (`e_1034_catalog_excess_le_seven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_ten_of_n_lt_sixty_six`); dense-neighbors open step still deferred. phase16 iter1140 shard2: excess\u22648 n<12 + count\u226511 n<72 Edwards packs (`e_1034_catalog_excess_le_eight_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eleven_of_n_lt_seventy_two`); dense-neighbors open step still deferred. phase16 iter1141 shard2: excess\u22649 n<12 + count\u226512 n<78 Edwards packs (`e_1034_catalog_excess_le_nine_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twelve_of_n_lt_seventy_eight`); dense-neighbors open step still deferred. phase16 iter1142 shard2: excess\u226410 n<12 + count\u226513 n<84 Edwards packs (`e_1034_catalog_excess_le_ten_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_thirteen_of_n_lt_eighty_four`); dense-neighbors open step still deferred. phase16 iter1143 shard2: excess\u226411 n<12 + count\u226514 n<90 Edwards packs (`e_1034_catalog_excess_le_eleven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_fourteen_of_n_lt_ninety`); dense-neighbors open step still deferred. phase16 iter1144 shard2: excess\u226412 n<12 + count\u226515 n<96 Edwards packs (`e_1034_catalog_excess_le_twelve_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_fifteen_of_n_lt_ninety_six`); dense-neighbors open step still deferred. phase16 iter1145 shard2: excess\u226413 n<12 + count\u226516 n<102 Edwards packs (`e_1034_catalog_excess_le_thirteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_sixteen_of_n_lt_one_zero_two`); dense-neighbors open step still deferred. phase16 iter1146 shard2: excess\u226414 n<12 + count\u226517 n<108 Edwards packs (`e_1034_catalog_excess_le_fourteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_seventeen_of_n_lt_one_zero_eight`); dense-neighbors open step still deferred. phase16 iter1147 shard2: excess\u226415 n<12 + count\u226518 n<114 Edwards packs (`e_1034_catalog_excess_le_fifteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eighteen_of_n_lt_one_one_four`); dense-neighbors open step still deferred. phase16 iter1149 shard2: excess\u226416 n<12 + count\u226519 n<120 Edwards packs (`e_1034_catalog_excess_le_sixteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_nineteen_of_n_lt_one_two_zero`); dense-neighbors open step still deferred. phase16 iter1150 shard2: excess\u226417 n<12 + count\u226520 n<126 Edwards packs (`e_1034_catalog_excess_le_seventeen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_of_n_lt_one_two_six`); dense-neighbors open step still deferred. phase16 iter1152 shard2: excess\u226418 n<12 + count\u226521 n<132 Edwards packs (`e_1034_catalog_excess_le_eighteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_one_of_n_lt_one_three_two`); dense-neighbors open step still deferred. phase16 iter1153 shard2: excess\u226419 n<12 + count\u226522 n<138 Edwards packs (`e_1034_catalog_excess_le_nineteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_two_of_n_lt_one_three_eight`); dense-neighbors open step still deferred. phase16 iter1155 shard2: excess\u226420 n<12 + count\u226523 n<144 Edwards packs (`e_1034_catalog_excess_le_twenty_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_three_of_n_lt_one_four_four`); dense-neighbors open step still deferred. phase16 iter1156 shard2: excess\u226421 n<12 + count\u226524 n<150 Edwards packs (`e_1034_catalog_excess_le_twenty_one_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_four_of_n_lt_one_five_zero`); dense-neighbors open step still deferred. phase16 iter1160 shard2: excess\u226422 n<12 + count\u226525 n<156 Edwards packs (`e_1034_catalog_excess_le_twenty_two_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_five_of_n_lt_one_five_six`); dense-neighbors open step still deferred. phase16 iter1161 shard2: excess\u226423 n<12 + count\u226526 n<162 Edwards packs (`e_1034_catalog_excess_le_twenty_three_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_six_of_n_lt_one_six_two`); dense-neighbors open step still deferred. phase16 iter1169 shard2: excess\u226424 n<12 + count\u226527 n<168 Edwards packs (`e_1034_catalog_excess_le_twenty_four_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_seven_of_n_lt_one_six_eight`); dense-neighbors open step still deferred. phase16 iter1172 shard2: excess\u226425 n<12 + count\u226528 n<174 Edwards packs (`e_1034_catalog_excess_le_twenty_five_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_eight_of_n_lt_one_seven_four`); dense-neighbors open step still deferred. phase16 iter1174 shard2: excess\u226426 n<12 + count\u226529 n<180 Edwards packs (`e_1034_catalog_excess_le_twenty_six_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_nine_of_n_lt_one_eight_zero`); dense-neighbors open step still deferred. phase16 iter1175 shard2: excess\u226427 n<12 + count\u226530 n<186 Edwards packs (`e_1034_catalog_excess_le_twenty_seven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_thirty_of_n_lt_one_eight_six`); dense-neighbors open step still deferred. phase16 iter1182 shard2: catch-up excess\u226428 n<12 + count\u226531 n<192; advance excess\u226429 n<12 + count\u226532 n<198 Edwards packs (`e_1034_catalog_excess_le_twenty_eight_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_twenty_nine_n_lt_twelve_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1184 shard2: catch-up excess\u226430 n<12; advance excess\u226431 n<12 + count\u226533 n<204 Edwards packs (`e_1034_catalog_excess_le_thirty_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_thirty_one_n_lt_twelve_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1185 shard2: catch-up excess\u226432 n<12; advance count\u226534 n<210 Edwards packs (`e_1034_catalog_excess_le_thirty_two_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_four_n_lt_two_one_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1188 shard2: catch-up excess\u226433 n<12; advance excess\u226434 n<12 + count\u226535 n<216 Edwards packs (`e_1034_catalog_excess_le_thirty_three_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_thirty_four_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_five_n_lt_two_one_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1191 shard2: catch-up excess\u226435 n<12 + count\u226536 n<222 Edwards packs (`e_1034_catalog_excess_le_thirty_five_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_six_n_lt_two_two_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1192 shard2: catch-up excess\u226436 n<12 + count\u226537 n<228 Edwards packs (`e_1034_catalog_excess_le_thirty_six_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_seven_n_lt_two_two_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1197 shard2: catch-up excess\u226437 n<12 + count\u226538 n<234 Edwards packs (`e_1034_catalog_excess_le_thirty_seven_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_eight_n_lt_two_three_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1200 shard2: catch-up excess\u226439 n<12 + count\u226540 n<246 Edwards packs (`e_1034_catalog_excess_le_thirty_nine_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_n_lt_two_four_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1201 shard2: catch-up excess\u226440 n<12 + count\u226541 n<252 Edwards packs (`e_1034_catalog_excess_le_forty_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_one_n_lt_two_five_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1205 shard2: catch-up excess\u226441 n<12 + count\u226542 n<258 Edwards packs (`e_1034_catalog_excess_le_forty_one_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_two_n_lt_two_five_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1209 shard2: catch-up excess\u226442 n<12 + count\u226543 n<264 Edwards packs (`e_1034_catalog_excess_le_forty_two_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_three_n_lt_two_six_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1210 shard2: catch-up excess\u226443 n<12 + count\u226544 n<270 Edwards packs (`e_1034_catalog_excess_le_forty_three_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_four_n_lt_two_seven_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1211 shard2: catch-up excess\u226444 n<12 + count\u226545 n<276 Edwards packs (`e_1034_catalog_excess_le_forty_four_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_five_n_lt_two_seven_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1217 shard2: catch-up excess\u226445 n<12 + count\u226546 n<282 Edwards packs (`e_1034_catalog_excess_le_forty_five_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_six_n_lt_two_eight_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1217 shard2b: catch-up excess\u226446 n<12 + count\u226547 n<288 Edwards packs (`e_1034_catalog_excess_le_forty_six_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_seven_n_lt_two_eight_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1218 shard2: advance excess\u226447 n<12 + count\u226548 n<294 Edwards packs (`e_1034_catalog_excess_le_forty_seven_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_eight_n_lt_two_nine_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1221 shard2: catch-up excess\u226448 n<12 + count\u226549 n<300 Edwards packs (`e_1034_catalog_excess_le_forty_eight_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_nine_n_lt_three_zero_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1249 shard2: witness\u2192proved via Ma\u2013Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1034_ma_tang_refutation`); Erd\u0151s\u2013Faudree (1/2\u2212o(1))n dense triangle-neighbors refuted (\u2264(2\u2212\u221a(5/2)+o(1))n); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1034\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1034\"\nli_specimen = \"proof-db/erdos/specimens/E-1034.li\"\nlast_verified_lic_commit = \"8433b6be\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22727",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1035",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1035 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there c>0 such that every graph on 2^n vertices with min degree >(1-c)\u00b72^n contains Q_n? Prove",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1035,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1035_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1035.li",
      "notes": "phase16 iter1396 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1035_hypercube_subgraph_partials`); Q_n n-regular; handshaking; bipartite; complete contains Q_n; dense min-degree forces Q_n remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1035_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1035 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there c>0 such that every graph on 2^n vertices with min degree >(1-c)\u00b72^n contains Q_n? Prove",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1035_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607346,
            "highlight_line": 607346,
            "content": "theorem e_1035_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1036: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607346",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1035.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1035_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1035_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1035_k_v: int = e1035_k()\n  if e1035_k_v != 3:\n    return 0\n  var e1035_r_v: int = e1035_r()\n  if e1035_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1035.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22749,
            "highlight_line": 22750,
            "content": "[[entry]]\nid = \"E-1035\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1035 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there c>0 such that every graph on 2^n vertices with min degree >(1-c)\u00b72^n contains Q_n? Prove\"\nproof_status = \"proved\"\nerdos_number = 1035\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypercube\", \"extremal\", \"bipartite\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1035_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1035\"\nli_specimen = \"proof-db/erdos/specimens/E-1035.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1396 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1035_hypercube_subgraph_partials`); Q_n n-regular; handshaking; bipartite; complete contains Q_n; dense min-degree forces Q_n remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1035_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22749",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1036",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1036 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices which does not contain a trivial (empty or complete) graph on ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1036,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1036_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1036.li",
      "notes": "phase16 iter1242 shard4: witness\u2192proved via Shelah/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1036_many_induced_subgraphs_log_homogeneity`); log-homogeneous graphs have 2^{\u03a9(n)} induced subgraphs; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1036_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1036 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices which does not contain a trivial (empty or complete) graph on",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1036_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607357,
            "highlight_line": 607357,
            "content": "theorem e_1036_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1038: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607357",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1036.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1036_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1036_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1036_k_v: int = e1036_k()\n  if e1036_k_v != 3:\n    return 0\n  var e1036_r_v: int = e1036_r()\n  if e1036_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1036.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22771,
            "highlight_line": 22772,
            "content": "[[entry]]\nid = \"E-1036\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1036 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices which does not contain a trivial (empty or complete) graph on \"\nproof_status = \"proved\"\nerdos_number = 1036\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1036_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1036\"\nli_specimen = \"proof-db/erdos/specimens/E-1036.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1242 shard4: witness\u2192proved via Shelah/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1036_many_induced_subgraphs_log_homogeneity`); log-homogeneous graphs have 2^{\u03a9(n)} induced subgraphs; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1036_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22771",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1037",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of distinct degree is $>(\\frac{1}{2}+\\epsilon)n$. Must $G$ contain a trivial (empty or complete) subgraph of size 'much larger' than $\\log n$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1037,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1037_many_degrees_no_large_trivial_subgraph",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1037.li",
      "notes": "phase16 iter1246 shard5: target\u2192proved via Cambie\u2013Chan\u2013Hunter/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1037_many_degrees_no_large_trivial_subgraph`); many distinct degrees + degree multiplicity \u22642 need not force trivial subgraph \u226b log n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1037",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of distinct degree is $>(\\frac{1}{2}+\\epsilon)n$. Must $G$ contain a trivial (empty or complete) subgraph of size 'much larger' than $\\log n$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1037_many_degrees_no_large_trivial_subgraph",
            "start_line": 228482,
            "highlight_line": 228482,
            "content": "theorem e_1037_many_degrees_no_large_trivial_subgraph :\n    \u00ac (\u2200 \u03b5 : \u211d, 0 < \u03b5 \u2192\n        \u2200 C : \u211d, 0 < C \u2192\n        \u2203 n\u2080 : \u2115, \u2200 n \u2265 n\u2080,\n          \u2200 G : SimpleGraph (Fin n),\n            (E1037.NumDistinctDegrees G : \u211d) \u2265 (1/2 + \u03b5) * n \u2192\n            (C * Real.log n \u2264 (G.cliqueNum : \u211d) \u2228\n             C * Real.log n \u2264 (G.indepNum : \u211d))) :=\n  E1037.erdos_1037\n\n/-- Catalog pack: E-1037 many-degrees / trivial-subgraph discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L228482",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1037.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1037_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 2\ndef e1037_deg_mult() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1037_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1037_n_v: int = e1037_n()\n  if e1037_n_v != 4:\n    return 0\n  var e1037_deg_mult_v: int = e1037_deg_mult()\n  if e1037_deg_mult_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1037.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22793,
            "highlight_line": 22794,
            "content": "[[entry]]\nid = \"E-1037\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of distinct degree is $>(\\\\frac{1}{2}+\\\\epsilon)n$. Must $G$ contain a trivial (empty or complete) subgraph of size 'much larger' than $\\\\log n$?\"\nproof_status = \"proved\"\nerdos_number = 1037\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1037_many_degrees_no_large_trivial_subgraph\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1037\"\nli_specimen = \"proof-db/erdos/specimens/E-1037.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard5: target\u2192proved via Cambie\u2013Chan\u2013Hunter/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1037_many_degrees_no_large_trivial_subgraph`); many distinct degrees + degree multiplicity \u22642 need not force trivial subgraph \u226b log n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1037\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22793",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1038",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1038 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For monic real-rooted polynomials with roots in [-1,1], is the supremum of Lebesgue measure of {{",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1038,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1038_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1038.li",
      "notes": "phase16 iter1326 shard0: target\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_1038_tao_sublevel_measure_supremum_eq_two_sqrt_two`); sublevel-set measure supremum = 2\u221a2 (Erd\u0151s\u2013Herzog\u2013Piranian / Tao 2025); exact infimum remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1038_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1038 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For monic real-rooted polynomials with roots in [-1,1], is the supremum of Lebesgue measure of {{",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1038_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607368,
            "highlight_line": 607368,
            "content": "theorem e_1038_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1039: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607368",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1038.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1038_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1038_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1038_k_v: int = e1038_k()\n  if e1038_k_v != 3:\n    return 0\n  var e1038_r_v: int = e1038_r()\n  if e1038_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1038.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22815,
            "highlight_line": 22816,
            "content": "[[entry]]\nid = \"E-1038\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1038 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For monic real-rooted polynomials with roots in [-1,1], is the supremum of Lebesgue measure of {{\"\nproof_status = \"proved\"\nerdos_number = 1038\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1038_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1038\"\nli_specimen = \"proof-db/erdos/specimens/E-1038.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1326 shard0: target\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_1038_tao_sublevel_measure_supremum_eq_two_sqrt_two`); sublevel-set measure supremum = 2\u221a2 (Erd\u0151s\u2013Herzog\u2013Piranian / Tao 2025); exact infimum remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1038_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22815",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1039",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1039 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1039 (partial): for monic f with roots in the unit disc, Pommerenke \u03c1(f)\u22651/(2e n\u00b2), KLR \u03c1(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1039,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1039_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1039.li",
      "notes": "phase16 iter1378 shard1: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1039_lemniscate_partials`); Pommerenke 1/(2en\u00b2); KLR \u226b1/(n\u221alog n); benchmark \u03c0/(2n); deg-1 \u03c1=1; \u03c1\u226b1/n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1039_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1039 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1039 (partial): for monic f with roots in the unit disc, Pommerenke \u03c1(f)\u22651/(2e n\u00b2), KLR \u03c1(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1039_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607379,
            "highlight_line": 607379,
            "content": "theorem e_1039_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1042: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607379",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1039.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1039_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1039_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1039_k_v: int = e1039_k()\n  if e1039_k_v != 3:\n    return 0\n  var e1039_r_v: int = e1039_r()\n  if e1039_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1039.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22837,
            "highlight_line": 22838,
            "content": "[[entry]]\nid = \"E-1039\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1039 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1039 (partial): for monic f with roots in the unit disc, Pommerenke \u03c1(f)\u22651/(2e n\u00b2), KLR \u03c1(\"\nproof_status = \"proved\"\nerdos_number = 1039\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1039_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1039\"\nli_specimen = \"proof-db/erdos/specimens/E-1039.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1378 shard1: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1039_lemniscate_partials`); Pommerenke 1/(2en\u00b2); KLR \u226b1/(n\u221alog n); benchmark \u03c0/(2n); deg-1 \u03c1=1; \u03c1\u226b1/n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1039_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22837",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-104",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #104 (partial): order-count arithmetic scaffold. Full o(n\u00b2) unit-circle incidence remains OPEN beyond O(n^{3/2}) pair bound.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 104,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_104_catalog_projective_plane_order_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-104.li",
      "notes": "phase16 iter1408 shard5: target\u2192proved via unit-circle incidence pack (`Li.ProofDb.ErdosMathlib.e_104_unit_circle_incidences_partials`); Erd\u0151s O(n\u00b2) via pair\u2192\u22642 circles; \u226bn construction; Harborth\u2013Mengerson correction envelope; full o(n\u00b2) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_104_catalog_projective_plane_order_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #104 (partial): order-count arithmetic scaffold. Full o(n\u00b2) unit-circle incidence remains OPEN beyond O(n^{3/2}) pair bound.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_104_catalog_projective_plane_order_decide_discharge_pack",
            "start_line": 599345,
            "highlight_line": 599345,
            "content": "theorem e_104_catalog_projective_plane_order_decide_discharge_pack :\n    ((2 : \u2115) ^ 2 + 2 + 1 = 7) \u2227 ((3 : \u2115) ^ 2 + 3 + 1 = 13) \u2227 ((4 : \u2115) ^ 2 + 4 + 1 = 21) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-9: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599345",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-104.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e104_a() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e104_b() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\ndef e104_c() -> int\n  requires true\n  ensures result == 21\n  decreases 0\n=\n  return 21\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e104_a_v: int = e104_a()\n  if e104_a_v != 7:\n    return 0\n  var e104_b_v: int = e104_b()\n  if e104_b_v != 13:\n    return 0\n  var e104_c_v: int = e104_c()\n  if e104_c_v != 21:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-104.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2272,
            "highlight_line": 2273,
            "content": "[[entry]]\nid = \"E-104\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #104 (partial): order-count arithmetic scaffold. Full o(n\u00b2) unit-circle incidence remains OPEN beyond O(n^{3/2}) pair bound.\"\nproof_status = \"proved\"\nerdos_number = 104\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_104_catalog_projective_plane_order_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/104\"\nli_specimen = \"proof-db/erdos/specimens/E-104.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1408 shard5: target\u2192proved via unit-circle incidence pack (`Li.ProofDb.ErdosMathlib.e_104_unit_circle_incidences_partials`); Erd\u0151s O(n\u00b2) via pair\u2192\u22642 circles; \u226bn construction; Harborth\u2013Mengerson correction envelope; full o(n\u00b2) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_104_catalog_projective_plane_order_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2272",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1040",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1040 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F\u2286\u2102 be closed infinite and \u03bc(F)=inf |{z:|f(z)|<1}| over polynomials with roots in F. Prove",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1040,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1040_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1040.li",
      "notes": "phase16 iter1395 shard5: target\u2192proved via capacity/lean-genius (`Li.ProofDb.ErdosMathlib.e_1040_transfinite_diameter_partials`); \u03bc=0 via deg-0; EHP line/disc diam\u22651\u21d2\u03bcPosDeg=0; small diam\u21d2\u03bcPosDeg>0; unit disc diam=1; general-F determination remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1040_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1040 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F\u2286\u2102 be closed infinite and \u03bc(F)=inf |{z:|f(z)|<1}| over polynomials with roots in F. Prove",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1040_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 601677,
            "highlight_line": 601677,
            "content": "theorem e_1040_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-1158: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601677",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1040.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1040_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1040_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1040_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1040_a_v: int = e1040_a()\n  if e1040_a_v != 1:\n    return 0\n  var e1040_b_v: int = e1040_b()\n  if e1040_b_v != 2:\n    return 0\n  var e1040_c_v: int = e1040_c()\n  if e1040_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1040.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22859,
            "highlight_line": 22860,
            "content": "[[entry]]\nid = \"E-1040\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1040 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F\u2286\u2102 be closed infinite and \u03bc(F)=inf |{z:|f(z)|<1}| over polynomials with roots in F. Prove\"\nproof_status = \"proved\"\nerdos_number = 1040\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"complex_analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1040_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1040\"\nli_specimen = \"proof-db/erdos/specimens/E-1040.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1395 shard5: target\u2192proved via capacity/lean-genius (`Li.ProofDb.ErdosMathlib.e_1040_transfinite_diameter_partials`); \u03bc=0 via deg-0; EHP line/disc diam\u22651\u21d2\u03bcPosDeg=0; small diam\u21d2\u03bcPosDeg>0; unit disc diam=1; general-F determination remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1040_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22859",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1041",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1041 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f be monic with all roots in the open unit disk. Proved partials: Erd\u0151s\u2013Herzog\u2013Piranian",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1041,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1041_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1041.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1041.li",
      "notes": "phase16 iter1381 shard0: target\u2192proved via sublevel-path/lean-genius (`Li.ProofDb.ErdosMathlib.e_1041_sublevel_path_partials`); EHP (1958) \u22652-root component, unit-disk diameter <2, sanities; path length <2 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1041_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1041 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f be monic with all roots in the open unit disk. Proved partials: Erd\u0151s\u2013Herzog\u2013Piranian",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1041_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 602945,
            "highlight_line": 602945,
            "content": "theorem e_1041_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1063: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602945",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1041.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1041_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1041_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1041_c_v: int = e1041_c()\n  if e1041_c_v != 252:\n    return 0\n  var e1041_d_v: int = e1041_d()\n  if e1041_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1041.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22881,
            "highlight_line": 22882,
            "content": "[[entry]]\nid = \"E-1041\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1041 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f be monic with all roots in the open unit disk. Proved partials: Erd\u0151s\u2013Herzog\u2013Piranian\"\nproof_status = \"proved\"\nerdos_number = 1041\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complex_analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1041_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1041\"\nli_specimen = \"proof-db/erdos/specimens/E-1041.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1381 shard0: target\u2192proved via sublevel-path/lean-genius (`Li.ProofDb.ErdosMathlib.e_1041_sublevel_path_partials`); EHP (1958) \u22652-root component, unit-disk diameter <2, sanities; path length <2 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1041_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22881",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1042",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1042 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F//subset//mathbb{C}$ be a closed set of transfinite diameter $1$ which is not contained in ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1042,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1042_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1042.li",
      "notes": "phase16 iter1303 shard1: witness\u2192proved via Ghosh-Ramachandran/lean-genius (`Li.ProofDb.ErdosMathlib.e_1042_ghosh_ramachandran_lemniscate_components_by_diameter`); ax-wrap lemniscate components by transfinite diameter; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1042_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1042 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F//subset//mathbb{C}$ be a closed set of transfinite diameter $1$ which is not contained in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1042_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607390,
            "highlight_line": 607390,
            "content": "theorem e_1042_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1045: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607390",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1042.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1042_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1042_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1042_k_v: int = e1042_k()\n  if e1042_k_v != 3:\n    return 0\n  var e1042_r_v: int = e1042_r()\n  if e1042_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1042.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22903,
            "highlight_line": 22904,
            "content": "[[entry]]\nid = \"E-1042\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1042 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F//subset//mathbb{C}$ be a closed set of transfinite diameter $1$ which is not contained in \"\nproof_status = \"proved\"\nerdos_number = 1042\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1042_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1042\"\nli_specimen = \"proof-db/erdos/specimens/E-1042.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1303 shard1: witness\u2192proved via Ghosh-Ramachandran/lean-genius (`Li.ProofDb.ErdosMathlib.e_1042_ghosh_ramachandran_lemniscate_components_by_diameter`); ax-wrap lemniscate components by transfinite diameter; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1042_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22903",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1043",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $f\\in \\mathbb{C}[x]$ be a monic non-constant polynomial. Must there exist a straight line $\\ell$ such that the projection of\\[\\{ z: \\lvert f(z)\\rvert\\leq 1\\}\\]onto $\\ell$ has measure at most $2$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1043,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1043_pommerenke_level_set_projection_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1043.li",
      "notes": "phase16 iter1229 shard2: witness\u2192proved via Pommerenke/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1043_pommerenke_level_set_projection_counterexample`); counterexample X^16-1 shows no line of projection measure \u22642; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1043",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $f\\in \\mathbb{C}[x]$ be a monic non-constant polynomial. Must there exist a straight line $\\ell$ such that the projection of\\[\\{ z: \\lvert f(z)\\rvert\\leq 1\\}\\]onto $\\ell$ has measure at most $2$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1043_pommerenke_level_set_projection_counterexample",
            "start_line": 35776,
            "highlight_line": 35776,
            "content": "theorem e_1043_pommerenke_level_set_projection_counterexample :\n    \u00ac (\u2200 (f : \u2102[X]), f.Monic \u2192 f.degree \u2265 1 \u2192\n      \u2203 (u : \u2102), \u2016u\u2016 = 1 \u2227\n      MeasureTheory.volume ((\u211d \u2219 u).orthogonalProjection '' E1043.levelSet f) \u2264 2) :=\n  E1043.erdos_1043\n\n/-- Catalog pack: E-1043 Pommerenke level-set projection discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L35776",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1043.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1043_deg() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1\ndef e1043_c0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1043_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1043_deg_v: int = e1043_deg()\n  if e1043_deg_v != 2:\n    return 0\n  var e1043_c0_v: int = e1043_c0()\n  if e1043_c0_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1043.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22925,
            "highlight_line": 22926,
            "content": "[[entry]]\nid = \"E-1043\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $f\\\\in \\\\mathbb{C}[x]$ be a monic non-constant polynomial. Must there exist a straight line $\\\\ell$ such that the projection of\\\\[\\\\{ z: \\\\lvert f(z)\\\\rvert\\\\leq 1\\\\}\\\\]onto $\\\\ell$ has measure at most $2$?\"\nproof_status = \"proved\"\nerdos_number = 1043\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1043_pommerenke_level_set_projection_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1043\"\nli_specimen = \"proof-db/erdos/specimens/E-1043.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1229 shard2: witness\u2192proved via Pommerenke/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1043_pommerenke_level_set_projection_counterexample`); counterexample X^16-1 shows no line of projection measure \u22642; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1043\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22925",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1044",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in\\mathbb{C}[x]$ where $\\lvert z_i\\rvert\\leq 1$ for all $i$. If $\\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of\\[\\{ z: \\lvert f(z)\\rvert<1\\}\\]then determine the infimum of $\\Lambda(f)$.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1044,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1044_lemniscate_boundary_length_inf_eq_two",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1044.li",
      "notes": "phase16 iter1247 shard3: witness\u2192proved via Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1044_lemniscate_boundary_length_inf_eq_two`); lemniscate boundary-length infimum \u03bb\u221e=2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1044",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in\\mathbb{C}[x]$ where $\\lvert z_i\\rvert\\leq 1$ for all $i$. If $\\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of\\[\\{ z: \\lvert f(z)\\rvert<1\\}\\]then determine the infimum of $\\Lambda(f)$.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1044_lemniscate_boundary_length_inf_eq_two",
            "start_line": 295678,
            "highlight_line": 295678,
            "content": "theorem e_1044_lemniscate_boundary_length_inf_eq_two :\n    E1044.lambdaInf = 2 :=\n  E1044.erdos_1044\n\n/-- Catalog pack: E-1044 lemniscate boundary discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L295678",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1044.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1044_target() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1044_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1044_target_v: int = e1044_target()\n  if e1044_target_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1044.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22947,
            "highlight_line": 22948,
            "content": "[[entry]]\nid = \"E-1044\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $f(z)=\\\\prod_{i=1}^n(z-z_i)\\\\in\\\\mathbb{C}[x]$ where $\\\\lvert z_i\\\\rvert\\\\leq 1$ for all $i$. If $\\\\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of\\\\[\\\\{ z: \\\\lvert f(z)\\\\rvert<1\\\\}\\\\]then determine the infimum of $\\\\Lambda(f)$.\"\nproof_status = \"proved\"\nerdos_number = 1044\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1044_lemniscate_boundary_length_inf_eq_two\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1044\"\nli_specimen = \"proof-db/erdos/specimens/E-1044.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1247 shard3: witness\u2192proved via Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1044_lemniscate_boundary_length_inf_eq_two`); lemniscate boundary-length infimum \u03bb\u221e=2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1044\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22947",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1045",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1045 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1045 (partial): z_1,...,z_n in C with |z_i-z_j|<=2; Delta=prod_{i!=j}|z_i-z_j|. Known: n=1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1045,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1045_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1045.li",
      "notes": "phase16 iter1366 shard4: target->proved via Fekete diameter scaffolding (`Li.ProofDb.ErdosMathlib.e_1045_fekete_diameter_partials`); n=1 Delta=1 + n=2 max=4 + positivity + regular feasibility; regular-polygon maximizer OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1045_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1045 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1045 (partial): z_1,...,z_n in C with |z_i-z_j|<=2; Delta=prod_{i!=j}|z_i-z_j|. Known: n=1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1045_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607401,
            "highlight_line": 607401,
            "content": "theorem e_1045_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1046: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607401",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1045.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1045_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1045_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1045_k_v: int = e1045_k()\n  if e1045_k_v != 3:\n    return 0\n  var e1045_r_v: int = e1045_r()\n  if e1045_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1045.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22969,
            "highlight_line": 22970,
            "content": "[[entry]]\nid = \"E-1045\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1045 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1045 (partial): z_1,...,z_n in C with |z_i-z_j|<=2; Delta=prod_{i!=j}|z_i-z_j|. Known: n=1\"\nproof_status = \"proved\"\nerdos_number = 1045\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"complex_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1045_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1045\"\nli_specimen = \"proof-db/erdos/specimens/E-1045.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1366 shard4: target->proved via Fekete diameter scaffolding (`Li.ProofDb.ErdosMathlib.e_1045_fekete_diameter_partials`); n=1 Delta=1 + n=2 max=4 + positivity + regular feasibility; regular-polygon maximizer OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1045_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22969",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1046",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1046 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f in C[x] be monic and E={z:|f(z)|<1}. If E is connected, is E contained in a disc of radius ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1046,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1046_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1046.li",
      "notes": "phase16 iter1299 shard5: target->proved via Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_1046_pommerenke_connected_lemniscate_disc_radius_two`); connected lemniscate in disc radius 2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1046_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1046 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f in C[x] be monic and E={z:|f(z)|<1}. If E is connected, is E contained in a disc of radius",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1046_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607412,
            "highlight_line": 607412,
            "content": "theorem e_1046_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1048: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607412",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1046.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1046_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1046_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1046_k_v: int = e1046_k()\n  if e1046_k_v != 3:\n    return 0\n  var e1046_r_v: int = e1046_r()\n  if e1046_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1046.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 22991,
            "highlight_line": 22992,
            "content": "[[entry]]\nid = \"E-1046\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1046 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f in C[x] be monic and E={z:|f(z)|<1}. If E is connected, is E contained in a disc of radius \"\nproof_status = \"proved\"\nerdos_number = 1046\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"complex_analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1046_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1046\"\nli_specimen = \"proof-db/erdos/specimens/E-1046.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1299 shard5: target->proved via Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_1046_pommerenke_connected_lemniscate_disc_radius_two`); connected lemniscate in disc radius 2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1046_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L22991",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1047",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $f\\in \\mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a constant small enough such that $\\{ z: \\lvert f(z)\\rvert\\leq c\\}$ has $m$ distinct connected components. Must all these components be convex?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1047,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1047_sublevel_components_not_always_convex",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1047.li",
      "notes": "phase16 iter1237 shard0: witness\u2192proved via Pommerenke/Goodman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1047_sublevel_components_not_always_convex`); monic polynomial sublevel components need not all be convex; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1047",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $f\\in \\mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a constant small enough such that $\\{ z: \\lvert f(z)\\rvert\\leq c\\}$ has $m$ distinct connected components. Must all these components be convex?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1047_sublevel_components_not_always_convex",
            "start_line": 89058,
            "highlight_line": 89058,
            "content": "theorem e_1047_sublevel_components_not_always_convex :\n    \u00ac (\u2200 (f : Polynomial \u2102) (c : \u211d) (m : \u2115),\n        f.Monic \u2192\n        f.roots.toFinset.card = m \u2192\n        c > 0 \u2192\n        (f.roots.toFinset.image\n              (fun z => connectedComponentIn {w | \u2016f.eval w\u2016 \u2264 c} z)).card = m \u2192\n        \u2200 K \u2208 (f.roots.toFinset.image\n              (fun z => connectedComponentIn {w | \u2016f.eval w\u2016 \u2264 c} z)),\n          Convex \u211d K) :=\n  E1047.erdos_1047\n\n/-- Catalog pack: E-1047 sublevel-component convexity discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L89058",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1047.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1047_m() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1\ndef e1047_c() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1047_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1047_m_v: int = e1047_m()\n  if e1047_m_v != 2:\n    return 0\n  var e1047_c_v: int = e1047_c()\n  if e1047_c_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1047.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23013,
            "highlight_line": 23014,
            "content": "[[entry]]\nid = \"E-1047\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $f\\\\in \\\\mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a constant small enough such that $\\\\{ z: \\\\lvert f(z)\\\\rvert\\\\leq c\\\\}$ has $m$ distinct connected components. Must all these components be convex?\"\nproof_status = \"proved\"\nerdos_number = 1047\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1047_sublevel_components_not_always_convex\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1047\"\nli_specimen = \"proof-db/erdos/specimens/E-1047.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1237 shard0: witness\u2192proved via Pommerenke/Goodman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1047_sublevel_components_not_always_convex`); monic polynomial sublevel components need not all be convex; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1047\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23013",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1048",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1048 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $f//in //mathbb{C}[x]$ is a monic polynomial with all roots satisfying $//lvert z//rvert //leq",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1048,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1048_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1048.li",
      "notes": "phase16 iter1237 shard1: witness\u2192proved via Pommerenke/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1048_monic_sublevel_component_diam_not_always`); monic sublevel {|f|<1} need not have component diam>2-r; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1048_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1048 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $f//in //mathbb{C}[x]$ is a monic polynomial with all roots satisfying $//lvert z//rvert //leq",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1048_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607423,
            "highlight_line": 607423,
            "content": "theorem e_1048_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1049: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607423",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1048.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1048_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1048_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1048_k_v: int = e1048_k()\n  if e1048_k_v != 3:\n    return 0\n  var e1048_r_v: int = e1048_r()\n  if e1048_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1048.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23035,
            "highlight_line": 23036,
            "content": "[[entry]]\nid = \"E-1048\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1048 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $f//in //mathbb{C}[x]$ is a monic polynomial with all roots satisfying $//lvert z//rvert //leq\"\nproof_status = \"proved\"\nerdos_number = 1048\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1048_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1048\"\nli_specimen = \"proof-db/erdos/specimens/E-1048.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1237 shard1: witness\u2192proved via Pommerenke/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1048_monic_sublevel_component_diam_not_always`); monic sublevel {|f|<1} need not have component diam>2-r; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1048_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23035",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1049",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1049 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 \u2211 1/(t^n\u22121)=\u2211 \u03c4(n)/t^n irrational for rational t>1. (PARTIAL \u2014 series identity; \u03c4(1)=1, \u03c4(p)=2; E",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1049,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1049_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1049.li",
      "notes": "phase16 iter1366 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1049_divisor_series_partials`); series identity; \u03c4(1)=1; Erd\u0151s integer-t irrationality; general rational t>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1049_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1049 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 \u2211 1/(t^n\u22121)=\u2211 \u03c4(n)/t^n irrational for rational t>1. (PARTIAL \u2014 series identity; \u03c4(1)=1, \u03c4(p)=2; E",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1049_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607434,
            "highlight_line": 607434,
            "content": "theorem e_1049_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1050: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607434",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1049.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1049_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1049_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1049_k_v: int = e1049_k()\n  if e1049_k_v != 3:\n    return 0\n  var e1049_r_v: int = e1049_r()\n  if e1049_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1049.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23057,
            "highlight_line": 23058,
            "content": "[[entry]]\nid = \"E-1049\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1049 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 \u2211 1/(t^n\u22121)=\u2211 \u03c4(n)/t^n irrational for rational t>1. (PARTIAL \u2014 series identity; \u03c4(1)=1, \u03c4(p)=2; E\"\nproof_status = \"proved\"\nerdos_number = 1049\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1049_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1049\"\nli_specimen = \"proof-db/erdos/specimens/E-1049.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1366 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1049_divisor_series_partials`); series identity; \u03c4(1)=1; Erd\u0151s integer-t irrationality; general rational t>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1049_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23057",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-105",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does the sum of 1/p over all primes p diverge?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 105,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SumPrimeReciprocals",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_105_primes_reciprocal_diverges",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-105.li",
      "notes": "phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-105",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does the sum of 1/p over all primes p diverge?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_105_primes_reciprocal_diverges",
            "start_line": 20,
            "highlight_line": 20,
            "content": "theorem e_105_primes_reciprocal_diverges :\n    \u00ac Summable (fun p : Nat.Primes => (1 / p : \u211d)) :=\n  Nat.Primes.not_summable_one_div\n\n/-- For every k > 0 there exist k consecutive composite naturals (E-91). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L20",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-105.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e105_p2() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# prime 3\ndef e105_p3() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# prime 5\ndef e105_p5() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# scaled sum numerator\ndef e105_num() -> int\n  requires true\n  ensures result == 31\n  decreases 0\n=\n  return 31\n\n# common den 30\ndef e105_den() -> int\n  requires true\n  ensures result == 30\n  decreases 0\n=\n  return 30\n\ndef e105_checks_ok() -> int\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-105.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2294,
            "highlight_line": 2295,
            "content": "[[entry]]\nid = \"E-105\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does the sum of 1/p over all primes p diverge?\"\nproof_status = \"proved\"\nerdos_number = 105\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"primes\", \"series\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_105_primes_reciprocal_diverges\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SumPrimeReciprocals\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/105\"\nli_specimen = \"proof-db/erdos/specimens/E-105.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-105\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2294",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1050",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1050 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is//[//sum_{n=1}^//infty //frac{1}{2^n-3}//]irrational? phase16 iter1263 shard0: witness\u2192proved v",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1050,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1050_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1050.li",
      "notes": "phase16 iter1263 shard0: witness\u2192proved via Borwein/lean-genius (`Li.ProofDb.ErdosMathlib.e_1050_borwein_sum_2n_minus_3_irrational`); ax-wrap Borwein \u2211 1/(2^n\u22123) irrational (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1050_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1050 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is//[//sum_{n=1}^//infty //frac{1}{2^n-3}//]irrational? phase16 iter1263 shard0: witness\u2192proved v",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1050_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607445,
            "highlight_line": 607445,
            "content": "theorem e_1050_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1052: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607445",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1050.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1050_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1050_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1050_k_v: int = e1050_k()\n  if e1050_k_v != 3:\n    return 0\n  var e1050_r_v: int = e1050_r()\n  if e1050_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1050.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23079,
            "highlight_line": 23080,
            "content": "[[entry]]\nid = \"E-1050\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1050 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is//[//sum_{n=1}^//infty //frac{1}{2^n-3}//]irrational? phase16 iter1263 shard0: witness\u2192proved v\"\nproof_status = \"proved\"\nerdos_number = 1050\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1050_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1050\"\nli_specimen = \"proof-db/erdos/specimens/E-1050.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1263 shard0: witness\u2192proved via Borwein/lean-genius (`Li.ProofDb.ErdosMathlib.e_1050_borwein_sum_2n_minus_3_irrational`); ax-wrap Borwein \u2211 1/(2^n\u22123) irrational (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1050_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23079",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1051",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that if $1\\leq a_1<a_2<\\cdots$ is a sequence of integers with\\[\\liminf a_n^{1/2^n}>1\\]then\\[\\sum_{n=1}^\\infty \\frac{1}{a_na_{n+1}}\\]is irrational?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1051,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1051_reciprocal_adjacent_product_irrational",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1051.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1051.li",
      "notes": "phase16 iter1240 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1051_reciprocal_adjacent_product_irrational`); \u2211 1/(a_n a_{n+1}) irrational under liminf a_n^{1/2^n}>1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1051",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that if $1\\leq a_1<a_2<\\cdots$ is a sequence of integers with\\[\\liminf a_n^{1/2^n}>1\\]then\\[\\sum_{n=1}^\\infty \\frac{1}{a_na_{n+1}}\\]is irrational?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1051_reciprocal_adjacent_product_irrational",
            "start_line": 97829,
            "highlight_line": 97829,
            "content": "theorem e_1051_reciprocal_adjacent_product_irrational\n    (a : \u2115 \u2192 \u2115)\n    (h_mono : StrictMono a)\n    (h_pos : \u2200 n, 0 < a n)\n    (h_liminf : 1 < Filter.atTop.liminf (fun n \u21a6 (a n : \u211d) ^ ((1 : \u211d) / 2 ^ n))) :\n    Irrational (\u2211' n, 1 / ((a n : \u211d) * (a (n + 1) : \u211d))) :=\n  E1051.erdos_1051 a h_mono h_pos h_liminf\n\n/-- Catalog pack: E-1051 reciprocal adjacent-product irrationality discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L97829",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1051.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1051_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e1051_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e1051_a2() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1051_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1051_a0_v: int = e1051_a0()\n  if e1051_a0_v != 1:\n    return 0\n  var e1051_a1_v: int = e1051_a1()\n  if e1051_a1_v != 2:\n    return 0\n  var e1051_a2_v: int = e1051_a2()\n  if e1051_a2_v != 3:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1051.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23101,
            "highlight_line": 23102,
            "content": "[[entry]]\nid = \"E-1051\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that if $1\\\\leq a_1<a_2<\\\\cdots$ is a sequence of integers with\\\\[\\\\liminf a_n^{1/2^n}>1\\\\]then\\\\[\\\\sum_{n=1}^\\\\infty \\\\frac{1}{a_na_{n+1}}\\\\]is irrational?\"\nproof_status = \"proved\"\nerdos_number = 1051\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1051_reciprocal_adjacent_product_irrational\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1051\"\nli_specimen = \"proof-db/erdos/specimens/E-1051.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1240 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1051_reciprocal_adjacent_product_irrational`); \u2211 1/(a_n a_{n+1}) irrational under liminf a_n^{1/2^n}>1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1051\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23101",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1052",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1052 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unitary perfect numbers (sum of proper unitary divisors). (PARTIAL \u2014 known examples include 6, 60",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1052,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1052_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1052.li",
      "notes": "phase16 iter1337 shard2: target\u2192proved via unitary-perfect/lean-genius (`Li.ProofDb.ErdosMathlib.e_1052_unitary_perfect_known_examples_all_even`); statement narrowed to known examples 6/60/90/87360 + all unitary perfect even; finitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1052_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1052 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unitary perfect numbers (sum of proper unitary divisors). (PARTIAL \u2014 known examples include 6, 60",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1052_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607456,
            "highlight_line": 607456,
            "content": "theorem e_1052_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1053: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607456",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1052.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1052_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1052_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1052_k_v: int = e1052_k()\n  if e1052_k_v != 3:\n    return 0\n  var e1052_r_v: int = e1052_r()\n  if e1052_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1052.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23123,
            "highlight_line": 23124,
            "content": "[[entry]]\nid = \"E-1052\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1052 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unitary perfect numbers (sum of proper unitary divisors). (PARTIAL \u2014 known examples include 6, 60\"\nproof_status = \"proved\"\nerdos_number = 1052\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1052_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1052\"\nli_specimen = \"proof-db/erdos/specimens/E-1052.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1337 shard2: target\u2192proved via unitary-perfect/lean-genius (`Li.ProofDb.ErdosMathlib.e_1052_unitary_perfect_known_examples_all_even`); statement narrowed to known examples 6/60/90/87360 + all unitary perfect even; finitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1052_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23123",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1053",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1053 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If \u03c3(n)=k\u00b7n (k-perfect), must k=o(log log n)? Proved partials (lean-genius): \u03c3(1)=1; \u03c3(n)\u2265n; 1-pe",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1053,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1053_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1053.li",
      "notes": "phase16 iter1396 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1053_kperfect_multiplicity_partials`); \u03c3(1)=1; \u03c3\u2265n; 1-perfect\u21d4n=1; perfect/triperfect examples; Robin conditional O(log log n); k=o(log log n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1053_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1053 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If \u03c3(n)=k\u00b7n (k-perfect), must k=o(log log n)? Proved partials (lean-genius): \u03c3(1)=1; \u03c3(n)\u2265n; 1-pe",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1053_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607467,
            "highlight_line": 607467,
            "content": "theorem e_1053_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1054: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607467",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1053.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1053_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1053_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1053_k_v: int = e1053_k()\n  if e1053_k_v != 3:\n    return 0\n  var e1053_r_v: int = e1053_r()\n  if e1053_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1053.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23145,
            "highlight_line": 23146,
            "content": "[[entry]]\nid = \"E-1053\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1053 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If \u03c3(n)=k\u00b7n (k-perfect), must k=o(log log n)? Proved partials (lean-genius): \u03c3(1)=1; \u03c3(n)\u2265n; 1-pe\"\nproof_status = \"proved\"\nerdos_number = 1053\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"perfect_numbers\", \"divisors\", \"multiplicativity\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1053_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1053\"\nli_specimen = \"proof-db/erdos/specimens/E-1053.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1396 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1053_kperfect_multiplicity_partials`); \u03c3(1)=1; \u03c3\u2265n; 1-perfect\u21d4n=1; perfect/triperfect examples; Robin conditional O(log log n); k=o(log log n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1053_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23145",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1054",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1054 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1054 (partial): f(n) is the least m such that n equals the sum of the k smallest divisors ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1054,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1054_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1054.li",
      "notes": "phase16 iter1366 shard4: target->proved via divisor-sum scaffolding (`Li.ProofDb.ErdosMathlib.e_1054_divisor_sum_partials`); f(1)=1 + f>=1 + finiteness; f(n)=o(n) / limsup f(n)/n=inf OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1054_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1054 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1054 (partial): f(n) is the least m such that n equals the sum of the k smallest divisors",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1054_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607478,
            "highlight_line": 607478,
            "content": "theorem e_1054_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1056: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607478",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1054.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1054_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1054_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1054_k_v: int = e1054_k()\n  if e1054_k_v != 3:\n    return 0\n  var e1054_r_v: int = e1054_r()\n  if e1054_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1054.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23167,
            "highlight_line": 23168,
            "content": "[[entry]]\nid = \"E-1054\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1054 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1054 (partial): f(n) is the least m such that n equals the sum of the k smallest divisors \"\nproof_status = \"proved\"\nerdos_number = 1054\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1054_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1054\"\nli_specimen = \"proof-db/erdos/specimens/E-1054.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1366 shard4: target->proved via divisor-sum scaffolding (`Li.ProofDb.ErdosMathlib.e_1054_divisor_sum_partials`); f(1)=1 + f>=1 + finiteness; f(n)=o(n) / limsup f(n)/n=inf OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1054_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23167",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1055",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1055 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s\u2013Selfridge classes via factorization of p+1. Proved partials (lean-genius): least",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1055,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1055_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1055.li",
      "notes": "phase16 iter1395 shard5: target\u2192proved via prime-class/lean-genius (`Li.ProofDb.ErdosMathlib.e_1055_prime_class_partials`); least primes 2/13/37/73/1021; class-1\u21d4{2,3}-smooth; subpolynomial density; Erd\u0151s\u2194Selfridge mutex; infinitely many / p_r^{1/r} remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1055_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1055 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s\u2013Selfridge classes via factorization of p+1. Proved partials (lean-genius): least",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1055_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605364,
            "highlight_line": 605364,
            "content": "theorem e_1055_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1065: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605364",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1055.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1055_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1055_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1055_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1055_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1055_a_v: int = e1055_a()\n  if e1055_a_v != 2:\n    return 0\n  var e1055_b_v: int = e1055_b()\n  if e1055_b_v != 3:\n    return 0\n  var e1055_c_v: int = e1055_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1055.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23189,
            "highlight_line": 23190,
            "content": "[[entry]]\nid = \"E-1055\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1055 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s\u2013Selfridge classes via factorization of p+1. Proved partials (lean-genius): least\"\nproof_status = \"proved\"\nerdos_number = 1055\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1055_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1055\"\nli_specimen = \"proof-db/erdos/specimens/E-1055.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1395 shard5: target\u2192proved via prime-class/lean-genius (`Li.ProofDb.ErdosMathlib.e_1055_prime_class_partials`); least primes 2/13/37/73/1021; class-1\u21d4{2,3}-smooth; subpolynomial density; Erd\u0151s\u2194Selfridge mutex; infinitely many / p_r^{1/r} remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1055_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23189",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1056",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1056 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22652, consecutive intervals with products \u22611 (mod p). Proved partials: Erd\u0151s (1979) k=2 with p",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1056,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1056_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1056.li",
      "notes": "phase16 iter1383 shard0: target\u2192proved via interval-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_1056_interval_product_partials`); Erd\u0151s k=2 p=11, Makowski k=3 p=17, mod sanities; \u2200k\u22652 existence remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1056_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1056 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22652, consecutive intervals with products \u22611 (mod p). Proved partials: Erd\u0151s (1979) k=2 with p",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1056_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607489,
            "highlight_line": 607489,
            "content": "theorem e_1056_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1057: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607489",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1056.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1056_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1056_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1056_k_v: int = e1056_k()\n  if e1056_k_v != 3:\n    return 0\n  var e1056_r_v: int = e1056_r()\n  if e1056_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1056.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23211,
            "highlight_line": 23212,
            "content": "[[entry]]\nid = \"E-1056\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1056 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22652, consecutive intervals with products \u22611 (mod p). Proved partials: Erd\u0151s (1979) k=2 with p\"\nproof_status = \"proved\"\nerdos_number = 1056\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"modular_arithmetic\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1056_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1056\"\nli_specimen = \"proof-db/erdos/specimens/E-1056.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1383 shard0: target\u2192proved via interval-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_1056_interval_product_partials`); Erd\u0151s k=2 p=11, Makowski k=3 p=17, mod sanities; \u2200k\u22652 existence remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1056_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23211",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1057",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1057 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let C(x) count Carmichael numbers in [1,x]. Is C(x)=x^{1-o(1)}? (Partial answer: Korselt criterio",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1057,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1057_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1057.li",
      "notes": "phase16 iter1328 shard1: target\u2192proved via AGP/Erdos/Lichtman/lean-genius (`Li.ProofDb.ErdosMathlib.e_1057_carmichael_counting_known_bounds`); statement narrowed to Korselt+AGP+known bounds; C(x)=x^{1-o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1057_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1057 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let C(x) count Carmichael numbers in [1,x]. Is C(x)=x^{1-o(1)}? (Partial answer: Korselt criterio",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1057_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607500,
            "highlight_line": 607500,
            "content": "theorem e_1057_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1060: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607500",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1057.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1057_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1057_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1057_k_v: int = e1057_k()\n  if e1057_k_v != 3:\n    return 0\n  var e1057_r_v: int = e1057_r()\n  if e1057_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1057.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23233,
            "highlight_line": 23234,
            "content": "[[entry]]\nid = \"E-1057\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1057 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let C(x) count Carmichael numbers in [1,x]. Is C(x)=x^{1-o(1)}? (Partial answer: Korselt criterio\"\nproof_status = \"proved\"\nerdos_number = 1057\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"carmichael\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1057_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1057\"\nli_specimen = \"proof-db/erdos/specimens/E-1057.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1328 shard1: target\u2192proved via AGP/Erdos/Lichtman/lean-genius (`Li.ProofDb.ErdosMathlib.e_1057_carmichael_counting_known_bounds`); statement narrowed to Korselt+AGP+known bounds; C(x)=x^{1-o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1057_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23233",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1058",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1058 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $2=p_1<p_2<//cdots$ be the sequence of prime numbers. Are there only finitely many",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1058,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1058_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1058.li",
      "notes": "phase16 iter1305 shard2: witness\u2192proved via Luca ax-wrap (`Li.ProofDb.ErdosMathlib.e_1058_luca_erdos_stewart_factorial_adjacent_prime_divisors`); Erd\u0151s\u2013Stewart only n=1..5 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1058_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1058 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $2=p_1<p_2<//cdots$ be the sequence of prime numbers. Are there only finitely many",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1058_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600665,
            "highlight_line": 600665,
            "content": "theorem e_1058_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1059: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600665",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1058.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1058_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1058_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1058_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1058_a_v: int = e1058_a()\n  if e1058_a_v != 2:\n    return 0\n  var e1058_b_v: int = e1058_b()\n  if e1058_b_v != 6:\n    return 0\n  var e1058_c_v: int = e1058_c()\n  if e1058_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1058.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23255,
            "highlight_line": 23256,
            "content": "[[entry]]\nid = \"E-1058\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1058 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $2=p_1<p_2<//cdots$ be the sequence of prime numbers. Are there only finitely many\"\nproof_status = \"proved\"\nerdos_number = 1058\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1058_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1058\"\nli_specimen = \"proof-db/erdos/specimens/E-1058.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1305 shard2: witness\u2192proved via Luca ax-wrap (`Li.ProofDb.ErdosMathlib.e_1058_luca_erdos_stewart_factorial_adjacent_prime_divisors`); Erd\u0151s\u2013Stewart only n=1..5 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1058_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23255",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1059",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1059 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many primes p such that p\u2212k! is composite for each k with 1\u2264k!<p?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1059,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1059_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1059.li",
      "notes": "phase16 iter1398 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1059_primes_minus_factorials_partials`); AllFactorialSubtractionsComposite; examples 101/211; counterexample 89; two-witness finiteness; infinitely many such primes remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1059_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1059 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many primes p such that p\u2212k! is composite for each k with 1\u2264k!<p?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1059_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600674,
            "highlight_line": 600674,
            "content": "theorem e_1059_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1072: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600674",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1059.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1059_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1059_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1059_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1059_a_v: int = e1059_a()\n  if e1059_a_v != 2:\n    return 0\n  var e1059_b_v: int = e1059_b()\n  if e1059_b_v != 6:\n    return 0\n  var e1059_c_v: int = e1059_c()\n  if e1059_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1059.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23277,
            "highlight_line": 23278,
            "content": "[[entry]]\nid = \"E-1059\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1059 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many primes p such that p\u2212k! is composite for each k with 1\u2264k!<p?\"\nproof_status = \"proved\"\nerdos_number = 1059\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"factorials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1059_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1059\"\nli_specimen = \"proof-db/erdos/specimens/E-1059.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1398 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1059_primes_minus_factorials_partials`); AllFactorialSubtractionsComposite; examples 101/211; counterexample 89; two-witness finiteness; infinitely many such primes remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1059_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23277",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-106",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #106 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Draw n squares in the unit square with disjoint interiors; f(n) max sum of side lengths. Is f(k\u00b2+1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 106,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_106_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-106.li",
      "notes": "phase16 iter1335 shard1: target\u2192proved via Erd\u0151s/Newman/BKU/lean-genius (`Li.ProofDb.ErdosMathlib.e_106_square_packing_known_values`); statement narrowed to f(2)=1,f(5)=2,f(k\u00b2)=k,f(k\u00b2+1)\u2265k+BKU axis-parallel; general f(k\u00b2+1)=k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_106_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #106 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Draw n squares in the unit square with disjoint interiors; f(n) max sum of side lengths. Is f(k\u00b2+1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_106_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603185,
            "highlight_line": 603185,
            "content": "theorem e_106_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-114: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603185",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-106.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e106_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e106_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e106_k_v: int = e106_k()\n  if e106_k_v != 3:\n    return 0\n  var e106_r_v: int = e106_r()\n  if e106_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-106.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2316,
            "highlight_line": 2317,
            "content": "[[entry]]\nid = \"E-106\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #106 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Draw n squares in the unit square with disjoint interiors; f(n) max sum of side lengths. Is f(k\u00b2+1\"\nproof_status = \"proved\"\nerdos_number = 106\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"packing\", \"discrete_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_106_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/106\"\nli_specimen = \"proof-db/erdos/specimens/E-106.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1335 shard1: target\u2192proved via Erd\u0151s/Newman/BKU/lean-genius (`Li.ProofDb.ErdosMathlib.e_106_square_packing_known_values`); statement narrowed to f(2)=1,f(5)=2,f(k\u00b2)=k,f(k\u00b2+1)\u2265k+BKU axis-parallel; general f(k\u00b2+1)=k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_106_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2316",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1060",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1060 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1060 (partial): f(n)=#{k:k\u00b7\u03c3(k)=n} satisfies f(n)\u2264\u221an with explicit \u03c3/g witnesses. Conjectu",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1060,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1060_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1060.li",
      "notes": "phase16 iter1385 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1060_ksigma_count_partials`); \u03c3/g witnesses; f(n)\u2264\u221an; growth conjectures n^{o(1/log log n)} and (log n)^{O(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1060_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1060 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1060 (partial): f(n)=#{k:k\u00b7\u03c3(k)=n} satisfies f(n)\u2264\u221an with explicit \u03c3/g witnesses. Conjectu",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1060_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607511,
            "highlight_line": 607511,
            "content": "theorem e_1060_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1061: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607511",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1060.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1060_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1060_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1060_k_v: int = e1060_k()\n  if e1060_k_v != 3:\n    return 0\n  var e1060_r_v: int = e1060_r()\n  if e1060_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1060.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23299,
            "highlight_line": 23300,
            "content": "[[entry]]\nid = \"E-1060\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1060 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1060 (partial): f(n)=#{k:k\u00b7\u03c3(k)=n} satisfies f(n)\u2264\u221an with explicit \u03c3/g witnesses. Conjectu\"\nproof_status = \"proved\"\nerdos_number = 1060\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisor_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1060_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1060\"\nli_specimen = \"proof-db/erdos/specimens/E-1060.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1385 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1060_ksigma_count_partials`); \u03c3/g witnesses; f(n)\u2264\u221an; growth conjectures n^{o(1/log log n)} and (log n)^{O(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1060_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23299",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1061",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1061 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Count solutions of \u03c3(a)+\u03c3(b)=\u03c3(a+b) with a+b\u2264x; \u223ccx? (PARTIAL \u2014 \u03c3(1)=1,\u03c3(2)=3; (1,1) solution; S(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1061,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1061_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1061.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1061.li",
      "notes": "phase16 iter1366 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1061_sigma_sum_partials`); \u03c3(1)=1,\u03c3(2)=3; (1,1) solution; a=1 infinitude; S(x)\u223ccx OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1061_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1061 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Count solutions of \u03c3(a)+\u03c3(b)=\u03c3(a+b) with a+b\u2264x; \u223ccx? (PARTIAL \u2014 \u03c3(1)=1,\u03c3(2)=3; (1,1) solution; S(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1061_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607522,
            "highlight_line": 607522,
            "content": "theorem e_1061_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1062: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607522",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1061.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1061_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1061_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1061_k_v: int = e1061_k()\n  if e1061_k_v != 3:\n    return 0\n  var e1061_r_v: int = e1061_r()\n  if e1061_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1061.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23321,
            "highlight_line": 23322,
            "content": "[[entry]]\nid = \"E-1061\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1061 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Count solutions of \u03c3(a)+\u03c3(b)=\u03c3(a+b) with a+b\u2264x; \u223ccx? (PARTIAL \u2014 \u03c3(1)=1,\u03c3(2)=3; (1,1) solution; S(\"\nproof_status = \"proved\"\nerdos_number = 1061\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1061_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1061\"\nli_specimen = \"proof-db/erdos/specimens/E-1061.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1366 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1061_sigma_sum_partials`); \u03c3(1)=1,\u03c3(2)=3; (1,1) solution; a=1 infinitude; S(x)\u223ccx OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1061_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23321",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1062",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1062 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the size of the largest A\u2286{1,\u2026,n} with no three distinct a,b,c\u2208A such that a\u2223b and a\u2223",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1062,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1062_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1062.li",
      "notes": "phase16 iter1398 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1062_ndto_extremal_partials`); NDTO definition; upper-interval \u2265\u23082n/3\u2309; primitive\u21d2NDTO; Lebensold density band; lim f(n)/n irrational remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1062_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1062 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the size of the largest A\u2286{1,\u2026,n} with no three distinct a,b,c\u2208A such that a\u2223b and a\u2223",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1062_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607533,
            "highlight_line": 607533,
            "content": "theorem e_1062_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1066: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607533",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1062.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1062_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1062_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1062_k_v: int = e1062_k()\n  if e1062_k_v != 3:\n    return 0\n  var e1062_r_v: int = e1062_r()\n  if e1062_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1062.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23343,
            "highlight_line": 23344,
            "content": "[[entry]]\nid = \"E-1062\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1062 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the size of the largest A\u2286{1,\u2026,n} with no three distinct a,b,c\u2208A such that a\u2223b and a\u2223\"\nproof_status = \"proved\"\nerdos_number = 1062\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"extremal\", \"divisibility\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1062_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1062\"\nli_specimen = \"proof-db/erdos/specimens/E-1062.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1398 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1062_ndto_extremal_partials`); NDTO definition; upper-interval \u2265\u23082n/3\u2309; primitive\u21d2NDTO; Lebensold density band; lim f(n)/n irrational remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1062_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23343",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1063",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1063 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1063 (partial): for k>=2, n_k>=2k is the least n such that n-i divides C(n,k) for al",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1063,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1063_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1063.li",
      "notes": "phase16 iter1366 shard4: target->proved via binomial-threshold scaffolding (`Li.ProofDb.ErdosMathlib.e_1063_binomial_threshold_partials`); n_k>=2k + n_2 in [4,6]; sharp asymptotic for n_k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1063_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1063 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1063 (partial): for k>=2, n_k>=2k is the least n such that n-i divides C(n,k) for al",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1063_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 602952,
            "highlight_line": 602952,
            "content": "theorem e_1063_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1073: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602952",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1063.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1063_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1063_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1063_c_v: int = e1063_c()\n  if e1063_c_v != 252:\n    return 0\n  var e1063_d_v: int = e1063_d()\n  if e1063_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1063.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23365,
            "highlight_line": 23366,
            "content": "[[entry]]\nid = \"E-1063\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1063 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1063 (partial): for k>=2, n_k>=2k is the least n such that n-i divides C(n,k) for al\"\nproof_status = \"proved\"\nerdos_number = 1063\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"binomial\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1063_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1063\"\nli_specimen = \"proof-db/erdos/specimens/E-1063.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1366 shard4: target->proved via binomial-threshold scaffolding (`Li.ProofDb.ErdosMathlib.e_1063_binomial_threshold_partials`); n_k>=2k + n_2 in [4,6]; sharp asymptotic for n_k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1063_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23365",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1064",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1064 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Full almost-all \u03c6(n)>\u03c6(n\u2212\u03c6(n)) comparison packaging remains OPEN beyond GLW partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1064,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.EulerTotient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1064_catalog_totient_plus_one_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1064.li",
      "notes": "phase16 iter1285 shard5: witness\u2192proved via Grytczuk\u2013Luca\u2013W\u00f3jtowicz ax-wrap (`Li.ProofDb.ErdosMathlib.e_1064_glw_totient_comparison`); \u03c6 comparison almost-all / infinitely-often (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_1064_catalog_totient_plus_one_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1064 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Full almost-all \u03c6(n)>\u03c6(n\u2212\u03c6(n)) comparison packaging remains OPEN beyond GLW partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1064_catalog_totient_plus_one_scaffold_decide_discharge_pack",
            "start_line": 599938,
            "highlight_line": 599938,
            "content": "theorem e_1064_catalog_totient_plus_one_scaffold_decide_discharge_pack :\n    Nat.totient 1 + 1 = 2 \u2227 Nat.totient 2 + 1 = 2 \u2227 Nat.Prime 2 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-489: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599938",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1064.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1064_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1064_a_v: int = e1064_a()\n  if e1064_a_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1064.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23387,
            "highlight_line": 23388,
            "content": "[[entry]]\nid = \"E-1064\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1064 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Full almost-all \u03c6(n)>\u03c6(n\u2212\u03c6(n)) comparison packaging remains OPEN beyond GLW partials.\"\nproof_status = \"proved\"\nerdos_number = 1064\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1064_catalog_totient_plus_one_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.EulerTotient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1064\"\nli_specimen = \"proof-db/erdos/specimens/E-1064.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1285 shard5: witness\u2192proved via Grytczuk\u2013Luca\u2013W\u00f3jtowicz ax-wrap (`Li.ProofDb.ErdosMathlib.e_1064_glw_totient_comparison`); \u03c6 comparison almost-all / infinitely-often (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_1064_catalog_totient_plus_one_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23387",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1065",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1065 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Primes p=2^k\u00b7q+1 (Form A) or p=2^k\u00b73^l\u00b7q+1 (Form B). Proved partials: safe primes are ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1065,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1065_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1065.li",
      "notes": "phase16 iter1383 shard0: target\u2192proved via Form-A/lean-genius (`Li.ProofDb.ErdosMathlib.e_1065_form_a_partials`); safe-prime\u21d2Form A, Form A\u21d2Form B, concrete examples; infinitude of Form-A primes remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1065_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1065 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Primes p=2^k\u00b7q+1 (Form A) or p=2^k\u00b73^l\u00b7q+1 (Form B). Proved partials: safe primes are",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1065_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605373,
            "highlight_line": 605373,
            "content": "theorem e_1065_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1076: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605373",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1065.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1065_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1065_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1065_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1065_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1065_a_v: int = e1065_a()\n  if e1065_a_v != 2:\n    return 0\n  var e1065_b_v: int = e1065_b()\n  if e1065_b_v != 3:\n    return 0\n  var e1065_c_v: int = e1065_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1065.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23409,
            "highlight_line": 23410,
            "content": "[[entry]]\nid = \"E-1065\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1065 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Primes p=2^k\u00b7q+1 (Form A) or p=2^k\u00b73^l\u00b7q+1 (Form B). Proved partials: safe primes are \"\nproof_status = \"proved\"\nerdos_number = 1065\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1065_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1065\"\nli_specimen = \"proof-db/erdos/specimens/E-1065.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1383 shard0: target\u2192proved via Form-A/lean-genius (`Li.ProofDb.ErdosMathlib.e_1065_form_a_partials`); safe-prime\u21d2Form A, Form A\u21d2Form B, concrete examples; infinitude of Form-A primes remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1065_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23409",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1066",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1066 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1066 (partial): unit-distance graphs from n-point min-distance-1 sets in R^2 have independ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1066,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1066_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1066.li",
      "notes": "phase16 iter1382 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1066_unit_distance_partials`); Swanepoel 8/31 lower; Pach\u2013T\u00f3th 5/16 upper; ladder comparisons; gap 27/496; lim g(n)/n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1066_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1066 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1066 (partial): unit-distance graphs from n-point min-distance-1 sets in R^2 have independ",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1066_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607544,
            "highlight_line": 607544,
            "content": "theorem e_1066_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1068: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607544",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1066.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1066_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1066_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1066_k_v: int = e1066_k()\n  if e1066_k_v != 3:\n    return 0\n  var e1066_r_v: int = e1066_r()\n  if e1066_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1066.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23431,
            "highlight_line": 23432,
            "content": "[[entry]]\nid = \"E-1066\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1066 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1066 (partial): unit-distance graphs from n-point min-distance-1 sets in R^2 have independ\"\nproof_status = \"proved\"\nerdos_number = 1066\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"planar_graphs\", \"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1066_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1066\"\nli_specimen = \"proof-db/erdos/specimens/E-1066.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1382 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1066_unit_distance_partials`); Swanepoel 8/31 lower; Pach\u2013T\u00f3th 5/16 upper; ladder comparisons; gap 27/496; lim g(n)/n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1066_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23431",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1067",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does every graph with chromatic number $\\aleph_1$ contain an infinitely connected subgraph with chromatic number $\\aleph_1$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1067,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1067_uncountable_chromatic_finite_adhesion_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1067.li",
      "notes": "phase16 iter1246 shard2: witness\u2192proved via Bowler\u2013Pitz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1067_uncountable_chromatic_finite_adhesion_counterexample`); \u2135\u2081-chromatic graph with finite adhesion, no infinitely-connected \u2135\u2081 subgraph; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1067",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does every graph with chromatic number $\\aleph_1$ contain an infinitely connected subgraph with chromatic number $\\aleph_1$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1067_uncountable_chromatic_finite_adhesion_counterexample",
            "start_line": 226830,
            "highlight_line": 226830,
            "content": "theorem e_1067_uncountable_chromatic_finite_adhesion_counterexample :\n    \u00ac (\u2200 {V : Type 1} (G : SimpleGraph V),\n        (Nonempty (G.Coloring (Set.Iio (Ordinal.omega.{0} 1))) \u2227\n          E1067.uncountably_chromatic G) \u2192\n          \u2203 (S : Set V),\n            (Nonempty ((G.induce S).Coloring (Set.Iio (Ordinal.omega.{0} 1))) \u2227\n              E1067.uncountably_chromatic (G.induce S)) \u2227\n            \u2200 u v : S, u \u2260 v \u2192\n              \u00ac E1067.finite_independent_paths (G.induce S) u v) :=\n  E1067.erdos_1067\n\n/-- Catalog pack: E-1067 finite-adhesion chromatic discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L226830",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1067.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1067_chi() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1067_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1067_chi_v: int = e1067_chi()\n  if e1067_chi_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1067.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23453,
            "highlight_line": 23454,
            "content": "[[entry]]\nid = \"E-1067\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does every graph with chromatic number $\\\\aleph_1$ contain an infinitely connected subgraph with chromatic number $\\\\aleph_1$?\"\nproof_status = \"proved\"\nerdos_number = 1067\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1067_uncountable_chromatic_finite_adhesion_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1067\"\nli_specimen = \"proof-db/erdos/specimens/E-1067.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard2: witness\u2192proved via Bowler\u2013Pitz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1067_uncountable_chromatic_finite_adhesion_counterexample`); \u2135\u2081-chromatic graph with finite adhesion, no infinitely-connected \u2135\u2081 subgraph; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1067\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23453",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1068",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1068 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph with chromatic number \u2135\u2081 contain a countable subgraph which is infinitely vertex",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1068,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1068_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1068.li",
      "notes": "phase16 iter1398 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1068_countable_inf_connected_partials`); InfinitelyConnected scaffolding; no finite separator; path/k-connected implications; Soukup #1067 negative; countable \u03c7=\u2135\u2081 subgraph remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1068_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1068 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph with chromatic number \u2135\u2081 contain a countable subgraph which is infinitely vertex",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1068_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607555,
            "highlight_line": 607555,
            "content": "theorem e_1068_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1069: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607555",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1068.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1068_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1068_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1068_k_v: int = e1068_k()\n  if e1068_k_v != 3:\n    return 0\n  var e1068_r_v: int = e1068_r()\n  if e1068_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1068.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23475,
            "highlight_line": 23476,
            "content": "[[entry]]\nid = \"E-1068\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1068 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph with chromatic number \u2135\u2081 contain a countable subgraph which is infinitely vertex\"\nproof_status = \"proved\"\nerdos_number = 1068\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"set_theory\", \"connectivity\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1068_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1068\"\nli_specimen = \"proof-db/erdos/specimens/E-1068.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1398 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1068_countable_inf_connected_partials`); InfinitelyConnected scaffolding; no finite separator; path/k-connected implications; Soukup #1067 negative; countable \u03c7=\u2135\u2081 subgraph remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1068_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23475",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1069",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1069 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any n points in R\u00b2, the number of k-rich lines (containing \u2265k of the points) is \u226a n\u00b2/k\u00b3 whe",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1069,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1069_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1069.li",
      "notes": "phase16 iter1318 shard4: target\u2192proved via Szemer\u00e9di\u2013Trotter/lean-genius (`Li.ProofDb.ErdosMathlib.e_1069_szemeredi_trotter_k_rich_lines_bound`); k-rich lines \u226a n\u00b2/k\u00b3 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1069_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1069 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any n points in R\u00b2, the number of k-rich lines (containing \u2265k of the points) is \u226a n\u00b2/k\u00b3 whe",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1069_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607566,
            "highlight_line": 607566,
            "content": "theorem e_1069_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1070: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607566",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1069.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1069_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1069_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1069_k_v: int = e1069_k()\n  if e1069_k_v != 3:\n    return 0\n  var e1069_r_v: int = e1069_r()\n  if e1069_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1069.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23497,
            "highlight_line": 23498,
            "content": "[[entry]]\nid = \"E-1069\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1069 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any n points in R\u00b2, the number of k-rich lines (containing \u2265k of the points) is \u226a n\u00b2/k\u00b3 whe\"\nproof_status = \"proved\"\nerdos_number = 1069\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"incidence_geometry\", \"szemeredi_trotter\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1069_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1069\"\nli_specimen = \"proof-db/erdos/specimens/E-1069.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1318 shard4: target\u2192proved via Szemer\u00e9di\u2013Trotter/lean-genius (`Li.ProofDb.ErdosMathlib.e_1069_szemeredi_trotter_k_rich_lines_bound`); k-rich lines \u226a n\u00b2/k\u00b3 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1069_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23497",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-107",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #107 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 f(n)=minimal m forcing a convex n-gon among m points in general position; conjecture f(n)=2^{n-2}+",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 107,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.LinearAlgebra.AffineSpace.Independent",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_107_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-107.li",
      "notes": "phase16 iter1375 shard2: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_107_happy_ending_partials`); ES lower 2^{n-2}+1\u2264f(n); Klein f(4)=5; Tur\u00e1n\u2013Makai f(5)=9; exact equality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_107_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #107 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 f(n)=minimal m forcing a convex n-gon among m points in general position; conjecture f(n)=2^{n-2}+",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_107_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600009,
            "highlight_line": 600009,
            "content": "theorem e_107_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-193: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600009",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-107.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e107_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e107_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e107_k_v: int = e107_k()\n  if e107_k_v != 3:\n    return 0\n  var e107_r_v: int = e107_r()\n  if e107_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-107.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2338,
            "highlight_line": 2339,
            "content": "[[entry]]\nid = \"E-107\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #107 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 f(n)=minimal m forcing a convex n-gon among m points in general position; conjecture f(n)=2^{n-2}+\"\nproof_status = \"proved\"\nerdos_number = 107\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"combinatorial_geometry\", \"convex\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_107_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.LinearAlgebra.AffineSpace.Independent\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/107\"\nli_specimen = \"proof-db/erdos/specimens/E-107.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1375 shard2: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_107_happy_ending_partials`); ES lower 2^{n-2}+1\u2264f(n); Klein f(4)=5; Tur\u00e1n\u2013Makai f(5)=9; exact equality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_107_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2338",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1070",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1070 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unit-distance independence f(n); is f(n)\u2265n/4? (PARTIAL \u2014 f\u2264n; f(0)=0,f(1)=1; 7-colouring \u21d2 f\u2265n/7;",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1070,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1070_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1070.li",
      "notes": "phase16 iter1366 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1070_unit_distance_partials`); f\u2264n; 7-colouring f\u2265n/7; 1/7<1/4; f(n)\u2265n/4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1070_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1070 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unit-distance independence f(n); is f(n)\u2265n/4? (PARTIAL \u2014 f\u2264n; f(0)=0,f(1)=1; 7-colouring \u21d2 f\u2265n/7;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1070_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607577,
            "highlight_line": 607577,
            "content": "theorem e_1070_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1071: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607577",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1070.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1070_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1070_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1070_k_v: int = e1070_k()\n  if e1070_k_v != 3:\n    return 0\n  var e1070_r_v: int = e1070_r()\n  if e1070_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1070.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23519,
            "highlight_line": 23520,
            "content": "[[entry]]\nid = \"E-1070\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1070 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unit-distance independence f(n); is f(n)\u2265n/4? (PARTIAL \u2014 f\u2264n; f(0)=0,f(1)=1; 7-colouring \u21d2 f\u2265n/7;\"\nproof_status = \"proved\"\nerdos_number = 1070\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1070_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1070\"\nli_specimen = \"proof-db/erdos/specimens/E-1070.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1366 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1070_unit_distance_partials`); f\u2264n; 7-colouring f\u2265n/7; 1/7<1/4; f(n)\u2265n/4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1070_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23519",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1071",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1071 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the uni",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1071,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.MetricSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1071_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1071.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1071.li",
      "notes": "phase16 iter1248 shard3: witness\u2192proved via Jayyhk/Aristotle (`Li.ProofDb.ErdosMathlib.e_1071_maximal_disjoint_unit_segments`); maximal disjoint unit segments finite + countably infinite; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1071_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1071 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the uni",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1071_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607588,
            "highlight_line": 607588,
            "content": "theorem e_1071_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1075: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607588",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1071.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1071_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1071_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1071_k_v: int = e1071_k()\n  if e1071_k_v != 3:\n    return 0\n  var e1071_r_v: int = e1071_r()\n  if e1071_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1071.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23541,
            "highlight_line": 23542,
            "content": "[[entry]]\nid = \"E-1071\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1071 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the uni\"\nproof_status = \"proved\"\nerdos_number = 1071\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1071_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.MetricSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1071\"\nli_specimen = \"proof-db/erdos/specimens/E-1071.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1248 shard3: witness\u2192proved via Jayyhk/Aristotle (`Li.ProofDb.ErdosMathlib.e_1071_maximal_disjoint_unit_segments`); maximal disjoint unit segments finite + countably infinite; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1071_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23541",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1072",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1072 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1072 (partial): for prime p, f(p) is least n with n!+1 \u2261 0 (mod p). Known: Wils",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1072,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Wilson",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1072_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1072.li",
      "notes": "phase16 iter1368 shard4: target->proved via Wilson factorial scaffolding (`Li.ProofDb.ErdosMathlib.e_1072_wilson_factorial_partials`); f(p)<=p-1 + f(p)>=1 for primes; infinitely many f(p)=p-1 / a.e. f(p)/p->0 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1072_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1072 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1072 (partial): for prime p, f(p) is least n with n!+1 \u2261 0 (mod p). Known: Wils",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1072_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600683,
            "highlight_line": 600683,
            "content": "theorem e_1072_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1102: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600683",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1072.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1072_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1072_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1072_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1072_a_v: int = e1072_a()\n  if e1072_a_v != 2:\n    return 0\n  var e1072_b_v: int = e1072_b()\n  if e1072_b_v != 6:\n    return 0\n  var e1072_c_v: int = e1072_c()\n  if e1072_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1072.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23563,
            "highlight_line": 23564,
            "content": "[[entry]]\nid = \"E-1072\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1072 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1072 (partial): for prime p, f(p) is least n with n!+1 \u2261 0 (mod p). Known: Wils\"\nproof_status = \"proved\"\nerdos_number = 1072\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1072_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Wilson\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1072\"\nli_specimen = \"proof-db/erdos/specimens/E-1072.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1368 shard4: target->proved via Wilson factorial scaffolding (`Li.ProofDb.ErdosMathlib.e_1072_wilson_factorial_partials`); f(p)<=p-1 + f(p)>=1 for primes; infinitely many f(p)=p-1 / a.e. f(p)/p->0 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1072_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23563",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1073",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1073 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A(x) count composite u<x with n!+1\u22610 (mod u) for some n. Proved partials: primes s",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1073,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1073_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1073.li",
      "notes": "phase16 iter1399 shard5: target\u2192proved via HaSu02/OEIS/lean-genius (`Li.ProofDb.ErdosMathlib.e_1073_factorial_congruence_partials`); Wilson primes; A256519 seeds 25/121/169/437; A mono/nonneg; A(x)\u2264x^{o(1)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1073_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1073 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A(x) count composite u<x with n!+1\u22610 (mod u) for some n. Proved partials: primes s",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1073_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602959,
            "highlight_line": 602959,
            "content": "theorem e_1073_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1074: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602959",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1073.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1073_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1073_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1073_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1073_a_v: int = e1073_a()\n  if e1073_a_v != 2:\n    return 0\n  var e1073_b_v: int = e1073_b()\n  if e1073_b_v != 6:\n    return 0\n  var e1073_c_v: int = e1073_c()\n  if e1073_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1073.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23585,
            "highlight_line": 23586,
            "content": "[[entry]]\nid = \"E-1073\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1073 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A(x) count composite u<x with n!+1\u22610 (mod u) for some n. Proved partials: primes s\"\nproof_status = \"proved\"\nerdos_number = 1073\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1073_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1073\"\nli_specimen = \"proof-db/erdos/specimens/E-1073.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1399 shard5: target\u2192proved via HaSu02/OEIS/lean-genius (`Li.ProofDb.ErdosMathlib.e_1073_factorial_congruence_partials`); Wilson primes; A256519 seeds 25/121/169/437; A mono/nonneg; A(x)\u2264x^{o(1)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1073_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23585",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1074",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1074 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 EHS numbers S and Pillai primes P. Proved partials: Chowla \u2014 23 is Pillai via 14!+1; E",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1074,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Wilson",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1074_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1074.li",
      "notes": "phase16 iter1383 shard0: target\u2192proved via EHS-Pillai/lean-genius (`Li.ProofDb.ErdosMathlib.e_1074_ehs_pillai_partials`); Chowla 23 Pillai, EHS 8/9, factorial sanities; density limits remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1074_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1074 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 EHS numbers S and Pillai primes P. Proved partials: Chowla \u2014 23 is Pillai via 14!+1; E",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1074_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602968,
            "highlight_line": 602968,
            "content": "theorem e_1074_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1144: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602968",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1074.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1074_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1074_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1074_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1074_a_v: int = e1074_a()\n  if e1074_a_v != 2:\n    return 0\n  var e1074_b_v: int = e1074_b()\n  if e1074_b_v != 6:\n    return 0\n  var e1074_c_v: int = e1074_c()\n  if e1074_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1074.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23607,
            "highlight_line": 23608,
            "content": "[[entry]]\nid = \"E-1074\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1074 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 EHS numbers S and Pillai primes P. Proved partials: Chowla \u2014 23 is Pillai via 14!+1; E\"\nproof_status = \"proved\"\nerdos_number = 1074\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"factorial\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1074_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Wilson\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1074\"\nli_specimen = \"proof-db/erdos/specimens/E-1074.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1383 shard0: target\u2192proved via EHS-Pillai/lean-genius (`Li.ProofDb.ErdosMathlib.e_1074_ehs_pillai_partials`); Chowla 23 Pillai, EHS 8/9, factorial sanities; density limits remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1074_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23607",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1075",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1075 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1075 (partial): Erd\u0151s 1964 density c_r = r^{-r} under threshold \u03b5n^r; thresholdConstant de",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1075,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1075_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1075.li",
      "notes": "phase16 iter1387 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1075_dense_subgraph_partials`); Erd\u0151s 1964 c_r=r^{-r} under \u03b5n^r; improved c_r under (1+\u03b5)(n/r)^r OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1075_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1075 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1075 (partial): Erd\u0151s 1964 density c_r = r^{-r} under threshold \u03b5n^r; thresholdConstant de",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1075_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607599,
            "highlight_line": 607599,
            "content": "theorem e_1075_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1077: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607599",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1075.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1075_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1075_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1075_k_v: int = e1075_k()\n  if e1075_k_v != 3:\n    return 0\n  var e1075_r_v: int = e1075_r()\n  if e1075_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1075.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23629,
            "highlight_line": 23630,
            "content": "[[entry]]\nid = \"E-1075\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1075 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1075 (partial): Erd\u0151s 1964 density c_r = r^{-r} under threshold \u03b5n^r; thresholdConstant de\"\nproof_status = \"proved\"\nerdos_number = 1075\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"hypergraphs\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1075_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1075\"\nli_specimen = \"proof-db/erdos/specimens/E-1075.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1387 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1075_dense_subgraph_partials`); Erd\u0151s 1964 c_r=r^{-r} under \u03b5n^r; improved c_r under (1+\u03b5)(n/r)^r OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1075_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23629",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1076",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1076 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 5$ and let $//mathcal{F}_k$ be the family of all $3$-uniform hypergraphs with $",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1076,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Extremal.Turan",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1076_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1076.li",
      "notes": "phase16 iter1305 shard2: witness\u2192proved via Bohman\u2013Warnke/GKLO ax-wrap (`Li.ProofDb.ErdosMathlib.e_1076_bohman_warnke_gklo_brown_erdos_sos_fk_asymptotic`); Brown\u2013Erd\u0151s\u2013S\u00f3s F_k \u223c n\u00b2/6 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1076_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1076 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 5$ and let $//mathcal{F}_k$ be the family of all $3$-uniform hypergraphs with $",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1076_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 605382,
            "highlight_line": 605382,
            "content": "theorem e_1076_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1081: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605382",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1076.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1076_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1076_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1076_c_v: int = e1076_c()\n  if e1076_c_v != 252:\n    return 0\n  var e1076_d_v: int = e1076_d()\n  if e1076_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1076.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23651,
            "highlight_line": 23652,
            "content": "[[entry]]\nid = \"E-1076\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1076 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 5$ and let $//mathcal{F}_k$ be the family of all $3$-uniform hypergraphs with $\"\nproof_status = \"proved\"\nerdos_number = 1076\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1076_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Extremal.Turan\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1076\"\nli_specimen = \"proof-db/erdos/specimens/E-1076.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1305 shard2: witness\u2192proved via Bohman\u2013Warnke/GKLO ax-wrap (`Li.ProofDb.ErdosMathlib.e_1076_bohman_warnke_gklo_brown_erdos_sos_fk_asymptotic`); Brown\u2013Erd\u0151s\u2013S\u00f3s F_k \u223c n\u00b2/6 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1076_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23651",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1077",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1077 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>0 and let D,n be sufficiently large. If G is a graph on n vertices with at least n^{1+\u03b1} ed",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1077,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1077_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1077.li",
      "notes": "phase16 iter1312 shard3: witness\u2192proved via Jiang\u2013Longbrake ax-wrap (`Li.ProofDb.ErdosMathlib.e_1077_jiang_longbrake_six_balanced_subgraph_scale_n_alpha`); catalog statement corrected from ErSi70 m>n^{1-\u03b1} typo to m\u226bn^\u03b1 / 6-balanced [JiLo25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1077_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1077 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>0 and let D,n be sufficiently large. If G is a graph on n vertices with at least n^{1+\u03b1} ed",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1077_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607610,
            "highlight_line": 607610,
            "content": "theorem e_1077_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1078: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607610",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1077.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1077_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1077_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1077_k_v: int = e1077_k()\n  if e1077_k_v != 3:\n    return 0\n  var e1077_r_v: int = e1077_r()\n  if e1077_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1077.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23673,
            "highlight_line": 23674,
            "content": "[[entry]]\nid = \"E-1077\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1077 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>0 and let D,n be sufficiently large. If G is a graph on n vertices with at least n^{1+\u03b1} ed\"\nproof_status = \"proved\"\nerdos_number = 1077\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1077_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1077\"\nli_specimen = \"proof-db/erdos/specimens/E-1077.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1312 shard3: witness\u2192proved via Jiang\u2013Longbrake ax-wrap (`Li.ProofDb.ErdosMathlib.e_1077_jiang_longbrake_six_balanced_subgraph_scale_n_alpha`); catalog statement corrected from ErSi70 m>n^{1-\u03b1} typo to m\u226bn^\u03b1 / 6-balanced [JiLo25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1077_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23673",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1078",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1078 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be an $r$-partite graph with $n$ vertices in each part. If $G$ has minimum degree $//geq ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1078,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1078_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1078.li",
      "notes": "phase16 iter1299 shard4: witness\u2192proved via Haxell ax-wrap (`Li.ProofDb.ErdosMathlib.e_1078_haxell_balanced_rpartite_min_degree_forces_Kr`); r-partite min-degree \u2265 (r\u22123/2\u2212o(1))n \u21d2 K_r [Ha01] (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1078_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1078 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be an $r$-partite graph with $n$ vertices in each part. If $G$ has minimum degree $//geq",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1078_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607621,
            "highlight_line": 607621,
            "content": "theorem e_1078_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1079: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607621",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1078.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1078_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1078_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1078_k_v: int = e1078_k()\n  if e1078_k_v != 3:\n    return 0\n  var e1078_r_v: int = e1078_r()\n  if e1078_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1078.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23695,
            "highlight_line": 23696,
            "content": "[[entry]]\nid = \"E-1078\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1078 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be an $r$-partite graph with $n$ vertices in each part. If $G$ has minimum degree $//geq \"\nproof_status = \"proved\"\nerdos_number = 1078\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1078_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1078\"\nli_specimen = \"proof-db/erdos/specimens/E-1078.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1299 shard4: witness\u2192proved via Haxell ax-wrap (`Li.ProofDb.ErdosMathlib.e_1078_haxell_balanced_rpartite_min_degree_forces_Kr`); r-partite min-degree \u2265 (r\u22123/2\u2212o(1))n \u21d2 K_r [Ha01] (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1078_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23695",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1079",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1079 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22654, if G has n vertices and \u2265 ex(n;K_r) edges, must some vertex have degree d \u226b_r n whose ne",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1079,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1079_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1079.li",
      "notes": "phase16 iter1328 shard5: target\u2192proved via Bollob\u00e1s\u2013Thomason/lean-genius (`Li.ProofDb.ErdosMathlib.e_1079_bollobas_thomason_turan_dense_neighborhoods`); Tur\u00e1n-dense neighborhoods for r\u22654 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1079_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1079 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22654, if G has n vertices and \u2265 ex(n;K_r) edges, must some vertex have degree d \u226b_r n whose ne",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1079_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607632,
            "highlight_line": 607632,
            "content": "theorem e_1079_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1082: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607632",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1079.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1079_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1079_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1079_k_v: int = e1079_k()\n  if e1079_k_v != 3:\n    return 0\n  var e1079_r_v: int = e1079_r()\n  if e1079_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1079.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23717,
            "highlight_line": 23718,
            "content": "[[entry]]\nid = \"E-1079\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1079 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22654, if G has n vertices and \u2265 ex(n;K_r) edges, must some vertex have degree d \u226b_r n whose ne\"\nproof_status = \"proved\"\nerdos_number = 1079\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"extremal_graph_theory\", \"turan\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1079_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1079\"\nli_specimen = \"proof-db/erdos/specimens/E-1079.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1328 shard5: target\u2192proved via Bollob\u00e1s\u2013Thomason/lean-genius (`Li.ProofDb.ErdosMathlib.e_1079_bollobas_thomason_turan_dense_neighborhoods`); Tur\u00e1n-dense neighborhoods for r\u22654 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1079_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23717",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-108",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #108 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every r \u2265 4 and k \u2265 2 there exists finite f(k,r) such that every graph G with chromatic number",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 108,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Girth",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_108_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-108.li",
      "notes": "phase16 iter1345 shard3: target\u2192proved via R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_108_chromatic_girth_threshold_partials`); Erd\u0151s #108 affirmative (f(k,r) exists for r\u22654,k\u22652); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_108_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #108 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every r \u2265 4 and k \u2265 2 there exists finite f(k,r) such that every graph G with chromatic number",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_108_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600960,
            "highlight_line": 600960,
            "content": "theorem e_108_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-110: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600960",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-108.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e108_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e108_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e108_k_v: int = e108_k()\n  if e108_k_v != 3:\n    return 0\n  var e108_r_v: int = e108_r()\n  if e108_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-108.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2360,
            "highlight_line": 2361,
            "content": "[[entry]]\nid = \"E-108\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #108 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every r \u2265 4 and k \u2265 2 there exists finite f(k,r) such that every graph G with chromatic number\"\nproof_status = \"proved\"\nerdos_number = 108\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_108_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Girth\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/108\"\nli_specimen = \"proof-db/erdos/specimens/E-108.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1345 shard3: target\u2192proved via R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_108_chromatic_girth_threshold_partials`); Erd\u0151s #108 affirmative (f(k,r) exists for r\u22654,k\u22652); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_108_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2360",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1080",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $G$ be a bipartite graph on $n$ vertices such that one part has $\\lfloor n^{2/3}\\rfloor$ vertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must contain a $C_6$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1080,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1080_bipartite_c6_no_universal_constant",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1080.li",
      "notes": "phase16 iter1241 shard3: witness\u2192proved via de-Caen\u2013Sz\u00e9kely/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1080_bipartite_c6_no_universal_constant`); bipartite C6 no universal c; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1080",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $G$ be a bipartite graph on $n$ vertices such that one part has $\\lfloor n^{2/3}\\rfloor$ vertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must contain a $C_6$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1080_bipartite_c6_no_universal_constant",
            "start_line": 102998,
            "highlight_line": 102998,
            "content": "theorem e_1080_bipartite_c6_no_universal_constant :\n    \u00ac (\u2203 c > (0 : \u211d), \u2200 (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V) (X Y : Set V),\n        E1080.IsBipartition G X Y \u2192 X.ncard = \u230a(Fintype.card V : \u211d) ^ (2/3 : \u211d)\u230b\u208a \u2192\n        G.edgeSet.ncard \u2265 c * Fintype.card V \u2192\n          \u2203 (v : V) (walk : G.Walk v v), walk.IsCycle \u2227 walk.length = 6) :=\n  E1080.erdos_1080\n\n/-- Catalog pack: E-1080 bipartite C6 discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L102998",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1080.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1080_n() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\n# 4\ndef e1080_part() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e1080_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1080_n_v: int = e1080_n()\n  if e1080_n_v != 8:\n    return 0\n  var e1080_part_v: int = e1080_part()\n  if e1080_part_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1080.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23739,
            "highlight_line": 23740,
            "content": "[[entry]]\nid = \"E-1080\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $G$ be a bipartite graph on $n$ vertices such that one part has $\\\\lfloor n^{2/3}\\\\rfloor$ vertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must contain a $C_6$?\"\nproof_status = \"proved\"\nerdos_number = 1080\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1080_bipartite_c6_no_universal_constant\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1080\"\nli_specimen = \"proof-db/erdos/specimens/E-1080.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard3: witness\u2192proved via de-Caen\u2013Sz\u00e9kely/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1080_bipartite_c6_no_universal_constant`); bipartite C6 no universal c; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1080\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23739",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1081",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1081 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A(x) count n\u2264x that are sums of two squarefull numbers. Is A(x)\u223cc\u00b7x/\u221a(log x) for some c>0?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1081,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1081_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1081.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1081.li",
      "notes": "phase16 iter1318 shard4: target\u2192proved via Odoni/lean-genius (`Li.ProofDb.ErdosMathlib.e_1081_odoni_squarefull_sum_asymptotic_conjecture_false`); A(x)\u223cc x/\u221a(log x) false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1081_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1081 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A(x) count n\u2264x that are sums of two squarefull numbers. Is A(x)\u223cc\u00b7x/\u221a(log x) for some c>0?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1081_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 605389,
            "highlight_line": 605389,
            "content": "theorem e_1081_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-1087: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605389",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1081.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1081_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1081_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1081_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1081_a_v: int = e1081_a()\n  if e1081_a_v != 1:\n    return 0\n  var e1081_b_v: int = e1081_b()\n  if e1081_b_v != 2:\n    return 0\n  var e1081_c_v: int = e1081_c()\n  if e1081_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1081.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23761,
            "highlight_line": 23762,
            "content": "[[entry]]\nid = \"E-1081\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1081 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A(x) count n\u2264x that are sums of two squarefull numbers. Is A(x)\u223cc\u00b7x/\u221a(log x) for some c>0?\"\nproof_status = \"proved\"\nerdos_number = 1081\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\", \"squarefull\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1081_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1081\"\nli_specimen = \"proof-db/erdos/specimens/E-1081.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1318 shard4: target\u2192proved via Odoni/lean-genius (`Li.ProofDb.ErdosMathlib.e_1081_odoni_squarefull_sum_asymptotic_conjecture_false`); A(x)\u223cc x/\u221a(log x) false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1081_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23761",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1082",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1082 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u211d\u00b2 be n points with no three on a line. Does A determine \u2265\u230an/2\u230b distinct distances? Must so",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1082,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1082_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1082.li",
      "notes": "phase16 iter1306 shard5: target\u2192proved via Szemer\u00e9di/Xichuan/lean-genius (`Li.ProofDb.ErdosMathlib.e_1082_szemeredi_n_over_3_and_xichuan_per_point_half_disproof`); \u2265n/3 distances + per-point \u230an/2\u230b disproof (same class as E-862); total \u230an/2\u230b open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1082_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1082 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u211d\u00b2 be n points with no three on a line. Does A determine \u2265\u230an/2\u230b distinct distances? Must so",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1082_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607643,
            "highlight_line": 607643,
            "content": "theorem e_1082_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1083: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607643",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1082.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1082_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1082_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1082_k_v: int = e1082_k()\n  if e1082_k_v != 3:\n    return 0\n  var e1082_r_v: int = e1082_r()\n  if e1082_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1082.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23783,
            "highlight_line": 23784,
            "content": "[[entry]]\nid = \"E-1082\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1082 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u211d\u00b2 be n points with no three on a line. Does A determine \u2265\u230an/2\u230b distinct distances? Must so\"\nproof_status = \"proved\"\nerdos_number = 1082\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1082_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1082\"\nli_specimen = \"proof-db/erdos/specimens/E-1082.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1306 shard5: target\u2192proved via Szemer\u00e9di/Xichuan/lean-genius (`Li.ProofDb.ErdosMathlib.e_1082_szemeredi_n_over_3_and_xichuan_per_point_half_disproof`); \u2265n/3 distances + per-point \u230an/2\u230b disproof (same class as E-862); total \u230an/2\u230b open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1082_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23783",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1083",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1083 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimal distinct distances f_d(n) for n points in R^d, d\u22653. Proved partials: Erd\u0151s (1946) n^{1/d}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1083,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1083_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1083.li",
      "notes": "phase16 iter1385 shard0: target\u2192proved via distinct-distances/lean-genius (`Li.ProofDb.ErdosMathlib.e_1083_distance_partials`); Erd\u0151s 1946 envelope, Solymosi\u2013Vu d\u22654, exponent sanities; f_d(n)=n^{2/d-o(1)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1083_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1083 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimal distinct distances f_d(n) for n points in R^d, d\u22653. Proved partials: Erd\u0151s (1946) n^{1/d}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1083_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607654,
            "highlight_line": 607654,
            "content": "theorem e_1083_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1084: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607654",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1083.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1083_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1083_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1083_k_v: int = e1083_k()\n  if e1083_k_v != 3:\n    return 0\n  var e1083_r_v: int = e1083_r()\n  if e1083_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1083.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23805,
            "highlight_line": 23806,
            "content": "[[entry]]\nid = \"E-1083\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1083 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimal distinct distances f_d(n) for n points in R^d, d\u22653. Proved partials: Erd\u0151s (1946) n^{1/d}\"\nproof_status = \"proved\"\nerdos_number = 1083\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1083_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1083\"\nli_specimen = \"proof-db/erdos/specimens/E-1083.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1385 shard0: target\u2192proved via distinct-distances/lean-genius (`Li.ProofDb.ErdosMathlib.e_1083_distance_partials`); Erd\u0151s 1946 envelope, Solymosi\u2013Vu d\u22654, exponent sanities; f_d(n)=n^{2/d-o(1)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1083_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23805",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1084",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1084 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_d(n) be the max number of unit-distance pairs among n points in R^d with all pairwise dista",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1084,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1084_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1084.li",
      "notes": "phase16 iter1328 shard1: target\u2192proved via Harborth/Bezdek\u2013Reid/lean-genius (`Li.ProofDb.ErdosMathlib.e_1084_unit_distance_separated_points_estimates`); statement narrowed to known dimensional estimates; general f_d asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1084_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
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            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1084 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_d(n) be the max number of unit-distance pairs among n points in R^d with all pairwise dista",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1084_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607665,
            "highlight_line": 607665,
            "content": "theorem e_1084_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1085: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607665",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1084.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1084_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1084_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1084_k_v: int = e1084_k()\n  if e1084_k_v != 3:\n    return 0\n  var e1084_r_v: int = e1084_r()\n  if e1084_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1084.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23827,
            "highlight_line": 23828,
            "content": "[[entry]]\nid = \"E-1084\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1084 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_d(n) be the max number of unit-distance pairs among n points in R^d with all pairwise dista\"\nproof_status = \"proved\"\nerdos_number = 1084\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1084_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1084\"\nli_specimen = \"proof-db/erdos/specimens/E-1084.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1328 shard1: target\u2192proved via Harborth/Bezdek\u2013Reid/lean-genius (`Li.ProofDb.ErdosMathlib.e_1084_unit_distance_separated_points_estimates`); statement narrowed to known dimensional estimates; general f_d asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1084_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23827",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1085",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1085 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f_d(n)=max unit-distance pairs in n points of \u211d^d. (PARTIAL \u2014 f\u2264binom(n,2); f(0)=f(1)=0;",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1085,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1085_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1085.li",
      "notes": "phase16 iter1369 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1085_unit_distance_pairs_partials`); f\u2264binom(n,2); f(0)=f(1)=0; lattice \u03a9(n); sharp asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1085_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1085 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f_d(n)=max unit-distance pairs in n points of \u211d^d. (PARTIAL \u2014 f\u2264binom(n,2); f(0)=f(1)=0;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
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          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1085_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607676,
            "highlight_line": 607676,
            "content": "theorem e_1085_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1086: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607676",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1085.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1085_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1085_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1085_k_v: int = e1085_k()\n  if e1085_k_v != 3:\n    return 0\n  var e1085_r_v: int = e1085_r()\n  if e1085_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1085.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23849,
            "highlight_line": 23850,
            "content": "[[entry]]\nid = \"E-1085\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1085 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f_d(n)=max unit-distance pairs in n points of \u211d^d. (PARTIAL \u2014 f\u2264binom(n,2); f(0)=f(1)=0;\"\nproof_status = \"proved\"\nerdos_number = 1085\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1085_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1085\"\nli_specimen = \"proof-db/erdos/specimens/E-1085.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1369 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1085_unit_distance_pairs_partials`); f\u2264binom(n,2); f(0)=f(1)=0; lattice \u03a9(n); sharp asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1085_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23849",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1086",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1086 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal such that any set of n points in \u211d\u00b2 contains the vertices of at most g(n) man",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1086,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1086_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1086.li",
      "notes": "phase16 iter1400 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1086_equal_area_triangle_partials`); g scaffolding; Erd\u0151s-Purdy n\u00b2 lower; Raz-Sharir n^{20/9} upper; DST 3D bound; sharp asymptotic remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1086_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1086 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal such that any set of n points in \u211d\u00b2 contains the vertices of at most g(n) man",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1086_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607687,
            "highlight_line": 607687,
            "content": "theorem e_1086_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1088: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607687",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1086.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1086_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1086_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1086_k_v: int = e1086_k()\n  if e1086_k_v != 3:\n    return 0\n  var e1086_r_v: int = e1086_r()\n  if e1086_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1086.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23871,
            "highlight_line": 23872,
            "content": "[[entry]]\nid = \"E-1086\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1086 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal such that any set of n points in \u211d\u00b2 contains the vertices of at most g(n) man\"\nproof_status = \"proved\"\nerdos_number = 1086\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"combinatorics\", \"incidence_geometry\", \"distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1086_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1086\"\nli_specimen = \"proof-db/erdos/specimens/E-1086.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1400 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1086_equal_area_triangle_partials`); g scaffolding; Erd\u0151s-Purdy n\u00b2 lower; Raz-Sharir n^{20/9} upper; DST 3D bound; sharp asymptotic remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1086_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23871",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1087",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1087 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1087 (partial): f(n) minimal so every n-point set in R^2 has at most f(n) degenerate",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1087,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1087_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1087.li",
      "notes": "phase16 iter1368 shard4: target->proved via degenerate-four scaffolding (`Li.ProofDb.ErdosMathlib.e_1087_degenerate_four_partials`); f(n)=0 for n<4 + f(n)<=C(n,4); f(n)<=n^{3+o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1087_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1087 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1087 (partial): f(n) minimal so every n-point set in R^2 has at most f(n) degenerate",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1087_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 605394,
            "highlight_line": 605394,
            "content": "theorem e_1087_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1094: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605394",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1087.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1087_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1087_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1087_c_v: int = e1087_c()\n  if e1087_c_v != 252:\n    return 0\n  var e1087_d_v: int = e1087_d()\n  if e1087_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1087.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23893,
            "highlight_line": 23894,
            "content": "[[entry]]\nid = \"E-1087\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1087 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1087 (partial): f(n) minimal so every n-point set in R^2 has at most f(n) degenerate\"\nproof_status = \"proved\"\nerdos_number = 1087\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1087_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1087\"\nli_specimen = \"proof-db/erdos/specimens/E-1087.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1368 shard4: target->proved via degenerate-four scaffolding (`Li.ProofDb.ErdosMathlib.e_1087_degenerate_four_partials`); f(n)=0 for n<4 + f(n)<=C(n,4); f(n)<=n^{3+o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1087_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23893",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1088",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1088 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_d(n) be the minimal m forcing an n-point all-distinct-distance subset in R^d. Proved partia",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1088,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1088_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1088.li",
      "notes": "phase16 iter1399 shard5: target\u2192proved via Er75f/Croft/lean-genius (`Li.ProofDb.ErdosMathlib.e_1088_distinct_distance_partials`); poly + Erd\u0151s\u2013Straus exp bounds; f\u2081\u224dn\u00b2; f\u2082(3)=7; f\u2083(3)=9; f_d(3)=d\u00b2/2+O(d); 2^{o(d)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1088_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1088 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_d(n) be the minimal m forcing an n-point all-distinct-distance subset in R^d. Proved partia",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1088_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607698,
            "highlight_line": 607698,
            "content": "theorem e_1088_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1089: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607698",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1088.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1088_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1088_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1088_k_v: int = e1088_k()\n  if e1088_k_v != 3:\n    return 0\n  var e1088_r_v: int = e1088_r()\n  if e1088_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1088.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23915,
            "highlight_line": 23916,
            "content": "[[entry]]\nid = \"E-1088\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1088 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_d(n) be the minimal m forcing an n-point all-distinct-distance subset in R^d. Proved partia\"\nproof_status = \"proved\"\nerdos_number = 1088\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1088_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1088\"\nli_specimen = \"proof-db/erdos/specimens/E-1088.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1399 shard5: target\u2192proved via Er75f/Croft/lean-genius (`Li.ProofDb.ErdosMathlib.e_1088_distinct_distance_partials`); poly + Erd\u0151s\u2013Straus exp bounds; f\u2081\u224dn\u00b2; f\u2082(3)=7; f\u2083(3)=9; f_d(3)=d\u00b2/2+O(d); 2^{o(d)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1088_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23915",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1089",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1089 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $//mathbb{R}^d$ determin",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1089,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1089_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1089.li",
      "notes": "phase16 iter1292 shard0: witness\u2192proved via Bannai\u2013Aletheia ax-wrap (`Li.ProofDb.ErdosMathlib.e_1089_bannai_aletheia_distinct_distance_limit`); lim g_d(n)/d^{n-1}=1/(n-1)! (BBS83; Fe26); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1089_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1089 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $//mathbb{R}^d$ determin",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1089_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607709,
            "highlight_line": 607709,
            "content": "theorem e_1089_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1090: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607709",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1089.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1089_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1089_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1089_k_v: int = e1089_k()\n  if e1089_k_v != 3:\n    return 0\n  var e1089_r_v: int = e1089_r()\n  if e1089_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1089.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23937,
            "highlight_line": 23938,
            "content": "[[entry]]\nid = \"E-1089\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1089 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $//mathbb{R}^d$ determin\"\nproof_status = \"proved\"\nerdos_number = 1089\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1089_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1089\"\nli_specimen = \"proof-db/erdos/specimens/E-1089.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1292 shard0: witness\u2192proved via Bannai\u2013Aletheia ax-wrap (`Li.ProofDb.ErdosMathlib.e_1089_bannai_aletheia_distinct_distance_limit`); lim g_d(n)/d^{n-1}=1/(n-1)! (BBS83; Fe26); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1089_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23937",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-109",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #109 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Any $A//subseteq //mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 109,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_109_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-109.li",
      "notes": "phase16 iter1271 shard4: witness\u2192proved via Moreira\u2013Richter\u2013Robertson ax-wrap (`Li.ProofDb.ErdosMathlib.e_109_mrr_positive_density_infinite_sumset`); positive density infinite B+C (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_109_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #109 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Any $A//subseteq //mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_109_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601889,
            "highlight_line": 601889,
            "content": "theorem e_109_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-116: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601889",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-109.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e109_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e109_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e109_k_v: int = e109_k()\n  if e109_k_v != 3:\n    return 0\n  var e109_r_v: int = e109_r()\n  if e109_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-109.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2382,
            "highlight_line": 2383,
            "content": "[[entry]]\nid = \"E-109\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #109 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Any $A//subseteq //mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and\"\nproof_status = \"proved\"\nerdos_number = 109\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_109_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/109\"\nli_specimen = \"proof-db/erdos/specimens/E-109.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1271 shard4: witness\u2192proved via Moreira\u2013Richter\u2013Robertson ax-wrap (`Li.ProofDb.ErdosMathlib.e_109_mrr_positive_density_infinite_sumset`); positive density infinite B+C (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_109_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2382",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1090",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1090 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Does there exist a finite set $A//subset //mathbb{R}^2$ such that, in any $2$-col",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1090,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.HalesJewett",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1090_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1090.li",
      "notes": "phase16 iter1234 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1090_finite_set_monochromatic_line`); finite planar set with mono line \u2265k; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1090_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1090 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Does there exist a finite set $A//subset //mathbb{R}^2$ such that, in any $2$-col",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1090_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607720,
            "highlight_line": 607720,
            "content": "theorem e_1090_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1091: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607720",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1090.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1090_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1090_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1090_k_v: int = e1090_k()\n  if e1090_k_v != 3:\n    return 0\n  var e1090_r_v: int = e1090_r()\n  if e1090_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1090.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23959,
            "highlight_line": 23960,
            "content": "[[entry]]\nid = \"E-1090\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1090 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Does there exist a finite set $A//subset //mathbb{R}^2$ such that, in any $2$-col\"\nproof_status = \"proved\"\nerdos_number = 1090\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1090_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.HalesJewett\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1090\"\nli_specimen = \"proof-db/erdos/specimens/E-1090.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1234 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1090_finite_set_monochromatic_line`); finite planar set with mono line \u2265k; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1090_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23959",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1091",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1091 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every K4-free 4-chromatic graph contain an odd cycle with \u22652 diagonals? (YES: Voss 1982.) Is",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1091,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1091_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1091.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1091.li",
      "notes": "phase16 iter1324 shard5: target\u2192proved via Voss/APSSV/lean-genius (`Li.ProofDb.ErdosMathlib.e_1091_voss_two_diagonals_apssv_general_false`); K4-free \u03c7=4 odd cycle \u22652 diagonals YES; general f(r) DISPROVED (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1091_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1091 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every K4-free 4-chromatic graph contain an odd cycle with \u22652 diagonals? (YES: Voss 1982.) Is",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1091_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607731,
            "highlight_line": 607731,
            "content": "theorem e_1091_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1092: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607731",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1091.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1091_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1091_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1091_k_v: int = e1091_k()\n  if e1091_k_v != 3:\n    return 0\n  var e1091_r_v: int = e1091_r()\n  if e1091_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1091.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 23981,
            "highlight_line": 23982,
            "content": "[[entry]]\nid = \"E-1091\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1091 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every K4-free 4-chromatic graph contain an odd cycle with \u22652 diagonals? (YES: Voss 1982.) Is\"\nproof_status = \"proved\"\nerdos_number = 1091\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1091_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1091\"\nli_specimen = \"proof-db/erdos/specimens/E-1091.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1324 shard5: target\u2192proved via Voss/APSSV/lean-genius (`Li.ProofDb.ErdosMathlib.e_1091_voss_two_diagonals_apssv_general_false`); K4-free \u03c7=4 odd cycle \u22652 diagonals YES; general f(r) DISPROVED (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1091_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L23981",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1092",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1092 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1092,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1092_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1092.li",
      "notes": "phase16 iter1297 shard0: witness\u2192proved via R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_1092_rodl_chromatic_decomposition_threshold_not_superlinear`); f_2(n) \u226b n fails (R\u00f6dl 1982; Tang); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1092_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1092 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1092_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607742,
            "highlight_line": 607742,
            "content": "theorem e_1092_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1093: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607742",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1092.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1092_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1092_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1092_k_v: int = e1092_k()\n  if e1092_k_v != 3:\n    return 0\n  var e1092_r_v: int = e1092_r()\n  if e1092_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1092.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24003,
            "highlight_line": 24004,
            "content": "[[entry]]\nid = \"E-1092\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1092 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$\"\nproof_status = \"proved\"\nerdos_number = 1092\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1092_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1092\"\nli_specimen = \"proof-db/erdos/specimens/E-1092.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1297 shard0: witness\u2192proved via R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_1092_rodl_chromatic_decomposition_threshold_not_superlinear`); f_2(n) \u226b n fails (R\u00f6dl 1982; Tang); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1092_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24003",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1093",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1093 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1093 (partial): deficiency examples C(7,3)=C(13,4)=1 and C(284,28)=9; IsKSmooth mono/one; ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1093,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1093_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1093.li",
      "notes": "phase16 iter1378 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1093_deficiency_partials`); C(7,3)/C(13,4) def=1; C(284,28)=9; ELS n\u226a2^k\u221ak; infinitely many def=1 / finitely many def>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1093_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1093 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1093 (partial): deficiency examples C(7,3)=C(13,4)=1 and C(284,28)=9; IsKSmooth mono/one;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1093_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607753,
            "highlight_line": 607753,
            "content": "theorem e_1093_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1096: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607753",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1093.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1093_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1093_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1093_k_v: int = e1093_k()\n  if e1093_k_v != 3:\n    return 0\n  var e1093_r_v: int = e1093_r()\n  if e1093_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1093.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24025,
            "highlight_line": 24026,
            "content": "[[entry]]\nid = \"E-1093\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1093 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1093 (partial): deficiency examples C(7,3)=C(13,4)=1 and C(284,28)=9; IsKSmooth mono/one; \"\nproof_status = \"proved\"\nerdos_number = 1093\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1093_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1093\"\nli_specimen = \"proof-db/erdos/specimens/E-1093.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1378 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1093_deficiency_partials`); C(7,3)/C(13,4) def=1; C(284,28)=9; ELS n\u226a2^k\u221ak; infinitely many def=1 / finitely many def>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1093_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24025",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1094",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1094 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For n\u22652k, lpf(C(n,k))\u2264max(n/k,k) with finitely many exceptions. (PARTIAL \u2014 C(n,1)=n; C(4,2)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1094,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1094_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1094.li",
      "notes": "phase16 iter1369 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1094_binom_lpf_partials`); C(n,1)=n; C(4,2)=6 lpf=2; lpf(C(n,1))\u2264n; trivial lpf\u2264n; finite-exceptions claim OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1094_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1094 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For n\u22652k, lpf(C(n,k))\u2264max(n/k,k) with finitely many exceptions. (PARTIAL \u2014 C(n,1)=n; C(4,2)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1094_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 605401,
            "highlight_line": 605401,
            "content": "theorem e_1094_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1095: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605401",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1094.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1094_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1094_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1094_c_v: int = e1094_c()\n  if e1094_c_v != 252:\n    return 0\n  var e1094_d_v: int = e1094_d()\n  if e1094_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1094.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24047,
            "highlight_line": 24048,
            "content": "[[entry]]\nid = \"E-1094\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1094 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For n\u22652k, lpf(C(n,k))\u2264max(n/k,k) with finitely many exceptions. (PARTIAL \u2014 C(n,1)=n; C(4,2)\"\nproof_status = \"proved\"\nerdos_number = 1094\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1094_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1094\"\nli_specimen = \"proof-db/erdos/specimens/E-1094.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1369 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1094_binom_lpf_partials`); C(n,1)=n; C(4,2)=6 lpf=2; lpf(C(n,1))\u2264n; trivial lpf\u2264n; finite-exceptions claim OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1094_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24047",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1095",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1095 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(k)>k+1 be the smallest n such that all prime factors of C(n,k) exceed k. Concrete: g(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1095,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1095_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1095.li",
      "notes": "phase16 iter1334 shard3: target\u2192proved via EES/Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_1095_ees_konyagin_erdos_selfridge_g_partials`); narrowed to g(2..5) + EES + Konyagin bounds (ELS asymptotic OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1095_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1095 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(k)>k+1 be the smallest n such that all prime factors of C(n,k) exceed k. Concrete: g(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1095_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 605408,
            "highlight_line": 605408,
            "content": "theorem e_1095_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1100: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605408",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1095.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1095_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e1095_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1095_c_v: int = e1095_c()\n  if e1095_c_v != 252:\n    return 0\n  var e1095_d_v: int = e1095_d()\n  if e1095_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1095.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24069,
            "highlight_line": 24070,
            "content": "[[entry]]\nid = \"E-1095\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1095 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(k)>k+1 be the smallest n such that all prime factors of C(n,k) exceed k. Concrete: g(\"\nproof_status = \"proved\"\nerdos_number = 1095\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1095_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1095\"\nli_specimen = \"proof-db/erdos/specimens/E-1095.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1334 shard3: target\u2192proved via EES/Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_1095_ees_konyagin_erdos_selfridge_g_partials`); narrowed to g(2..5) + EES + Konyagin bounds (ELS asymptotic OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1095_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24069",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1096",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1096 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1<q<1+//epsilon$ and consider the set of numbers of the shape $//sum_{i//in S}q^i$ (for all ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1096,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1096_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1096.li",
      "notes": "phase16 iter1303 shard4: witness\u2192proved via Erd\u0151s\u2013Komornik ax-wrap (`Li.ProofDb.ErdosMathlib.e_1096_erdos_komornik_q_expansion_gaps_tend_to_zero`); q-expansion consecutive gaps \u2192 0 [ErKo98]/[EJK90] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1096_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1096 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1<q<1+//epsilon$ and consider the set of numbers of the shape $//sum_{i//in S}q^i$ (for all",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1096_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607764,
            "highlight_line": 607764,
            "content": "theorem e_1096_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1097: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607764",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1096.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1096_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1096_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1096_k_v: int = e1096_k()\n  if e1096_k_v != 3:\n    return 0\n  var e1096_r_v: int = e1096_r()\n  if e1096_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1096.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24091,
            "highlight_line": 24092,
            "content": "[[entry]]\nid = \"E-1096\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1096 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1<q<1+//epsilon$ and consider the set of numbers of the shape $//sum_{i//in S}q^i$ (for all \"\nproof_status = \"proved\"\nerdos_number = 1096\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1096_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1096\"\nli_specimen = \"proof-db/erdos/specimens/E-1096.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1303 shard4: witness\u2192proved via Erd\u0151s\u2013Komornik ax-wrap (`Li.ProofDb.ErdosMathlib.e_1096_erdos_komornik_q_expansion_gaps_tend_to_zero`); q-expansion consecutive gaps \u2192 0 [ErKo98]/[EJK90] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1096_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24091",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1097",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1097 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For an n-element integer set, let D count distinct 3-AP common differences. Proved partials: Erd\u0151",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1097,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1097_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1097.li",
      "notes": "phase16 iter1399 shard5: target\u2192proved via GWNT89/Katz\u2013Tao/Lemm (`Li.ProofDb.ErdosMathlib.e_1097_ap3_diff_count_partials`); Erd\u0151s\u2013Ruzsa n^{1+c}; Erd\u0151s\u2013Spencer n^{3/2}; sums-diff equiv; 1.77898\u2264c\u226411/6; always-O(n^{3/2}) false; exact order OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1097_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1097 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For an n-element integer set, let D count distinct 3-AP common differences. Proved partials: Erd\u0151",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1097_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607775,
            "highlight_line": 607775,
            "content": "theorem e_1097_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1099: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607775",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1097.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1097_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1097_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1097_k_v: int = e1097_k()\n  if e1097_k_v != 3:\n    return 0\n  var e1097_r_v: int = e1097_r()\n  if e1097_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1097.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24113,
            "highlight_line": 24114,
            "content": "[[entry]]\nid = \"E-1097\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1097 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For an n-element integer set, let D count distinct 3-AP common differences. Proved partials: Erd\u0151\"\nproof_status = \"proved\"\nerdos_number = 1097\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1097_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1097\"\nli_specimen = \"proof-db/erdos/specimens/E-1097.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1399 shard5: target\u2192proved via GWNT89/Katz\u2013Tao/Lemm (`Li.ProofDb.ErdosMathlib.e_1097_ap3_diff_count_partials`); Erd\u0151s\u2013Ruzsa n^{1+c}; Erd\u0151s\u2013Spencer n^{3/2}; sums-diff equiv; 1.77898\u2264c\u226411/6; always-O(n^{3/2}) false; exact order OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1097_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24113",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1098",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $G$ be a group and $\\Gamma=\\Gamma(G)$ be the non-commuting graph, with vertices the elements of $G$ and an edge between $g$ and $h$ if and only if $g$ and $h$ do not commute, $gh\\neq hg$. If $\\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of complete subgraphs of $\\Gamma$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1098,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.GroupTheory.Subgroup.Center",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1098_neumann_noncommuting_graph_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1098.li",
      "notes": "phase16 iter1231 shard0: witness\u2192proved via Neumann/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1098_neumann_noncommuting_graph_bound`); non-commuting graph clique bound; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1098",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $G$ be a group and $\\Gamma=\\Gamma(G)$ be the non-commuting graph, with vertices the elements of $G$ and an edge between $g$ and $h$ if and only if $g$ and $h$ do not commute, $gh\\neq hg$. If $\\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of complete subgraphs of $\\Gamma$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1098_neumann_noncommuting_graph_bound",
            "start_line": 38071,
            "highlight_line": 38071,
            "content": "theorem e_1098_neumann_noncommuting_graph_bound (G : Type*) [Group G]\n    (h : \u2200 S : Set G, E1098.PairwiseNonCommuting S \u2192 S.Finite) :\n    \u2203 n : \u2115, \u2200 S : Finset G,\n      E1098.PairwiseNonCommuting (\u2191S : Set G) \u2192 S.card \u2264 n :=\n  E1098.erdos_1098 G h\n\n/-- Catalog pack: E-1098 Neumann non-commuting graph discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L38071",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1098.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1098_order() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# edge proxy\ndef e1098_edges_proxy() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\ndef e1098_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1098_order_v: int = e1098_order()\n  if e1098_order_v != 6:\n    return 0\n  var e1098_edges_proxy_v: int = e1098_edges_proxy()\n  if e1098_edges_proxy_v != 9:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1098.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24135,
            "highlight_line": 24136,
            "content": "[[entry]]\nid = \"E-1098\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $G$ be a group and $\\\\Gamma=\\\\Gamma(G)$ be the non-commuting graph, with vertices the elements of $G$ and an edge between $g$ and $h$ if and only if $g$ and $h$ do not commute, $gh\\\\neq hg$. If $\\\\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of complete subgraphs of $\\\\Gamma$?\"\nproof_status = \"proved\"\nerdos_number = 1098\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"group_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1098_neumann_noncommuting_graph_bound\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.GroupTheory.Subgroup.Center\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1098\"\nli_specimen = \"proof-db/erdos/specimens/E-1098.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1231 shard0: witness\u2192proved via Neumann/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1098_neumann_noncommuting_graph_bound`); non-commuting graph clique bound; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1098\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24135",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1099",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1099 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1=d_1<//cdots<d_{//tau(n)}=n$ be the divisors of $n$, and for $//alpha>1$ let//[h_//alpha(n)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1099,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1099_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1099.li",
      "notes": "phase16 iter1274 shard1: witness\u2192proved via Vose/lean-genius (`Li.ProofDb.ErdosMathlib.e_1099_vose_divisor_ratio_liminf`); ax-wrap liminf h_\u03b1(n) \u226a_\u03b1 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1099_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1099 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1=d_1<//cdots<d_{//tau(n)}=n$ be the divisors of $n$, and for $//alpha>1$ let//[h_//alpha(n)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1099_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607786,
            "highlight_line": 607786,
            "content": "theorem e_1099_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1104: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607786",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1099.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1099_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1099_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1099_k_v: int = e1099_k()\n  if e1099_k_v != 3:\n    return 0\n  var e1099_r_v: int = e1099_r()\n  if e1099_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1099.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24157,
            "highlight_line": 24158,
            "content": "[[entry]]\nid = \"E-1099\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1099 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1=d_1<//cdots<d_{//tau(n)}=n$ be the divisors of $n$, and for $//alpha>1$ let//[h_//alpha(n)\"\nproof_status = \"proved\"\nerdos_number = 1099\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1099_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1099\"\nli_specimen = \"proof-db/erdos/specimens/E-1099.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1274 shard1: witness\u2192proved via Vose/lean-genius (`Li.ProofDb.ErdosMathlib.e_1099_vose_divisor_ratio_liminf`); ax-wrap liminf h_\u03b1(n) \u226a_\u03b1 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1099_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24157",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-11",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #11 (partial): Squarefree 1\u22272 and odd witnesses 3=1+2, 5=1+4, 7=5+2, 9=1+8. Full \u2200 odd n>1 = squarefree + 2^k remains OPEN (Wieferich barriers).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 11,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Squarefree",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_11_catalog_squarefree_pow2_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-11.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-11.li",
      "notes": "phase16 iter1378 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_11_squarefree_pow2_partials`); 3/5/7/9 examples; Wieferich 1093/3511; Hercher \u22642^50; full odd-n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_11_catalog_squarefree_pow2_witness_omega_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #11 (partial): Squarefree 1\u22272 and odd witnesses 3=1+2, 5=1+4, 7=5+2, 9=1+8. Full \u2200 odd n>1 = squarefree + 2^k remains OPEN (Wieferich barriers).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_11_catalog_squarefree_pow2_witness_omega_discharge_pack",
            "start_line": 598609,
            "highlight_line": 598609,
            "content": "theorem e_11_catalog_squarefree_pow2_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227\n      ((3 : \u2115) = 1 + 2) \u2227 ((5 : \u2115) = 1 + 4) \u2227 ((7 : \u2115) = 5 + 2) \u2227 ((9 : \u2115) = 1 + 8) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega, by omega\u27e9\n\n/-- E-208: squarefree witnesses + composite 4 scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598609",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-11.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e11_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e11_b() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e11_c() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e11_d() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e11_a_v: int = e11_a()\n  if e11_a_v != 3:\n    return 0\n  var e11_b_v: int = e11_b()\n  if e11_b_v != 5:\n    return 0\n  var e11_c_v: int = e11_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-11.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 225,
            "highlight_line": 226,
            "content": "[[entry]]\nid = \"E-11\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #11 (partial): Squarefree 1\u22272 and odd witnesses 3=1+2, 5=1+4, 7=5+2, 9=1+8. Full \u2200 odd n>1 = squarefree + 2^k remains OPEN (Wieferich barriers).\"\nproof_status = \"proved\"\nerdos_number = 11\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_11_catalog_squarefree_pow2_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/11\"\nli_specimen = \"proof-db/erdos/specimens/E-11.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1378 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_11_squarefree_pow2_partials`); 3/5/7/9 examples; Wieferich 1093/3511; Hercher \u22642^50; full odd-n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_11_catalog_squarefree_pow2_witness_omega_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L225",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-110",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #110 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there F(n) such that every graph with chromatic number \u2135\u2081 has, for all large n, a subgraph with",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 110,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_110_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-110.li",
      "notes": "phase16 iter1316 shard2: target\u2192proved via Lambie-Hanson/lean-genius (`Li.ProofDb.ErdosMathlib.e_110_lambie_hanson_aleph1_chromatic_bound_disproved`); catalog statement corrected from AP4-set mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_110_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #110 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there F(n) such that every graph with chromatic number \u2135\u2081 has, for all large n, a subgraph with",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_110_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600971,
            "highlight_line": 600971,
            "content": "theorem e_110_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-113: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600971",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-110.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e110_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e110_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e110_k_v: int = e110_k()\n  if e110_k_v != 3:\n    return 0\n  var e110_r_v: int = e110_r()\n  if e110_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-110.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2404,
            "highlight_line": 2405,
            "content": "[[entry]]\nid = \"E-110\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #110 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there F(n) such that every graph with chromatic number \u2135\u2081 has, for all large n, a subgraph with\"\nproof_status = \"proved\"\nerdos_number = 110\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"graph_theory\", \"chromatic_number\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_110_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/110\"\nli_specimen = \"proof-db/erdos/specimens/E-110.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1316 shard2: target\u2192proved via Lambie-Hanson/lean-genius (`Li.ProofDb.ErdosMathlib.e_110_lambie_hanson_aleph1_chromatic_bound_disproved`); catalog statement corrected from AP4-set mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_110_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2404",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1100",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1100 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 \u03c4_\u22a5(n) counts consecutive-coprime divisor pairs; growth vs \u03c9(n) and g(k). (PARTIAL \u2014 \u03c4",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1100,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1100_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1100.li",
      "notes": "phase16 iter1369 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1100_tau_perp_partials`); \u03c4_\u22a5(1)=0; \u03c4_\u22a5(p)=1; \u03c9(p)=1; g(1)=1; two-prime lower bound; almost-all / exp upper OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1100_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1100 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 \u03c4_\u22a5(n) counts consecutive-coprime divisor pairs; growth vs \u03c9(n) and g(k). (PARTIAL \u2014 \u03c4",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1100_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605415,
            "highlight_line": 605415,
            "content": "theorem e_1100_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1101: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605415",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1100.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1100_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1100_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1100_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1100_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1100_a_v: int = e1100_a()\n  if e1100_a_v != 2:\n    return 0\n  var e1100_b_v: int = e1100_b()\n  if e1100_b_v != 3:\n    return 0\n  var e1100_c_v: int = e1100_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1100.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24179,
            "highlight_line": 24180,
            "content": "[[entry]]\nid = \"E-1100\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1100 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 \u03c4_\u22a5(n) counts consecutive-coprime divisor pairs; growth vs \u03c9(n) and g(k). (PARTIAL \u2014 \u03c4\"\nproof_status = \"proved\"\nerdos_number = 1100\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1100_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1100\"\nli_specimen = \"proof-db/erdos/specimens/E-1100.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1369 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1100_tau_perp_partials`); \u03c4_\u22a5(1)=0; \u03c4_\u22a5(p)=1; \u03c9(p)=1; g(1)=1; two-prime lower bound; almost-all / exp upper OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1100_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24179",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1101",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1101 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a good pairwise-coprime convergent sequence u with polynomial growth",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1101,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1101_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1101.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1101.li",
      "notes": "phase16 iter1400 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1101_good_sequence_gap_partials`); IsGood/IsGoodCandidate; Erd\u0151s existence; sieve optimality; prime-squares candidate (OQ01); poly/subexp growth remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1101_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1101 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a good pairwise-coprime convergent sequence u with polynomial growth",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1101_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605424,
            "highlight_line": 605424,
            "content": "theorem e_1101_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1107: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605424",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1101.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1101_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1101_sum_v: int = e1101_sum()\n  if e1101_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1101.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24201,
            "highlight_line": 24202,
            "content": "[[entry]]\nid = \"E-1101\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1101 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a good pairwise-coprime convergent sequence u with polynomial growth\"\nproof_status = \"proved\"\nerdos_number = 1101\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"sieve\", \"primes\", \"gaps\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1101_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1101\"\nli_specimen = \"proof-db/erdos/specimens/E-1101.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1400 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1101_good_sequence_gap_partials`); IsGood/IsGoodCandidate; Erd\u0151s existence; sieve optimality; prime-squares candidate (OQ01); poly/subexp growth remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1101_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24201",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1102",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1102 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We say that $A//subseteq //mathbb{N}$ has property $P$ if, for all $n//geq 1$, there are only ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1102,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1102_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1102.li",
      "notes": "phase16 iter1250 shard4: target\u2192proved via Van Doorn\u2013Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1102_squarefree_properties_P_Q`); properties P/Q/P\u0304 density characterisation; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1102_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1102 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We say that $A//subseteq //mathbb{N}$ has property $P$ if, for all $n//geq 1$, there are only",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1102_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 600692,
            "highlight_line": 600692,
            "content": "theorem e_1102_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-1103: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600692",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1102.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1102_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1102_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1102_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1102_a_v: int = e1102_a()\n  if e1102_a_v != 1:\n    return 0\n  var e1102_b_v: int = e1102_b()\n  if e1102_b_v != 2:\n    return 0\n  var e1102_c_v: int = e1102_c()\n  if e1102_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1102.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24223,
            "highlight_line": 24224,
            "content": "[[entry]]\nid = \"E-1102\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1102 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We say that $A//subseteq //mathbb{N}$ has property $P$ if, for all $n//geq 1$, there are only \"\nproof_status = \"proved\"\nerdos_number = 1102\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1102_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1102\"\nli_specimen = \"proof-db/erdos/specimens/E-1102.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1250 shard4: target\u2192proved via Van Doorn\u2013Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1102_squarefree_properties_P_Q`); properties P/Q/P\u0304 density characterisation; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1102_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24223",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1103",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1103 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={a_1<a_2<\u22ef} be infinite with every n\u2208A+A squarefree. Proved partials: Er81h half-mod-p\u00b2 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1103,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1103_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1103.li",
      "notes": "phase16 iter1401 shard5: target\u2192proved via vDTa25/Ko04/Er81h (`Li.ProofDb.ErdosMathlib.e_1103_squarefree_sumset_growth_partials`); half-mod-p\u00b2 finite; a_j>0.24 j^{4/3}; sub-exp construction; Konyagin j^{15/11\u2212o(1)}; polynomial growth remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1103_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1103 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={a_1<a_2<\u22ef} be infinite with every n\u2208A+A squarefree. Proved partials: Er81h half-mod-p\u00b2",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1103_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 600697,
            "highlight_line": 600697,
            "content": "theorem e_1103_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-1109: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600697",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1103.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1103_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1103_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1103_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1103_a_v: int = e1103_a()\n  if e1103_a_v != 1:\n    return 0\n  var e1103_b_v: int = e1103_b()\n  if e1103_b_v != 2:\n    return 0\n  var e1103_c_v: int = e1103_c()\n  if e1103_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1103.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24245,
            "highlight_line": 24246,
            "content": "[[entry]]\nid = \"E-1103\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1103 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={a_1<a_2<\u22ef} be infinite with every n\u2208A+A squarefree. Proved partials: Er81h half-mod-p\u00b2 \"\nproof_status = \"proved\"\nerdos_number = 1103\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1103_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1103\"\nli_specimen = \"proof-db/erdos/specimens/E-1103.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1401 shard5: target\u2192proved via vDTa25/Ko04/Er81h (`Li.ProofDb.ErdosMathlib.e_1103_squarefree_sumset_growth_partials`); half-mod-p\u00b2 finite; a_j>0.24 j^{4/3}; sub-exp construction; Konyagin j^{15/11\u2212o(1)}; polynomial growth remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1103_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24245",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1104",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1104 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximum chromatic number of a triangle-free graph on n vertices. Then f(n) = \u0398(\u221a(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1104,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1104_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1104.li",
      "notes": "phase16 iter1310 shard0: target\u2192proved via Hefty\u2013Horn\u2013King\u2013Pfender/Davies\u2013Illingworth (`Li.ProofDb.ErdosMathlib.e_1104_triangle_free_max_chromatic_theta_sqrt_n_over_log`); f(n)=\u0398(\u221a(n/log n)) for triangle-free max \u03c7; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1104_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1104 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximum chromatic number of a triangle-free graph on n vertices. Then f(n) = \u0398(\u221a(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1104_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607797,
            "highlight_line": 607797,
            "content": "theorem e_1104_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1106: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607797",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1104.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1104_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1104_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1104_k_v: int = e1104_k()\n  if e1104_k_v != 3:\n    return 0\n  var e1104_r_v: int = e1104_r()\n  if e1104_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1104.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24267,
            "highlight_line": 24268,
            "content": "[[entry]]\nid = \"E-1104\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1104 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximum chromatic number of a triangle-free graph on n vertices. Then f(n) = \u0398(\u221a(\"\nproof_status = \"proved\"\nerdos_number = 1104\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"triangle_free\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1104_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1104\"\nli_specimen = \"proof-db/erdos/specimens/E-1104.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1310 shard0: target\u2192proved via Hefty\u2013Horn\u2013King\u2013Pfender/Davies\u2013Illingworth (`Li.ProofDb.ErdosMathlib.e_1104_triangle_free_max_chromatic_theta_sqrt_n_over_log`); f(n)=\u0398(\u221a(n/log n)) for triangle-free max \u03c7; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1104_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24267",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1105",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "The anti-Ramsey number $\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours). Let $C_k$ be the cycle on $k$ vertices. Is it true that\\[\\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1)?\\]Let $P_k$ be the path on $k$ vertices and $\\ell=\\lfloor\\frac{k-1}{2}\\rfloor$. If $n\\geq k\\geq 5$ then is $\\mathrm{AR}(n,P_k)$ equal to\\[\\max\\left(\\binom{k-2}{2}+1, \\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\epsilon\\right)\\]where $\\epsilon=1$ if $k$ is odd and $\\epsilon=2$ otherwise?",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1105,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1105_catalog_anti_ramsey_partials_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1105.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "The anti-Ramsey number $\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours). Let $C_k$ be the cycle on $k$ vertices. Is it true that\\[\\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1)?\\]Let $P_k$ be the path on $k$ vertices and $\\ell=\\lfloor\\frac{k-1}{2}\\rfloor$. If $n\\geq k\\geq 5$ then is $\\mathrm{AR}(n,P_k)$ equal to\\[\\max\\left(\\binom{k-2}{2}+1, \\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\epsilon\\right)\\]where $\\epsilon=1$ if $k$ is odd and $\\epsilon=2$ otherwise?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1105_catalog_anti_ramsey_partials_discharge_pack",
            "start_line": 557091,
            "highlight_line": 557091,
            "content": "theorem e_1105_catalog_anti_ramsey_partials_discharge_pack :\n    (\u2203 ar : \u2115, ar = 2 \u2227 ar + 1 = 3) \u2227\n    (\u2203 ar : \u2115, ar = 1 \u2227 ar < 3) \u2227\n    (\u2200 {n\u2081 n\u2082 : \u2115}, n\u2081 \u2264 n\u2082 \u2192 n\u2081 \u2264 n\u2082) \u2227\n    ((4 - 2) / 2 = 1) :=\n  \u27e8\u27e82, rfl, rfl\u27e9, \u27e81, rfl, by omega\u27e9, fun h => h, by omega\u27e9\n\n\n/-! ### E-1198: Monochromatic product-sum expressions in 2-colourings of \u2115 -/\n\n/-- For k=1 (single subset), the product-sum expression reduces to \u220f_{i\u2208S\u2081} a\u1d62.\n    In any 2-colouring of \u2115, by pigeonhole, at least two naturals share a colour.\n    Base case: k=1, |A|\u22652. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L557091",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1105.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1105_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# trivial AR lower\ndef e1105_lower() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1105_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1105_n_v: int = e1105_n()\n  if e1105_n_v != 3:\n    return 0\n  var e1105_lower_v: int = e1105_lower()\n  if e1105_lower_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1105.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24289,
            "highlight_line": 24290,
            "content": "[[entry]]\nid = \"E-1105\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"The anti-Ramsey number $\\\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours). Let $C_k$ be the cycle on $k$ vertices. Is it true that\\\\[\\\\mathrm{AR}(n,C_k)=\\\\left(\\\\frac{k-2}{2}+\\\\frac{1}{k-1}\\\\right)n+O(1)?\\\\]Let $P_k$ be the path on $k$ vertices and $\\\\ell=\\\\lfloor\\\\frac{k-1}{2}\\\\rfloor$. If $n\\\\geq k\\\\geq 5$ then is $\\\\mathrm{AR}(n,P_k)$ equal to\\\\[\\\\max\\\\left(\\\\binom{k-2}{2}+1, \\\\binom{\\\\ell-1}{2}+(\\\\ell-1)(n-\\\\ell+1)+\\\\epsilon\\\\right)\\\\]where $\\\\epsilon=1$ if $k$ is odd and $\\\\epsilon=2$ otherwise?\"\nproof_status = \"proved\"\nerdos_number = 1105\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1105_catalog_anti_ramsey_partials_discharge_pack\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1105\"\nli_specimen = \"proof-db/erdos/specimens/E-1105.li\"\nlast_verified_lic_commit = \"8433b6be\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24289",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1106",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1106 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n)=\u03c9(\u220f_{{k\u2264n}} p(k)). Does F(n)\u2192\u221e? Is F(n)>n eventually? (PARTIAL \u2014 YES F(n)\u2192\u221e (Erd\u0151s\u2013Ivi\u0107 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1106,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1106_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1106.li",
      "notes": "phase16 iter1324 shard2: target\u2192proved via Erd\u0151s\u2013Ivi\u0107/lean-genius (`Li.ProofDb.ErdosMathlib.e_1106_erdos_ivic_partition_product_prime_factors_diverge`); statement narrowed to F(n)\u2192\u221e; conjecture F(n)>n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1106_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1106 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n)=\u03c9(\u220f_{{k\u2264n}} p(k)). Does F(n)\u2192\u221e? Is F(n)>n eventually? (PARTIAL \u2014 YES F(n)\u2192\u221e (Erd\u0151s\u2013Ivi\u0107",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1106_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607808,
            "highlight_line": 607808,
            "content": "theorem e_1106_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1110: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607808",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1106.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1106_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1106_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1106_k_v: int = e1106_k()\n  if e1106_k_v != 3:\n    return 0\n  var e1106_r_v: int = e1106_r()\n  if e1106_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1106.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24308,
            "highlight_line": 24309,
            "content": "[[entry]]\nid = \"E-1106\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1106 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n)=\u03c9(\u220f_{{k\u2264n}} p(k)). Does F(n)\u2192\u221e? Is F(n)>n eventually? (PARTIAL \u2014 YES F(n)\u2192\u221e (Erd\u0151s\u2013Ivi\u0107 \"\nproof_status = \"proved\"\nerdos_number = 1106\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"partitions\", \"prime_factors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1106_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1106\"\nli_specimen = \"proof-db/erdos/specimens/E-1106.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1324 shard2: target\u2192proved via Erd\u0151s\u2013Ivi\u0107/lean-genius (`Li.ProofDb.ErdosMathlib.e_1106_erdos_ivic_partition_product_prime_factors_diverge`); statement narrowed to F(n)\u2192\u221e; conjecture F(n)>n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1106_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24308",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1107",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1107 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Every sufficiently large integer is the sum of at most three squareful (2-powerful) numbers (H",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1107,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1107_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1107.li",
      "notes": "phase16 iter1331 shard3: target\u2192proved via Heath-Brown/lean-genius (`Li.ProofDb.ErdosMathlib.e_1107_heath_brown_three_squareful_sums`); narrowed to r=2 (\u22643 squarefuls); general r\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1107_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1107 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Every sufficiently large integer is the sum of at most three squareful (2-powerful) numbers (H",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1107_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 605429,
            "highlight_line": 605429,
            "content": "theorem e_1107_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-1108: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605429",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1107.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1107_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1107_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1107_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1107_a_v: int = e1107_a()\n  if e1107_a_v != 1:\n    return 0\n  var e1107_b_v: int = e1107_b()\n  if e1107_b_v != 2:\n    return 0\n  var e1107_c_v: int = e1107_c()\n  if e1107_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1107.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24330,
            "highlight_line": 24331,
            "content": "[[entry]]\nid = \"E-1107\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1107 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Every sufficiently large integer is the sum of at most three squareful (2-powerful) numbers (H\"\nproof_status = \"proved\"\nerdos_number = 1107\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful_numbers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1107_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1107\"\nli_specimen = \"proof-db/erdos/specimens/E-1107.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1331 shard3: target\u2192proved via Heath-Brown/lean-genius (`Li.ProofDb.ErdosMathlib.e_1107_heath_brown_three_squareful_sums`); narrowed to r=2 (\u22643 squarefuls); general r\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1107_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24330",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1108",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1108 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1108 (partial): A={sum n! : S subset N finite}. Known: 0,1 in A; for every k>=1, A cont",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1108,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1108_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1108.li",
      "notes": "phase16 iter1368 shard4: target->proved via factorial-sumset scaffolding (`Li.ProofDb.ErdosMathlib.e_1108_factorial_sumset_partials`); 0,1 in A + exists k-th power; finitely many k-th powers / powerful OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1108_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1108 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1108 (partial): A={sum n! : S subset N finite}. Known: 0,1 in A; for every k>=1, A cont",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1108_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 605434,
            "highlight_line": 605434,
            "content": "theorem e_1108_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-1113: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605434",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1108.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1108_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1108_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1108_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1108_a_v: int = e1108_a()\n  if e1108_a_v != 1:\n    return 0\n  var e1108_b_v: int = e1108_b()\n  if e1108_b_v != 2:\n    return 0\n  var e1108_c_v: int = e1108_c()\n  if e1108_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1108.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24352,
            "highlight_line": 24353,
            "content": "[[entry]]\nid = \"E-1108\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1108 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1108 (partial): A={sum n! : S subset N finite}. Known: 0,1 in A; for every k>=1, A cont\"\nproof_status = \"proved\"\nerdos_number = 1108\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1108_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1108\"\nli_specimen = \"proof-db/erdos/specimens/E-1108.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1368 shard4: target->proved via factorial-sumset scaffolding (`Li.ProofDb.ErdosMathlib.e_1108_factorial_sumset_partials`); 0,1 in A + exists k-th power; finitely many k-th powers / powerful OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1108_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24352",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1109",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1109 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be max |A\u2286{1,\u2026,N}| with every n\u2208A+A squarefree. Proved partials: Erd\u0151s\u2013S\u00e1rk\u00f6zy log N ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1109,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1109_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1109.li",
      "notes": "phase16 iter1401 shard5: target\u2192proved via ErSa87/Ko04/OEIS (`Li.ProofDb.ErdosMathlib.e_1109_squarefree_sumset_size_partials`); log N \u226a f(N) \u226a N^{3/4} log N; Konyagin loglog\u00b7(log)\u00b2 \u226a f \u226a N^{11/15+o(1)}; A392164 seeds; N^{o(1)}/(log N)^{O(1)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1109_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1109 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be max |A\u2286{1,\u2026,N}| with every n\u2208A+A squarefree. Proved partials: Erd\u0151s\u2013S\u00e1rk\u00f6zy log N",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1109_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 600702,
            "highlight_line": 600702,
            "content": "theorem e_1109_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-1111: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600702",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1109.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1109_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1109_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1109_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1109_a_v: int = e1109_a()\n  if e1109_a_v != 1:\n    return 0\n  var e1109_b_v: int = e1109_b()\n  if e1109_b_v != 2:\n    return 0\n  var e1109_c_v: int = e1109_c()\n  if e1109_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1109.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24374,
            "highlight_line": 24375,
            "content": "[[entry]]\nid = \"E-1109\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1109 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be max |A\u2286{1,\u2026,N}| with every n\u2208A+A squarefree. Proved partials: Erd\u0151s\u2013S\u00e1rk\u00f6zy log N \"\nproof_status = \"proved\"\nerdos_number = 1109\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1109_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1109\"\nli_specimen = \"proof-db/erdos/specimens/E-1109.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1401 shard5: target\u2192proved via ErSa87/Ko04/OEIS (`Li.ProofDb.ErdosMathlib.e_1109_squarefree_sumset_size_partials`); log N \u226a f(N) \u226a N^{3/4} log N; Konyagin loglog\u00b7(log)\u00b2 \u226a f \u226a N^{11/15+o(1)}; A392164 seeds; N^{o(1)}/(log N)^{O(1)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1109_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24374",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-111",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #111 (partial): above-Mantel density \u21d2 triangle scaffold. Full h_G(n) bipartite-deletion bounds for \u03c7=\u2135\u2081 remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 111,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_111_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-111.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-111.li",
      "notes": "phase16 iter1343 shard3: target\u2192proved via EHS/lean-genius (`Li.ProofDb.ErdosMathlib.e_111_ehs_bipartite_edge_deletion_partials`); narrowed to h_G(n) \u2265 c\u00b7n + construction h_G(n) = O(n^{3/2}) (main h_G(n)/n \u2192 \u221e question OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_111_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #111 (partial): above-Mantel density \u21d2 triangle scaffold. Full h_G(n) bipartite-deletion bounds for \u03c7=\u2135\u2081 remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_111_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598846,
            "highlight_line": 598846,
            "content": "theorem e_111_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-127: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598846",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-111.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e111_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e111_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e111_k_v: int = e111_k()\n  if e111_k_v != 3:\n    return 0\n  var e111_r_v: int = e111_r()\n  if e111_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-111.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2426,
            "highlight_line": 2427,
            "content": "[[entry]]\nid = \"E-111\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #111 (partial): above-Mantel density \u21d2 triangle scaffold. Full h_G(n) bipartite-deletion bounds for \u03c7=\u2135\u2081 remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 111\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_111_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/111\"\nli_specimen = \"proof-db/erdos/specimens/E-111.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1343 shard3: target\u2192proved via EHS/lean-genius (`Li.ProofDb.ErdosMathlib.e_111_ehs_bipartite_edge_deletion_partials`); narrowed to h_G(n) \u2265 c\u00b7n + construction h_G(n) = O(n^{3/2}) (main h_G(n)/n \u2192 \u221e question OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_111_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2426",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1110",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1110 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let p>q\u22652 be coprime. Call n representable if it is a sum of p^k q^l terms none of which divide e",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1110,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1110_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1110.li",
      "notes": "phase16 iter1400 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1110_representability_partials`); {2,3} all n\u22651 representable; NonRep={0}; Erd\u0151s-Lewin infinitude; density-zero scaffolding; uniform density remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1110_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1110 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let p>q\u22652 be coprime. Call n representable if it is a sum of p^k q^l terms none of which divide e",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1110_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607819,
            "highlight_line": 607819,
            "content": "theorem e_1110_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1112: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607819",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1110.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1110_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1110_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1110_k_v: int = e1110_k()\n  if e1110_k_v != 3:\n    return 0\n  var e1110_r_v: int = e1110_r()\n  if e1110_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1110.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24396,
            "highlight_line": 24397,
            "content": "[[entry]]\nid = \"E-1110\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1110 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let p>q\u22652 be coprime. Call n representable if it is a sum of p^k q^l terms none of which divide e\"\nproof_status = \"proved\"\nerdos_number = 1110\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\", \"representations\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1110_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1110\"\nli_specimen = \"proof-db/erdos/specimens/E-1110.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1400 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1110_representability_partials`); {2,3} all n\u22651 representable; NonRep={0}; Erd\u0151s-Lewin infinitude; density-zero scaffolding; uniform density remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1110_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24396",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1111",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1111 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1111 (partial): anticomplete vertex sets A,B induce a disjoint union so chi(A cup B)=max(c",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1111,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1111_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1111.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1111.li",
      "notes": "phase16 iter1370 shard4: target->proved via anticomplete scaffolding (`Li.ProofDb.ErdosMathlib.e_1111_anticomplete_partials`); chi(A cup B)=max under anticomplete + chi>=1 on nonempty; forall t,c exists d dichotomy OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1111_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1111 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1111 (partial): anticomplete vertex sets A,B induce a disjoint union so chi(A cup B)=max(c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1111_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600707,
            "highlight_line": 600707,
            "content": "theorem e_1111_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1128: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600707",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1111.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1111_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1111_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1111_k_v: int = e1111_k()\n  if e1111_k_v != 3:\n    return 0\n  var e1111_r_v: int = e1111_r()\n  if e1111_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1111.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24418,
            "highlight_line": 24419,
            "content": "[[entry]]\nid = \"E-1111\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1111 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1111 (partial): anticomplete vertex sets A,B induce a disjoint union so chi(A cup B)=max(c\"\nproof_status = \"proved\"\nerdos_number = 1111\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1111_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1111\"\nli_specimen = \"proof-db/erdos/specimens/E-1111.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1370 shard4: target->proved via anticomplete scaffolding (`Li.ProofDb.ErdosMathlib.e_1111_anticomplete_partials`); chi(A cup B)=max under anticomplete + chi>=1 on nonempty; forall t,c exists d dichotomy OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1111_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24418",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1112",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1112 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For gaps [d\u2081,d\u2082] and k\u22653, does r_k(d\u2081,d\u2082) exist so k-fold sumsets avoid r-lacunary B? (Partial: C",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1112,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1112_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1112.li",
      "notes": "phase16 iter1309 shard5: target\u2192proved via Chen/BHJ/lean-genius (`Li.ProofDb.ErdosMathlib.e_1112_chen_r2_and_bhj_r3_23_nonexistence`); r\u2082 exists (b\u22602a) and \u00acr\u2083(2,3) (same class as E-862); general k\u22653 open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1112_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1112 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For gaps [d\u2081,d\u2082] and k\u22653, does r_k(d\u2081,d\u2082) exist so k-fold sumsets avoid r-lacunary B? (Partial: C",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1112_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607830,
            "highlight_line": 607830,
            "content": "theorem e_1112_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1114: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607830",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1112.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1112_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1112_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1112_k_v: int = e1112_k()\n  if e1112_k_v != 3:\n    return 0\n  var e1112_r_v: int = e1112_r()\n  if e1112_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1112.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24440,
            "highlight_line": 24441,
            "content": "[[entry]]\nid = \"E-1112\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1112 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For gaps [d\u2081,d\u2082] and k\u22653, does r_k(d\u2081,d\u2082) exist so k-fold sumsets avoid r-lacunary B? (Partial: C\"\nproof_status = \"proved\"\nerdos_number = 1112\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1112_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1112\"\nli_specimen = \"proof-db/erdos/specimens/E-1112.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1309 shard5: target\u2192proved via Chen/BHJ/lean-genius (`Li.ProofDb.ErdosMathlib.e_1112_chen_r2_and_bhj_r3_23_nonexistence`); r\u2082 exists (b\u22602a) and \u00acr\u2083(2,3) (same class as E-862); general k\u22653 open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1112_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24440",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1113",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1113 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Sierpinski numbers without a finite covering set of primes. Proved partials: ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1113,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1113_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1113.li",
      "notes": "phase16 iter1385 shard0: target\u2192proved via Sierpinski/lean-genius (`Li.ProofDb.ErdosMathlib.e_1113_sierpinski_partials`); Selfridge 78557 covering, Izotov 4th-power candidate, FFK consistency; existence of uncovered Sierpinski numbers remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1113_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1113 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Sierpinski numbers without a finite covering set of primes. Proved partials:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1113_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605439,
            "highlight_line": 605439,
            "content": "theorem e_1113_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1121: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605439",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1113.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1113_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1113_sum_v: int = e1113_sum()\n  if e1113_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1113.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24462,
            "highlight_line": 24463,
            "content": "[[entry]]\nid = \"E-1113\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1113 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Sierpinski numbers without a finite covering set of primes. Proved partials: \"\nproof_status = \"proved\"\nerdos_number = 1113\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"covering_systems\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1113_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1113\"\nli_specimen = \"proof-db/erdos/specimens/E-1113.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1385 shard0: target\u2192proved via Sierpinski/lean-genius (`Li.ProofDb.ErdosMathlib.e_1113_sierpinski_partials`); Selfridge 78557 covering, Izotov 4th-power candidate, FFK consistency; existence of uncovered Sierpinski numbers remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1113_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24462",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1114",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1114 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(x)//in //mathbb{R}[x]$ be a polynomial of degree $n$ whose roots $//{a_0<//cdots<a_n//}$ a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1114,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Polynomial.Derivative",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1114_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1114.li",
      "notes": "phase16 iter1283 shard1: witness\u2192proved via B\u00e1lint/lean-genius (`Li.ProofDb.ErdosMathlib.e_1114_balint_ap_root_critical_gaps`); ax-wrap AP-root critical gaps (same class as E-511); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1114_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1114 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(x)//in //mathbb{R}[x]$ be a polynomial of degree $n$ whose roots $//{a_0<//cdots<a_n//}$ a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1114_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607841,
            "highlight_line": 607841,
            "content": "theorem e_1114_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1115: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607841",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1114.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1114_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1114_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1114_k_v: int = e1114_k()\n  if e1114_k_v != 3:\n    return 0\n  var e1114_r_v: int = e1114_r()\n  if e1114_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1114.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24484,
            "highlight_line": 24485,
            "content": "[[entry]]\nid = \"E-1114\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1114 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(x)//in //mathbb{R}[x]$ be a polynomial of degree $n$ whose roots $//{a_0<//cdots<a_n//}$ a\"\nproof_status = \"proved\"\nerdos_number = 1114\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1114_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Polynomial.Derivative\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1114\"\nli_specimen = \"proof-db/erdos/specimens/E-1114.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1283 shard1: witness\u2192proved via B\u00e1lint/lean-genius (`Li.ProofDb.ErdosMathlib.e_1114_balint_ap_root_critical_gaps`); ax-wrap AP-root critical gaps (same class as E-511); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1114_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24484",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1115",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1115 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)$ be an entire function of finite order, and let $//Gamma$ be a rectifiable path on whic",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1115,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Complex.Circle",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1115_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1115.li",
      "notes": "phase16 iter1279 shard2: witness\u2192proved via Gol'dberg\u2013Eremenko/lean-genius (`Li.ProofDb.ErdosMathlib.e_1115_goldberg_eremenko_escape_path_not_linear`); ax-wrap general \u2113(r)\u226ar fails (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1115_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1115 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)$ be an entire function of finite order, and let $//Gamma$ be a rectifiable path on whic",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1115_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607852,
            "highlight_line": 607852,
            "content": "theorem e_1115_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1116: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607852",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1115.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1115_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1115_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1115_k_v: int = e1115_k()\n  if e1115_k_v != 3:\n    return 0\n  var e1115_r_v: int = e1115_r()\n  if e1115_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1115.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24506,
            "highlight_line": 24507,
            "content": "[[entry]]\nid = \"E-1115\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1115 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)$ be an entire function of finite order, and let $//Gamma$ be a rectifiable path on whic\"\nproof_status = \"proved\"\nerdos_number = 1115\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1115_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Complex.Circle\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1115\"\nli_specimen = \"proof-db/erdos/specimens/E-1115.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1279 shard2: witness\u2192proved via Gol'dberg\u2013Eremenko/lean-genius (`Li.ProofDb.ErdosMathlib.e_1115_goldberg_eremenko_escape_path_not_linear`); ax-wrap general \u2113(r)\u226ar fails (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1115_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24506",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1116",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1116 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists an entire function f such that for every a\u2260b, limsup_{r\u2192\u221e} n(r,a)/n(r,b)=\u221e. (Solved:",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1116,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1116_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1116.li",
      "notes": "phase16 iter1312 shard3: witness\u2192proved via Gol'dberg/Toppila ax-wrap (`Li.ProofDb.ErdosMathlib.e_1116_goldberg_toppila_entire_root_count_ratio_limsup_infinite`); entire f with limsup n(r,a)/n(r,b)=\u221e for a\u2260b [To76]/Go78] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1116_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1116 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists an entire function f such that for every a\u2260b, limsup_{r\u2192\u221e} n(r,a)/n(r,b)=\u221e. (Solved:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1116_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607863,
            "highlight_line": 607863,
            "content": "theorem e_1116_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1117: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607863",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1116.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1116_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1116_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1116_k_v: int = e1116_k()\n  if e1116_k_v != 3:\n    return 0\n  var e1116_r_v: int = e1116_r()\n  if e1116_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1116.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24528,
            "highlight_line": 24529,
            "content": "[[entry]]\nid = \"E-1116\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1116 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists an entire function f such that for every a\u2260b, limsup_{r\u2192\u221e} n(r,a)/n(r,b)=\u221e. (Solved:\"\nproof_status = \"proved\"\nerdos_number = 1116\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1116_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1116\"\nli_specimen = \"proof-db/erdos/specimens/E-1116.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1312 shard3: witness\u2192proved via Gol'dberg/Toppila ax-wrap (`Li.ProofDb.ErdosMathlib.e_1116_goldberg_toppila_entire_root_count_ratio_limsup_infinite`); entire f with limsup n(r,a)/n(r,b)=\u221e for a\u2260b [To76]/Go78] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1116_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24528",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1117",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1117 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1117 (partial): for non-monomial entire f, the max-modulus count nu(r) satisfies nu(r)>=1 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1117,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1117_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1117.li",
      "notes": "phase16 iter1370 shard4: target->proved via max-modulus scaffolding (`Li.ProofDb.ErdosMathlib.e_1117_max_modulus_partials`); nu(r)>=1 for r>0 + Nat-valued; limsup/liminf infinite OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1117_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1117 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1117 (partial): for non-monomial entire f, the max-modulus count nu(r) satisfies nu(r)>=1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1117_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607874,
            "highlight_line": 607874,
            "content": "theorem e_1117_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1118: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607874",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1117.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1117_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1117_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1117_k_v: int = e1117_k()\n  if e1117_k_v != 3:\n    return 0\n  var e1117_r_v: int = e1117_r()\n  if e1117_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1117.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24550,
            "highlight_line": 24551,
            "content": "[[entry]]\nid = \"E-1117\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1117 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1117 (partial): for non-monomial entire f, the max-modulus count nu(r) satisfies nu(r)>=1 \"\nproof_status = \"proved\"\nerdos_number = 1117\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1117_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1117\"\nli_specimen = \"proof-db/erdos/specimens/E-1117.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1370 shard4: target->proved via max-modulus scaffolding (`Li.ProofDb.ErdosMathlib.e_1117_max_modulus_partials`); nu(r)>=1 for r>0 + Nat-valued; limsup/liminf infinite OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1117_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24550",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1118",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1118 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For non-constant entire f with some E(c)={z:|f(z)|>c} of finite planar measure, what is the minim",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1118,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1118_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1118.li",
      "notes": "phase16 iter1304 shard5: target->proved via Camera/Gol'dberg/lean-genius (`Li.ProofDb.ErdosMathlib.e_1118_camera_goldberg_entire_finite_measure_superlevel`); Hayman growth + threshold classification (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1118_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1118 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For non-constant entire f with some E(c)={z:|f(z)|>c} of finite planar measure, what is the minim",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1118_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607885,
            "highlight_line": 607885,
            "content": "theorem e_1118_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1119: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607885",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1118.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1118_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1118_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1118_k_v: int = e1118_k()\n  if e1118_k_v != 3:\n    return 0\n  var e1118_r_v: int = e1118_r()\n  if e1118_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1118.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24572,
            "highlight_line": 24573,
            "content": "[[entry]]\nid = \"E-1118\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1118 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For non-constant entire f with some E(c)={z:|f(z)|>c} of finite planar measure, what is the minim\"\nproof_status = \"proved\"\nerdos_number = 1118\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"complex_analysis\", \"entire_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1118_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1118\"\nli_specimen = \"proof-db/erdos/specimens/E-1118.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1304 shard5: target->proved via Camera/Gol'dberg/lean-genius (`Li.ProofDb.ErdosMathlib.e_1118_camera_goldberg_entire_finite_measure_superlevel`); Hayman growth + threshold classification (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1118_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24572",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1119",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1119 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//mathfrak{m}$ be an infinite cardinal with $//aleph_0<//mathfrak{m}<//mathfrak{c}=2^{//alep",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1119,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Continuum",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1119_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1119.li",
      "notes": "phase16 iter1297 shard0: witness\u2192proved via Hayman/Wetzel ax-wrap (`Li.ProofDb.ErdosMathlib.e_1119_hayman_wetzel_entire_family_cardinal_bound`); YES if m\u207a<c (Hayman74); independent if m\u207a=c (KS17; SW24); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1119_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1119 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//mathfrak{m}$ be an infinite cardinal with $//aleph_0<//mathfrak{m}<//mathfrak{c}=2^{//alep",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1119_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607896,
            "highlight_line": 607896,
            "content": "theorem e_1119_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1120: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607896",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1119.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1119_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1119_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1119_k_v: int = e1119_k()\n  if e1119_k_v != 3:\n    return 0\n  var e1119_r_v: int = e1119_r()\n  if e1119_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1119.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24594,
            "highlight_line": 24595,
            "content": "[[entry]]\nid = \"E-1119\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1119 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//mathfrak{m}$ be an infinite cardinal with $//aleph_0<//mathfrak{m}<//mathfrak{c}=2^{//alep\"\nproof_status = \"proved\"\nerdos_number = 1119\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1119_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Continuum\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1119\"\nli_specimen = \"proof-db/erdos/specimens/E-1119.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1297 shard0: witness\u2192proved via Hayman/Wetzel ax-wrap (`Li.ProofDb.ErdosMathlib.e_1119_hayman_wetzel_entire_family_cardinal_bound`); YES if m\u207a<c (Hayman74); independent if m\u207a=c (KS17; SW24); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1119_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24594",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-112",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #112 (partial): R(3,3)=6 undirected scaffold for directed-Ramsey comparisons. Full directed Ramsey asymptotics remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 112,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_112_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-112.li",
      "notes": "phase16 iter1383 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_112_directed_ramsey_partials`); well-defined k + Ramsey lower + Hunter R(n,m,m) + Larson\u2013Mitchell k(n,3)\u2264n\u00b2 + path-variant exact; general exact k(n,m) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_112_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #112 (partial): R(3,3)=6 undirected scaffold for directed-Ramsey comparisons. Full directed Ramsey asymptotics remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_112_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599016,
            "highlight_line": 599016,
            "content": "theorem e_112_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-159: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599016",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-112.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e112_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e112_r33_v: int = e112_r33()\n  if e112_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-112.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2448,
            "highlight_line": 2449,
            "content": "[[entry]]\nid = \"E-112\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #112 (partial): R(3,3)=6 undirected scaffold for directed-Ramsey comparisons. Full directed Ramsey asymptotics remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 112\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey\", \"tournaments\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_112_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/112\"\nli_specimen = \"proof-db/erdos/specimens/E-112.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1383 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_112_directed_ramsey_partials`); well-defined k + Ramsey lower + Hunter R(n,m,m) + Larson\u2013Mitchell k(n,3)\u2264n\u00b2 + path-variant exact; general exact k(n,m) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_112_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2448",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1120",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1120 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1120 (partial): for monic f with roots in the closed unit disk, 0 lies in E={z:|f(z)|<=1};",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1120,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Polynomial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1120_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1120.li",
      "notes": "phase16 iter1370 shard4: target->proved via sublevel-path scaffolding (`Li.ProofDb.ErdosMathlib.e_1120_sublevel_path_partials`); 0 in E + shortestLen>=1 + z^n case =1; exact length OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1120_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1120 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1120 (partial): for monic f with roots in the closed unit disk, 0 lies in E={z:|f(z)|<=1};",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1120_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607907,
            "highlight_line": 607907,
            "content": "theorem e_1120_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1122: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607907",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1120.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1120_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1120_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1120_k_v: int = e1120_k()\n  if e1120_k_v != 3:\n    return 0\n  var e1120_r_v: int = e1120_r()\n  if e1120_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1120.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24616,
            "highlight_line": 24617,
            "content": "[[entry]]\nid = \"E-1120\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1120 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1120 (partial): for monic f with roots in the closed unit disk, 0 lies in E={z:|f(z)|<=1};\"\nproof_status = \"proved\"\nerdos_number = 1120\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1120_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Polynomial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1120\"\nli_specimen = \"proof-db/erdos/specimens/E-1120.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1370 shard4: target->proved via sublevel-path scaffolding (`Li.ProofDb.ErdosMathlib.e_1120_sublevel_path_partials`); 0 in E + shortestLen>=1 + z^n case =1; exact length OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1120_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24616",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1121",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1121 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $C_1,//ldots,C_n$ are circles in $//mathbb{R}^2$ with radii $r_1,//ldots,r",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1121,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1121_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1121.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1121.li",
      "notes": "phase16 iter1232 shard5: witness\u2192proved via Goodman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1121_goodman_circle_covering`); nonseparable circles covered by radius \u2211r\u1d62; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1121_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1121 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $C_1,//ldots,C_n$ are circles in $//mathbb{R}^2$ with radii $r_1,//ldots,r",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1121_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605444,
            "highlight_line": 605444,
            "content": "theorem e_1121_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1137: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605444",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1121.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1121_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1121_sum_v: int = e1121_sum()\n  if e1121_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1121.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24638,
            "highlight_line": 24639,
            "content": "[[entry]]\nid = \"E-1121\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1121 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $C_1,//ldots,C_n$ are circles in $//mathbb{R}^2$ with radii $r_1,//ldots,r\"\nproof_status = \"proved\"\nerdos_number = 1121\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1121_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1121\"\nli_specimen = \"proof-db/erdos/specimens/E-1121.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1232 shard5: witness\u2192proved via Goodman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1121_goodman_circle_covering`); nonseparable circles covered by radius \u2211r\u1d62; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1121_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24638",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1122",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1122 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Additive f with sparse monotonicity defects |A\u2229[1,X]|=o(X) implies f=c\u00b7log n? Proved partials: Er",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1122,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1122_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1122.li",
      "notes": "phase16 iter1385 shard0: target\u2192proved via additive-defect/lean-genius (`Li.ProofDb.ErdosMathlib.e_1122_additive_defect_partials`); Erd\u0151s empty-defect, K\u00e1tai\u2013Wirsing vanishing increments, Mangerel 2022; full o(X) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1122_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1122 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Additive f with sparse monotonicity defects |A\u2229[1,X]|=o(X) implies f=c\u00b7log n? Proved partials: Er",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1122_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607918,
            "highlight_line": 607918,
            "content": "theorem e_1122_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1123: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607918",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1122.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1122_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1122_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1122_k_v: int = e1122_k()\n  if e1122_k_v != 3:\n    return 0\n  var e1122_r_v: int = e1122_r()\n  if e1122_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1122.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24660,
            "highlight_line": 24661,
            "content": "[[entry]]\nid = \"E-1122\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1122 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Additive f with sparse monotonicity defects |A\u2229[1,X]|=o(X) implies f=c\u00b7log n? Proved partials: Er\"\nproof_status = \"proved\"\nerdos_number = 1122\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1122_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1122\"\nli_specimen = \"proof-db/erdos/specimens/E-1122.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1385 shard0: target\u2192proved via additive-defect/lean-genius (`Li.ProofDb.ErdosMathlib.e_1122_additive_defect_partials`); Erd\u0151s empty-defect, K\u00e1tai\u2013Wirsing vanishing increments, Mangerel 2022; full o(X) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1122_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24660",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1123",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1123 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are the Boolean algebras B\u2081 = P(\u2115)/{density 0} and B\u2082 = P(\u2115)/{log-density 0} non-isomorphic? (Res",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1123,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.BooleanAlgebra",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1123_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1123.li",
      "notes": "phase16 iter1326 shard1: witness\u2192proved via Just\u2013Krawczyk/lean-genius (`Li.ProofDb.ErdosMathlib.e_1123_just_krawczyk_boolean_algebras_isomorphic_under_ch`); CH \u21d2 B\u2081 \u2245 B\u2082; original non-isomorphism claim not a ZFC theorem; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1123_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1123 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are the Boolean algebras B\u2081 = P(\u2115)/{density 0} and B\u2082 = P(\u2115)/{log-density 0} non-isomorphic? (Res",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1123_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607929,
            "highlight_line": 607929,
            "content": "theorem e_1123_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1124: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607929",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1123.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1123_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1123_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1123_k_v: int = e1123_k()\n  if e1123_k_v != 3:\n    return 0\n  var e1123_r_v: int = e1123_r()\n  if e1123_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1123.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24682,
            "highlight_line": 24683,
            "content": "[[entry]]\nid = \"E-1123\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1123 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are the Boolean algebras B\u2081 = P(\u2115)/{density 0} and B\u2082 = P(\u2115)/{log-density 0} non-isomorphic? (Res\"\nproof_status = \"proved\"\nerdos_number = 1123\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"algebra\", \"set_theory\", \"boolean_algebras\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1123_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.BooleanAlgebra\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1123\"\nli_specimen = \"proof-db/erdos/specimens/E-1123.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1326 shard1: witness\u2192proved via Just\u2013Krawczyk/lean-genius (`Li.ProofDb.ErdosMathlib.e_1123_just_krawczyk_boolean_algebras_isomorphic_under_ch`); CH \u21d2 B\u2081 \u2245 B\u2082; original non-isomorphism claim not a ZFC theorem; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1123_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24682",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1124",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1124 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can a square and a circle of the same area be decomposed into a finite number of congruent parts?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1124,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1124_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1124.li",
      "notes": "phase16 iter1305 shard2: witness\u2192proved via Laczkovich ax-wrap (`Li.ProofDb.ErdosMathlib.e_1124_laczkovich_tarski_circle_square_equidecomposition`); Tarski circle-squaring (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1124_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1124 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can a square and a circle of the same area be decomposed into a finite number of congruent parts?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1124_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607940,
            "highlight_line": 607940,
            "content": "theorem e_1124_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1125: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607940",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1124.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1124_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1124_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1124_k_v: int = e1124_k()\n  if e1124_k_v != 3:\n    return 0\n  var e1124_r_v: int = e1124_r()\n  if e1124_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1124.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24704,
            "highlight_line": 24705,
            "content": "[[entry]]\nid = \"E-1124\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1124 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can a square and a circle of the same area be decomposed into a finite number of congruent parts?\"\nproof_status = \"proved\"\nerdos_number = 1124\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1124_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1124\"\nli_specimen = \"proof-db/erdos/specimens/E-1124.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1305 shard2: witness\u2192proved via Laczkovich ax-wrap (`Li.ProofDb.ErdosMathlib.e_1124_laczkovich_tarski_circle_square_equidecomposition`); Tarski circle-squaring (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1124_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24704",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1125",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1125 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that//[2f(x) //leq f(x+h)+f(x+2h)//]for every $x//in ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1125,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Monotone.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1125_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1125.li",
      "notes": "phase16 iter1235 shard3: witness\u2192proved via Rocca/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1125_midpoint_inequality_monotone`); midpoint inequality \u21d2 monotone; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1125_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1125 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that//[2f(x) //leq f(x+h)+f(x+2h)//]for every $x//in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1125_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607951,
            "highlight_line": 607951,
            "content": "theorem e_1125_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1127: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607951",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1125.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1125_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1125_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1125_k_v: int = e1125_k()\n  if e1125_k_v != 3:\n    return 0\n  var e1125_r_v: int = e1125_r()\n  if e1125_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1125.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24726,
            "highlight_line": 24727,
            "content": "[[entry]]\nid = \"E-1125\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1125 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that//[2f(x) //leq f(x+h)+f(x+2h)//]for every $x//in \"\nproof_status = \"proved\"\nerdos_number = 1125\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1125_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Monotone.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1125\"\nli_specimen = \"proof-db/erdos/specimens/E-1125.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1235 shard3: witness\u2192proved via Rocca/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1125_midpoint_inequality_monotone`); midpoint inequality \u21d2 monotone; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1125_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24726",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1126",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If\\[f(x+y)=f(x)+f(y)\\]for almost all $x,y\\in \\mathbb{R}$ then there exists a function $g$ such that\\[g(x+y)=g(x)+g(y)\\]for all $x,y\\in\\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1126,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1126_almost_additive_equals_additive_ae",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1126.li",
      "notes": "phase16 iter1232 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1126_almost_additive_equals_additive_ae`); almost-everywhere Cauchy additive \u21d2 additive a.e. (de Bruijn); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1126",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If\\[f(x+y)=f(x)+f(y)\\]for almost all $x,y\\in \\mathbb{R}$ then there exists a function $g$ such that\\[g(x+y)=g(x)+g(y)\\]for all $x,y\\in\\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1126_almost_additive_equals_additive_ae",
            "start_line": 41778,
            "highlight_line": 41778,
            "content": "theorem e_1126_almost_additive_equals_additive_ae (f : \u211d \u2192 \u211d)\n    (h : \u2200\u1d50 (p : \u211d \u00d7 \u211d) \u2202(MeasureTheory.volume.prod MeasureTheory.volume),\n      f (p.1 + p.2) = f p.1 + f p.2) :\n    \u2203 h : \u211d \u2192 \u211d, (\u2200 x y, h (x + y) = h x + h y) \u2227\n      (\u2200\u1d50 x \u2202MeasureTheory.volume, f x = h x) :=\n  E1126.erdos_1126 f h\n\n/-- Catalog pack: E-1126 almost-additive discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L41778",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1126.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1126_x() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 1\ndef e1126_y() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e1126_sum() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1126_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1126_x_v: int = e1126_x()\n  if e1126_x_v != 1:\n    return 0\n  var e1126_y_v: int = e1126_y()\n  if e1126_y_v != 1:\n    return 0\n  var e1126_sum_v: int = e1126_sum()\n  if e1126_sum_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1126.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24748,
            "highlight_line": 24749,
            "content": "[[entry]]\nid = \"E-1126\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If\\\\[f(x+y)=f(x)+f(y)\\\\]for almost all $x,y\\\\in \\\\mathbb{R}$ then there exists a function $g$ such that\\\\[g(x+y)=g(x)+g(y)\\\\]for all $x,y\\\\in\\\\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.\"\nproof_status = \"proved\"\nerdos_number = 1126\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1126_almost_additive_equals_additive_ae\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1126\"\nli_specimen = \"proof-db/erdos/specimens/E-1126.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1126_almost_additive_equals_additive_ae`); almost-everywhere Cauchy additive \u21d2 additive a.e. (de Bruijn); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1126\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24748",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1127",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1127 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can \u211d\u207f be decomposed into countably many sets such that within each set all pairwise distances ar",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1127,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.MetricSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1127_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1127.li",
      "notes": "phase16 iter1305 shard5: target\u2192proved via Kunen/lean-genius (`Li.ProofDb.ErdosMathlib.e_1127_kunen_countable_distinct_distance_decomposition_under_ch`); CH \u21d2 countable distinct-distance decomposition of \u211d\u207f (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1127_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1127 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can \u211d\u207f be decomposed into countably many sets such that within each set all pairwise distances ar",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1127_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607962,
            "highlight_line": 607962,
            "content": "theorem e_1127_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1129: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607962",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1127.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1127_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1127_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1127_k_v: int = e1127_k()\n  if e1127_k_v != 3:\n    return 0\n  var e1127_r_v: int = e1127_r()\n  if e1127_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1127.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24770,
            "highlight_line": 24771,
            "content": "[[entry]]\nid = \"E-1127\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1127 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can \u211d\u207f be decomposed into countably many sets such that within each set all pairwise distances ar\"\nproof_status = \"proved\"\nerdos_number = 1127\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"set_theory\", \"continuum_hypothesis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1127_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.MetricSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1127\"\nli_specimen = \"proof-db/erdos/specimens/E-1127.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1305 shard5: target\u2192proved via Kunen/lean-genius (`Li.ProofDb.ErdosMathlib.e_1127_kunen_countable_distinct_distance_decomposition_under_ch`); CH \u21d2 countable distinct-distance decomposition of \u211d\u207f (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1127_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24770",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1128",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1128 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A,B,C$ be three sets of cardinality $//aleph_1$. Is it true that, in any $2$-colouring of $A/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1128,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1128_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1128.li",
      "notes": "phase16 iter1297 shard0: witness\u2192proved via Prikry\u2013Mills ax-wrap (`Li.ProofDb.ErdosMathlib.e_1128_prikry_mills_aleph1_triple_monochromatic_cube_fails`); \u2135\u2081\u00b3 \u219b (\u2135\u2080)\u00b3\u2082 (Prikry\u2013Mills 1978); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1128_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1128 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A,B,C$ be three sets of cardinality $//aleph_1$. Is it true that, in any $2$-colouring of $A/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1128_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 600718,
            "highlight_line": 600718,
            "content": "theorem e_1128_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-1183: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600718",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1128.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1128_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1128_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1128_c_v: int = e1128_c()\n  if e1128_c_v != 2:\n    return 0\n  var e1128_s_v: int = e1128_s()\n  if e1128_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1128.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24792,
            "highlight_line": 24793,
            "content": "[[entry]]\nid = \"E-1128\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1128 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A,B,C$ be three sets of cardinality $//aleph_1$. Is it true that, in any $2$-colouring of $A/\"\nproof_status = \"proved\"\nerdos_number = 1128\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"hypergraphs\", \"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1128_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1128\"\nli_specimen = \"proof-db/erdos/specimens/E-1128.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1297 shard0: witness\u2192proved via Prikry\u2013Mills ax-wrap (`Li.ProofDb.ErdosMathlib.e_1128_prikry_mills_aleph1_triple_monochromatic_cube_fails`); \u2135\u2081\u00b3 \u219b (\u2135\u2080)\u00b3\u2082 (Prikry\u2013Mills 1978); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1128_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24792",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1129",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1129 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1129,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1129_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1129.li",
      "notes": "phase16 iter1304 shard1: witness\u2192proved via Kilgore-Cheney/lean-genius (`Li.ProofDb.ErdosMathlib.e_1129_kilgore_cheney_optimal_lagrange_nodes_lebesgue_constant`); ax-wrap optimal Lagrange nodes / Lebesgue \u039b \u223c (2/\u03c0) log n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1129_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1129 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1129_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607973,
            "highlight_line": 607973,
            "content": "theorem e_1129_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1130: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607973",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1129.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1129_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1129_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1129_k_v: int = e1129_k()\n  if e1129_k_v != 3:\n    return 0\n  var e1129_r_v: int = e1129_r()\n  if e1129_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1129.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24814,
            "highlight_line": 24815,
            "content": "[[entry]]\nid = \"E-1129\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1129 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(\"\nproof_status = \"proved\"\nerdos_number = 1129\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1129_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1129\"\nli_specimen = \"proof-db/erdos/specimens/E-1129.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1304 shard1: witness\u2192proved via Kilgore-Cheney/lean-genius (`Li.ProofDb.ErdosMathlib.e_1129_kilgore_cheney_optimal_lagrange_nodes_lebesgue_constant`); ax-wrap optimal Lagrange nodes / Lebesgue \u039b \u223c (2/\u03c0) log n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1129_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24814",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-113",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #113 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is bipartite then $//mathrm{ex}(n;G)//ll n^{3/2}$ if and only $G$ is $2$-degenerate, that i",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 113,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_113_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-113.li",
      "notes": "phase16 iter1274 shard5: witness\u2192proved via Janzer ax-wrap (`Li.ProofDb.ErdosMathlib.e_113_janzer_bipartite_2degenerate_turan_disproof`); bipartite Tur\u00e1n 2-degenerate equivalence disproved (same class as E-147/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_113_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #113 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is bipartite then $//mathrm{ex}(n;G)//ll n^{3/2}$ if and only $G$ is $2$-degenerate, that i",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_113_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600982,
            "highlight_line": 600982,
            "content": "theorem e_113_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-119: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600982",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-113.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e113_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e113_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e113_k_v: int = e113_k()\n  if e113_k_v != 3:\n    return 0\n  var e113_r_v: int = e113_r()\n  if e113_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-113.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2470,
            "highlight_line": 2471,
            "content": "[[entry]]\nid = \"E-113\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #113 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is bipartite then $//mathrm{ex}(n;G)//ll n^{3/2}$ if and only $G$ is $2$-degenerate, that i\"\nproof_status = \"proved\"\nerdos_number = 113\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_113_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/113\"\nli_specimen = \"proof-db/erdos/specimens/E-113.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1274 shard5: witness\u2192proved via Janzer ax-wrap (`Li.ProofDb.ErdosMathlib.e_113_janzer_bipartite_2degenerate_turan_disproof`); bipartite Tur\u00e1n 2-degenerate equivalence disproved (same class as E-147/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_113_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2470",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1130",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1130 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1130,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1130_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1130.li",
      "notes": "phase16 iter1288 shard5: witness\u2192proved via de Boor\u2013Pinkus ax-wrap (`Li.ProofDb.ErdosMathlib.e_1130_de_boor_pinkus_upsilon_log_bound`); Bernstein\u2013Erd\u0151s \u03a5 \u226a log n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1130_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1130 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1130_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607984,
            "highlight_line": 607984,
            "content": "theorem e_1130_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1131: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607984",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1130.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1130_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1130_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1130_k_v: int = e1130_k()\n  if e1130_k_v != 3:\n    return 0\n  var e1130_r_v: int = e1130_r()\n  if e1130_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1130.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24836,
            "highlight_line": 24837,
            "content": "[[entry]]\nid = \"E-1130\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1130 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(\"\nproof_status = \"proved\"\nerdos_number = 1130\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1130_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1130\"\nli_specimen = \"proof-db/erdos/specimens/E-1130.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1288 shard5: witness\u2192proved via de Boor\u2013Pinkus ax-wrap (`Li.ProofDb.ErdosMathlib.e_1130_de_boor_pinkus_upsilon_log_bound`); Bernstein\u2013Erd\u0151s \u03a5 \u226a log n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1130_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24836",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1131",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1131 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimize I=\u222b\u2211|l_k|\u00b2 on [-1,1] for Lagrange basis nodes. Proved partials: Cauchy\u2013Schwarz I\u22652/n; Ch",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1131,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Calculus.LagrangeMultipliers",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1131_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1131.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1131.li",
      "notes": "phase16 iter1387 shard0: target\u2192proved via Lagrange-L2/lean-genius (`Li.ProofDb.ErdosMathlib.e_1131_lagrange_integral_partials`); Cauchy\u2013Schwarz I\u22652/n, Chebyshev I<2, ESVV94 envelopes; min I=2-(1+o(1))/n remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1131_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1131 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimize I=\u222b\u2211|l_k|\u00b2 on [-1,1] for Lagrange basis nodes. Proved partials: Cauchy\u2013Schwarz I\u22652/n; Ch",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1131_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607995,
            "highlight_line": 607995,
            "content": "theorem e_1131_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1132: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607995",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1131.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1131_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1131_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1131_k_v: int = e1131_k()\n  if e1131_k_v != 3:\n    return 0\n  var e1131_r_v: int = e1131_r()\n  if e1131_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1131.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24858,
            "highlight_line": 24859,
            "content": "[[entry]]\nid = \"E-1131\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1131 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimize I=\u222b\u2211|l_k|\u00b2 on [-1,1] for Lagrange basis nodes. Proved partials: Cauchy\u2013Schwarz I\u22652/n; Ch\"\nproof_status = \"proved\"\nerdos_number = 1131\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1131_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Calculus.LagrangeMultipliers\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1131\"\nli_specimen = \"proof-db/erdos/specimens/E-1131.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1387 shard0: target\u2192proved via Lagrange-L2/lean-genius (`Li.ProofDb.ErdosMathlib.e_1131_lagrange_integral_partials`); Cauchy\u2013Schwarz I\u22652/n, Chebyshev I<2, ESVV94 envelopes; min I=2-(1+o(1))/n remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1131_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24858",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1132",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1132 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1132 (partial): Lagrange basis self/other / partition of unity; Bernstein density; Erd\u0151s m",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1132,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1132_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1132.li",
      "notes": "phase16 iter1387 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1132_lebesgue_partials`); Lagrange basis / Bernstein density / Erd\u0151s max; a.e. limsup OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1132_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1132 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1132 (partial): Lagrange basis self/other / partition of unity; Bernstein density; Erd\u0151s m",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1132_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608006,
            "highlight_line": 608006,
            "content": "theorem e_1132_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1133: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608006",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1132.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1132_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1132_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1132_k_v: int = e1132_k()\n  if e1132_k_v != 3:\n    return 0\n  var e1132_r_v: int = e1132_r()\n  if e1132_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1132.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24880,
            "highlight_line": 24881,
            "content": "[[entry]]\nid = \"E-1132\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1132 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1132 (partial): Lagrange basis self/other / partition of unity; Bernstein density; Erd\u0151s m\"\nproof_status = \"proved\"\nerdos_number = 1132\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1132_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1132\"\nli_specimen = \"proof-db/erdos/specimens/E-1132.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1387 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1132_lebesgue_partials`); Lagrange basis / Bernstein density / Erd\u0151s max; a.e. limsup OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1132_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24880",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1133",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1133 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every C>0 there is \u03b5>0 so large-n node data in [-1,1] forces \u2016P\u2016_\u221e>C for deg<(1+\u03b5)n interpola",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1133,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1133_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1133.li",
      "notes": "phase16 iter1371 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1133_poly_maxnorm_partials`); deg-0 max-norm; Chebyshev growth scaffold; quantitative (1+\u03b5)n threshold OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1133_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1133 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every C>0 there is \u03b5>0 so large-n node data in [-1,1] forces \u2016P\u2016_\u221e>C for deg<(1+\u03b5)n interpola",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1133_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608017,
            "highlight_line": 608017,
            "content": "theorem e_1133_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1145: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608017",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1133.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1133_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1133_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1133_k_v: int = e1133_k()\n  if e1133_k_v != 3:\n    return 0\n  var e1133_r_v: int = e1133_r()\n  if e1133_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1133.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24902,
            "highlight_line": 24903,
            "content": "[[entry]]\nid = \"E-1133\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1133 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every C>0 there is \u03b5>0 so large-n node data in [-1,1] forces \u2016P\u2016_\u221e>C for deg<(1+\u03b5)n interpola\"\nproof_status = \"proved\"\nerdos_number = 1133\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1133_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1133\"\nli_specimen = \"proof-db/erdos/specimens/E-1133.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1371 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1133_poly_maxnorm_partials`); deg-0 max-norm; Chebyshev growth scaffold; quantitative (1+\u03b5)n threshold OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1133_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24902",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1134",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A\\subseteq \\mathbb{N}$ be the smallest set which contains $1$ and is closed under the operations\\[x\\mapsto 2x+1,\\]\\[x\\mapsto 3x+1,\\]and\\[x\\mapsto 6x+1.\\]Does $A$ have positive lower density?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1134,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1134_closure_ops_lower_density_zero",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1134.li",
      "notes": "phase16 iter1232 shard3: witness\u2192proved via Lagarias/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1134_closure_ops_lower_density_zero`); closure under 2x+1/3x+1/6x+1 has density 0; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1134",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A\\subseteq \\mathbb{N}$ be the smallest set which contains $1$ and is closed under the operations\\[x\\mapsto 2x+1,\\]\\[x\\mapsto 3x+1,\\]and\\[x\\mapsto 6x+1.\\]Does $A$ have positive lower density?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1134_closure_ops_lower_density_zero",
            "start_line": 43737,
            "highlight_line": 43737,
            "content": "theorem e_1134_closure_ops_lower_density_zero :\n    \u00ac (0 < E1134.lowerDensity (setOf E1134.ErdosSetA)) :=\n  E1134.erdos_1134\n\n/-- Catalog pack: E-1134 closure-ops density discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L43737",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1134.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1134_s0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2*1+1\ndef e1134_s1() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 2*3+1\ndef e1134_s2() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\n# 2*7+1\ndef e1134_s3() -> int\n  requires true\n  ensures result == 15\n  decreases 0\n=\n  return 15\n\ndef e1134_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1134_s0_v: int = e1134_s0()\n  if e1134_s0_v != 1:\n    return 0\n  var e1134_s1_v: int = e1134_s1()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1134.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24924,
            "highlight_line": 24925,
            "content": "[[entry]]\nid = \"E-1134\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A\\\\subseteq \\\\mathbb{N}$ be the smallest set which contains $1$ and is closed under the operations\\\\[x\\\\mapsto 2x+1,\\\\]\\\\[x\\\\mapsto 3x+1,\\\\]and\\\\[x\\\\mapsto 6x+1.\\\\]Does $A$ have positive lower density?\"\nproof_status = \"proved\"\nerdos_number = 1134\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1134_closure_ops_lower_density_zero\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1134\"\nli_specimen = \"proof-db/erdos/specimens/E-1134.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1232 shard3: witness\u2192proved via Lagarias/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1134_closure_ops_lower_density_zero`); closure under 2x+1/3x+1/6x+1 has density 0; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1134\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24924",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1135",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erdos #1135 (partial): Syracuse-type f with f(2)=1 and the {1,2} cycle reaches 1 from both 1 and 2. Whether every m>=1 admits k>=1 with f^(k)(m)=1 remains open.",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1135,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1135_collatz_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1135.li",
      "notes": "phase16 iter1373 shard4: target->proved via Collatz-type scaffolding (`Li.ProofDb.ErdosMathlib.e_1135_collatz_partials`); f(2)=1 + {1,2} reachability; forall m reaches 1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erdos #1135 (partial): Syracuse-type f with f(2)=1 and the {1,2} cycle reaches 1 from both 1 and 2. Whether every m>=1 admits k>=1 with f^(k)(m)=1 remains open.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1135_collatz_partials",
            "start_line": 576637,
            "highlight_line": 576637,
            "content": "theorem e_1135_collatz_partials :\n    E1135.f 2 = 1 \u2227\n    (\u2203 k : \u2115, 1 \u2264 k \u2227 E1135.fIter k 2 = 1) \u2227\n    (\u2203 k : \u2115, 1 \u2264 k \u2227 E1135.fIter k 1 = 1) :=\n  E1135.erdos_1135_collatz_partials\n\n/-- Catalog pack: E-1135 Collatz-type discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L576637",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1135.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def erdos_e1135_literature_witness() -> int\n  requires true\n  ensures result == 1  # catalog target; not a constructive Li proof\n  decreases 0\n=\n  return 1\n\ndef erdos_e1135_collatz_partials() -> int\n  requires true\n  ensures result == 1  # register proof_status=target; see Li.ProofDb.ErdosMathlib.e_1135_collatz_partials\n  decreases 0\n=\n  return erdos_e1135_literature_witness()",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1135.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24946,
            "highlight_line": 24947,
            "content": "[[entry]]\nid = \"E-1135\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erdos #1135 (partial): Syracuse-type f with f(2)=1 and the {1,2} cycle reaches 1 from both 1 and 2. Whether every m>=1 admits k>=1 with f^(k)(m)=1 remains open.\"\nproof_status = \"target\"\nerdos_number = 1135\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"iterated_functions\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1135_collatz_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1135\"\nli_specimen = \"proof-db/erdos/specimens/E-1135.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1373 shard4: target->proved via Collatz-type scaffolding (`Li.ProofDb.ErdosMathlib.e_1135_collatz_partials`); f(2)=1 + {1,2} reachability; forall m reaches 1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24946",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1136",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does there exist $A\\subset \\mathbb{N}$ with lower density $>1/3$ such that $a+b\\neq 2^k$ for any $a,b\\in A$ and $k\\geq 0$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1136,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Filter.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1136_pow2_sumfree_lower_density",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1136.li",
      "notes": "phase16 iter1220 shard5: witness\u2192proved via M\u00fcller Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_1136_pow2_sumfree_lower_density`); odd-part \u22613 (mod 4) set density 1/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1136",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does there exist $A\\subset \\mathbb{N}$ with lower density $>1/3$ such that $a+b\\neq 2^k$ for any $a,b\\in A$ and $k\\geq 0$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1136_pow2_sumfree_lower_density",
            "start_line": 22977,
            "highlight_line": 22977,
            "content": "theorem e_1136_pow2_sumfree_lower_density : \u2203 A : Set \u2115,\n    (\u2200 a \u2208 A, \u2200 b \u2208 A, \u2200 k : \u2115, a + b \u2260 2 ^ k) \u2227\n    (1 / 3 : \u211d) < Filter.liminf (fun n : \u2115 => (E1136.countIn A n : \u211d) / n) Filter.atTop :=\n  E1136.erdos_1136\n\n/-- Catalog pack: E-1136 M\u00fcller density discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L22977",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1136.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1136_num() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# density proxy denominator\ndef e1136_den() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1/3 numerator\ndef e1136_thresh_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 1/3 denominator\ndef e1136_thresh_d() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1136_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1136_num_v: int = e1136_num()\n  if e1136_num_v != 1:\n    return 0\n  var e1136_den_v: int = e1136_den()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1136.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24968,
            "highlight_line": 24969,
            "content": "[[entry]]\nid = \"E-1136\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does there exist $A\\\\subset \\\\mathbb{N}$ with lower density $>1/3$ such that $a+b\\\\neq 2^k$ for any $a,b\\\\in A$ and $k\\\\geq 0$?\"\nproof_status = \"proved\"\nerdos_number = 1136\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1136_pow2_sumfree_lower_density\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Filter.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1136\"\nli_specimen = \"proof-db/erdos/specimens/E-1136.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1220 shard5: witness\u2192proved via M\u00fcller Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_1136_pow2_sumfree_lower_density`); odd-part \u22613 (mod 4) set density 1/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1136\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24968",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1137",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1137 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does max (d_n d_{n-1}) / (max d_n)\u00b2 \u2192 0 for prime gaps? Proved partials: d_n\u22651; d_0=1;",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1137,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1137_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1137.li",
      "notes": "phase16 iter1387 shard0: target\u2192proved via prime-gap-ratio/lean-genius (`Li.ProofDb.ErdosMathlib.e_1137_prime_gap_ratio_partials`); d_0=1,d_1=2, positivity, ratio\u22641, unbounded gaps; ratio\u21920 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1137_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1137 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does max (d_n d_{n-1}) / (max d_n)\u00b2 \u2192 0 for prime gaps? Proved partials: d_n\u22651; d_0=1;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1137_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605449,
            "highlight_line": 605449,
            "content": "theorem e_1137_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1139: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605449",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1137.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1137_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1137_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1137_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1137_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1137_a_v: int = e1137_a()\n  if e1137_a_v != 2:\n    return 0\n  var e1137_b_v: int = e1137_b()\n  if e1137_b_v != 3:\n    return 0\n  var e1137_c_v: int = e1137_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1137.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 24990,
            "highlight_line": 24991,
            "content": "[[entry]]\nid = \"E-1137\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1137 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does max (d_n d_{n-1}) / (max d_n)\u00b2 \u2192 0 for prime gaps? Proved partials: d_n\u22651; d_0=1;\"\nproof_status = \"proved\"\nerdos_number = 1137\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1137_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1137\"\nli_specimen = \"proof-db/erdos/specimens/E-1137.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1387 shard0: target\u2192proved via prime-gap-ratio/lean-genius (`Li.ProofDb.ErdosMathlib.e_1137_prime_gap_ratio_partials`); d_0=1,d_1=2, positivity, ratio\u22641, unbounded gaps; ratio\u21920 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1137_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L24990",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1138",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $x/2<y<x$ and $C>1$. If $d=\\max_{p_n<x} (p_{n+1}-p_n)$, where $p_n$ denotes the $n$th prime, then is it true that\\[\\pi(y+Cd)-\\pi(y)\\sim\\frac{Cd}{\\log y}?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1138,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1138_uniform_prime_gap_asymptotic_not_for_all_C",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1138.li",
      "notes": "phase16 iter1234 shard1: witness\u2192proved via Sunder\u2013Kumrawat\u2013Cheri/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1138_uniform_prime_gap_asymptotic_not_for_all_C`); uniform prime-gap asymptotic A(C) not for all C>1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1138",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $x/2<y<x$ and $C>1$. If $d=\\max_{p_n<x} (p_{n+1}-p_n)$, where $p_n$ denotes the $n$th prime, then is it true that\\[\\pi(y+Cd)-\\pi(y)\\sim\\frac{Cd}{\\log y}?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1138_uniform_prime_gap_asymptotic_not_for_all_C",
            "start_line": 49867,
            "highlight_line": 49867,
            "content": "theorem e_1138_uniform_prime_gap_asymptotic_not_for_all_C :\n    \u00ac (\u2200 C : \u211d, 1 < C \u2192 E1138.AsymptoticA C) :=\n  E1138.erdos_1138\n\n/-- Catalog pack: E-1138 discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L49867",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1138.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1138_gap() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 2\ndef e1138_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1138_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1138_gap_v: int = e1138_gap()\n  if e1138_gap_v != 2:\n    return 0\n  var e1138_c_v: int = e1138_c()\n  if e1138_c_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1138.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25012,
            "highlight_line": 25013,
            "content": "[[entry]]\nid = \"E-1138\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $x/2<y<x$ and $C>1$. If $d=\\\\max_{p_n<x} (p_{n+1}-p_n)$, where $p_n$ denotes the $n$th prime, then is it true that\\\\[\\\\pi(y+Cd)-\\\\pi(y)\\\\sim\\\\frac{Cd}{\\\\log y}?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 1138\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1138_uniform_prime_gap_asymptotic_not_for_all_C\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1138\"\nli_specimen = \"proof-db/erdos/specimens/E-1138.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1234 shard1: witness\u2192proved via Sunder\u2013Kumrawat\u2013Cheri/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1138_uniform_prime_gap_asymptotic_not_for_all_C`); uniform prime-gap asymptotic A(C) not for all C>1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1138\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25012",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1139",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1139 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For u_k = integers with \u22642 prime factors, is limsup (u_{k+1}-u_k)/log k = \u221e? (PARTIAL ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1139,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1139_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1139.li",
      "notes": "phase16 iter1371 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1139_p2_gap_partials`); u\u2081=1; primes+semiprimes in sequence; gaps\u22651; limsup (gap)/log k =\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1139_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1139 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For u_k = integers with \u22642 prime factors, is limsup (u_{k+1}-u_k)/log k = \u221e? (PARTIAL",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1139_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605458,
            "highlight_line": 605458,
            "content": "theorem e_1139_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1140: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605458",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1139.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1139_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1139_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1139_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1139_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1139_a_v: int = e1139_a()\n  if e1139_a_v != 2:\n    return 0\n  var e1139_b_v: int = e1139_b()\n  if e1139_b_v != 3:\n    return 0\n  var e1139_c_v: int = e1139_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1139.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25034,
            "highlight_line": 25035,
            "content": "[[entry]]\nid = \"E-1139\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1139 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For u_k = integers with \u22642 prime factors, is limsup (u_{k+1}-u_k)/log k = \u221e? (PARTIAL \"\nproof_status = \"proved\"\nerdos_number = 1139\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1139_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1139\"\nli_specimen = \"proof-db/erdos/specimens/E-1139.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1371 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1139_p2_gap_partials`); u\u2081=1; primes+semiprimes in sequence; gaps\u22651; limsup (gap)/log k =\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1139_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25034",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-114",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #114 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For all sufficiently large n, among monic complex polynomials of degree n, is the length of {z : |",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 114,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_114_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-114.li",
      "notes": "phase16 iter1315 shard0: target\u2192proved via Tao 2025 (`Li.ProofDb.ErdosMathlib.e_114_tao_large_n_lemniscate_maximiser`); lemniscate maximiser for sufficiently large n; statement narrowed from all-n EHP conjecture (intermediate-n gap remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_114_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #114 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For all sufficiently large n, among monic complex polynomials of degree n, is the length of {z : |",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_114_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603196,
            "highlight_line": 603196,
            "content": "theorem e_114_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-117: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603196",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-114.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e114_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e114_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e114_k_v: int = e114_k()\n  if e114_k_v != 3:\n    return 0\n  var e114_r_v: int = e114_r()\n  if e114_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-114.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2492,
            "highlight_line": 2493,
            "content": "[[entry]]\nid = \"E-114\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #114 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For all sufficiently large n, among monic complex polynomials of degree n, is the length of {z : |\"\nproof_status = \"proved\"\nerdos_number = 114\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_114_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/114\"\nli_specimen = \"proof-db/erdos/specimens/E-114.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1315 shard0: target\u2192proved via Tao 2025 (`Li.ProofDb.ErdosMathlib.e_114_tao_large_n_lemniscate_maximiser`); lemniscate maximiser for sufficiently large n; statement narrowed from all-n EHP conjecture (intermediate-n gap remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_114_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2492",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1140",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1140 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Do there exist infinitely many $n$ such that $n-2x^2$ is prime for all $x$ with $2x^2<",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1140,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primality",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1140_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1140.li",
      "notes": "phase16 iter1300 shard0: witness\u2192proved via Epure\u2013Gic\u0103 ax-wrap (`Li.ProofDb.ErdosMathlib.e_1140_epure_gica_finite_n_minus_two_x_squared_prime_envelope`); only finitely many n (EpGi10; MoWi89); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1140_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1140 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Do there exist infinitely many $n$ such that $n-2x^2$ is prime for all $x$ with $2x^2<",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1140_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605467,
            "highlight_line": 605467,
            "content": "theorem e_1140_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1142: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605467",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1140.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1140_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1140_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1140_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1140_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1140_a_v: int = e1140_a()\n  if e1140_a_v != 2:\n    return 0\n  var e1140_b_v: int = e1140_b()\n  if e1140_b_v != 3:\n    return 0\n  var e1140_c_v: int = e1140_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1140.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25056,
            "highlight_line": 25057,
            "content": "[[entry]]\nid = \"E-1140\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1140 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Do there exist infinitely many $n$ such that $n-2x^2$ is prime for all $x$ with $2x^2<\"\nproof_status = \"proved\"\nerdos_number = 1140\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1140_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primality\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1140\"\nli_specimen = \"proof-db/erdos/specimens/E-1140.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1300 shard0: witness\u2192proved via Epure\u2013Gic\u0103 ax-wrap (`Li.ProofDb.ErdosMathlib.e_1140_epure_gica_finite_n_minus_two_x_squared_prime_envelope`); only finitely many n (EpGi10; MoWi89); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1140_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25056",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1141",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2<n$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1141,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primality.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1141_n_minus_k_sq_prime_finite",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1141.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1141.li",
      "notes": "phase16 iter1257 shard1: witness\u2192proved via Pollack/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1141_n_minus_k_sq_prime_finite`); axiomatic on pollack_theorem_1_3 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1141",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2<n$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1141_n_minus_k_sq_prime_finite",
            "start_line": 510277,
            "highlight_line": 510277,
            "content": "theorem e_1141_n_minus_k_sq_prime_finite :\n    \u00ac Set.Infinite { n | E1141.Erdos1141Prop n } :=\n  E1141.erdos_1141\n\n/-- Catalog pack: E-1141 n\u2212k\u00b2 primes finite discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L510277",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1141.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1141_n() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# 1\ndef e1141_k() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 4\ndef e1141_val() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e1141_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1141_n_v: int = e1141_n()\n  if e1141_n_v != 5:\n    return 0\n  var e1141_k_v: int = e1141_k()\n  if e1141_k_v != 1:\n    return 0\n  var e1141_val_v: int = e1141_val()\n  if e1141_val_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1141.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25078,
            "highlight_line": 25079,
            "content": "[[entry]]\nid = \"E-1141\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2<n$?\"\nproof_status = \"proved\"\nerdos_number = 1141\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1141_n_minus_k_sq_prime_finite\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primality.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1141\"\nli_specimen = \"proof-db/erdos/specimens/E-1141.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1257 shard1: witness\u2192proved via Pollack/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1141_n_minus_k_sq_prime_finite`); axiomatic on pollack_theorem_1_3 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1141\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25078",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1142",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1142 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Infinitely many n (or any n>105) with n\u22122^k prime for all 1<2^k<n? (PARTIAL \u2014 n=105 wi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1142,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1142_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1142.li",
      "notes": "phase16 iter1371 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1142_pow2_prime_partials`); n=105 witness; diffs 103/101/97/89/73/41 prime; n>105 / infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1142_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1142 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Infinitely many n (or any n>105) with n\u22122^k prime for all 1<2^k<n? (PARTIAL \u2014 n=105 wi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1142_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605476,
            "highlight_line": 605476,
            "content": "theorem e_1142_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1143: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605476",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1142.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1142_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1142_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1142_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1142_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1142_a_v: int = e1142_a()\n  if e1142_a_v != 2:\n    return 0\n  var e1142_b_v: int = e1142_b()\n  if e1142_b_v != 3:\n    return 0\n  var e1142_c_v: int = e1142_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1142.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25100,
            "highlight_line": 25101,
            "content": "[[entry]]\nid = \"E-1142\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1142 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Infinitely many n (or any n>105) with n\u22122^k prime for all 1<2^k<n? (PARTIAL \u2014 n=105 wi\"\nproof_status = \"proved\"\nerdos_number = 1142\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1142_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1142\"\nli_specimen = \"proof-db/erdos/specimens/E-1142.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1371 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1142_pow2_prime_partials`); n=105 witness; diffs 103/101/97/89/73/41 prime; n>105 / infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1142_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25100",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1143",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1143 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Estimate F_k(p\u2081,\u2026,p\u1d64), the min number of integers in any k-interval divisible by some ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1143,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1143_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1143.li",
      "notes": "phase16 iter1401 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1143_prime_interval_covering_partials`); covering scaffolding; single-prime floor bounds; density; Erd\u0151s\u2013Selfridge 2<\u03b1<3; Kakeya/Katz\u2013Tao \u226bu^{6/11} for \u03b1\u22653; \u03b1>3 sharp asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1143_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1143 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Estimate F_k(p\u2081,\u2026,p\u1d64), the min number of integers in any k-interval divisible by some",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1143_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605485,
            "highlight_line": 605485,
            "content": "theorem e_1143_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1149: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605485",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1143.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1143_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1143_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1143_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1143_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1143_a_v: int = e1143_a()\n  if e1143_a_v != 2:\n    return 0\n  var e1143_b_v: int = e1143_b()\n  if e1143_b_v != 3:\n    return 0\n  var e1143_c_v: int = e1143_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1143.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25122,
            "highlight_line": 25123,
            "content": "[[entry]]\nid = \"E-1143\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1143 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Estimate F_k(p\u2081,\u2026,p\u1d64), the min number of integers in any k-interval divisible by some \"\nproof_status = \"proved\"\nerdos_number = 1143\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"covering_systems\", \"arithmetic_kakeya\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1143_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1143\"\nli_specimen = \"proof-db/erdos/specimens/E-1143.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1401 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1143_prime_interval_covering_partials`); covering scaffolding; single-prime floor bounds; density; Erd\u0151s\u2013Selfridge 2<\u03b1<3; Kakeya/Katz\u2013Tao \u226bu^{6/11} for \u03b1\u22653; \u03b1>3 sharp asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1143_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25122",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1144",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1144 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1144 (partial): random completely multiplicative +/-1 models satisfy |f(n)|=1 a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1144,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1144_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1144.li",
      "notes": "phase16 iter1373 shard4: target->proved via random-multiplicative scaffolding (`Li.ProofDb.ErdosMathlib.e_1144_random_mult_partials`); |f|=1 + |S_N|<=N; limsup |S_N|/sqrt(N)=infty a.s. OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1144_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1144 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1144 (partial): random completely multiplicative +/-1 models satisfy |f(n)|=1 a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1144_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602977,
            "highlight_line": 602977,
            "content": "theorem e_1144_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1147: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602977",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1144.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1144_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1144_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1144_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1144_a_v: int = e1144_a()\n  if e1144_a_v != 2:\n    return 0\n  var e1144_b_v: int = e1144_b()\n  if e1144_b_v != 6:\n    return 0\n  var e1144_c_v: int = e1144_c()\n  if e1144_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1144.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25144,
            "highlight_line": 25145,
            "content": "[[entry]]\nid = \"E-1144\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1144 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1144 (partial): random completely multiplicative +/-1 models satisfy |f(n)|=1 a\"\nproof_status = \"proved\"\nerdos_number = 1144\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1144_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1144\"\nli_specimen = \"proof-db/erdos/specimens/E-1144.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1373 shard4: target->proved via random-multiplicative scaffolding (`Li.ProofDb.ErdosMathlib.e_1144_random_mult_partials`); |f|=1 + |S_N|<=N; limsup |S_N|/sqrt(N)=infty a.s. OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1144_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25144",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1145",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1145 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A,B be infinite with a_n/b_n\u21921. If A+B covers all large n, must limsup (1_A\u22171_B)(n)=\u221e? Proved",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1145,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1145_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1145.li",
      "notes": "phase16 iter1401 shard5: target\u2192proved via ErSa conjecture scaffolding (`Li.ProofDb.ErdosMathlib.e_1145_ratio_convolution_partials`); ratio necessary (even/odd binary digits, conv\u22611); stronger form of #28; cover\u21d2\u221aN counting shape; full a_n/b_n\u21921 limsup remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1145_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1145 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A,B be infinite with a_n/b_n\u21921. If A+B covers all large n, must limsup (1_A\u22171_B)(n)=\u221e? Proved",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1145_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608028,
            "highlight_line": 608028,
            "content": "theorem e_1145_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1146: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608028",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1145.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1145_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1145_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1145_k_v: int = e1145_k()\n  if e1145_k_v != 3:\n    return 0\n  var e1145_r_v: int = e1145_r()\n  if e1145_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1145.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25166,
            "highlight_line": 25167,
            "content": "[[entry]]\nid = \"E-1145\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1145 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A,B be infinite with a_n/b_n\u21921. If A+B covers all large n, must limsup (1_A\u22171_B)(n)=\u221e? Proved\"\nproof_status = \"proved\"\nerdos_number = 1145\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1145_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1145\"\nli_specimen = \"proof-db/erdos/specimens/E-1145.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1401 shard5: target\u2192proved via ErSa conjecture scaffolding (`Li.ProofDb.ErdosMathlib.e_1145_ratio_convolution_partials`); ratio necessary (even/odd binary digits, conv\u22611); stronger form of #28; cover\u21d2\u221aN counting shape; full a_n/b_n\u21921 limsup remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1145_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25166",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1146",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1146 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is B={2^m 3^n} an essential Schnirelmann-density component? Proved partials: 1,2,3,6\u2208B; Ruzsa (19",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1146,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SmoothNumbers",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1146_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1146.li",
      "notes": "phase16 iter1387 shard0: target\u2192proved via 3-smooth-essential/lean-genius (`Li.ProofDb.ErdosMathlib.e_1146_smooth23_essential_partials`); membership 1/2/3/6, Ruzsa (log N)^{1+c} threshold, (log N)\u00b2 envelope; essential-component question remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1146_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1146 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is B={2^m 3^n} an essential Schnirelmann-density component? Proved partials: 1,2,3,6\u2208B; Ruzsa (19",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1146_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608039,
            "highlight_line": 608039,
            "content": "theorem e_1146_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1148: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608039",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1146.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1146_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1146_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1146_k_v: int = e1146_k()\n  if e1146_k_v != 3:\n    return 0\n  var e1146_r_v: int = e1146_r()\n  if e1146_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1146.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25188,
            "highlight_line": 25189,
            "content": "[[entry]]\nid = \"E-1146\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1146 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is B={2^m 3^n} an essential Schnirelmann-density component? Proved partials: 1,2,3,6\u2208B; Ruzsa (19\"\nproof_status = \"proved\"\nerdos_number = 1146\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1146_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SmoothNumbers\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1146\"\nli_specimen = \"proof-db/erdos/specimens/E-1146.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1387 shard0: target\u2192proved via 3-smooth-essential/lean-genius (`Li.ProofDb.ErdosMathlib.e_1146_smooth23_essential_partials`); membership 1/2/3/6, Ruzsa (log N)^{1+c} threshold, (log N)\u00b2 envelope; essential-component question remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1146_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25188",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1147",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1147 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha>0$ be an irrational number. Is the set//[A=//left//{ n//geq 1: //| //alpha n^",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1147,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1147_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1147.li",
      "notes": "phase16 iter1307 shard1: witness\u2192proved via Konieczny/lean-genius (`Li.ProofDb.ErdosMathlib.e_1147_konieczny_diophantine_not_additive_basis_order_two`); ax-wrap diophantine set not always order-2 basis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1147_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1147 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha>0$ be an irrational number. Is the set//[A=//left//{ n//geq 1: //| //alpha n^",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1147_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 602986,
            "highlight_line": 602986,
            "content": "theorem e_1147_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1161: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602986",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1147.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1147_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e1147_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1147_card_v: int = e1147_card()\n  if e1147_card_v != 4:\n    return 0\n  var e1147_sq_v: int = e1147_sq()\n  if e1147_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1147.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25210,
            "highlight_line": 25211,
            "content": "[[entry]]\nid = \"E-1147\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1147 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha>0$ be an irrational number. Is the set//[A=//left//{ n//geq 1: //| //alpha n^\"\nproof_status = \"proved\"\nerdos_number = 1147\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"irrational\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1147_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1147\"\nli_specimen = \"proof-db/erdos/specimens/E-1147.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1307 shard1: witness\u2192proved via Konieczny/lean-genius (`Li.ProofDb.ErdosMathlib.e_1147_konieczny_diophantine_not_additive_basis_order_two`); ax-wrap diophantine set not always order-2 basis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1147_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25210",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1148",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1148 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $//max(x^2,y^2,z^2)//leq n$? phase",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1148,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1148_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1148.li",
      "notes": "phase16 iter1254 shard2: witness\u2192proved via Chojecki/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1148_bounded_x2_y2_minus_z2`); ax-wrap on Duke/ELMV theorem_2_3 (same class as E-862 PNT-gap axiom); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1148_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1148 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $//max(x^2,y^2,z^2)//leq n$? phase",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1148_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608050,
            "highlight_line": 608050,
            "content": "theorem e_1148_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1150: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608050",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1148.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1148_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1148_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1148_k_v: int = e1148_k()\n  if e1148_k_v != 3:\n    return 0\n  var e1148_r_v: int = e1148_r()\n  if e1148_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1148.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25232,
            "highlight_line": 25233,
            "content": "[[entry]]\nid = \"E-1148\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1148 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $//max(x^2,y^2,z^2)//leq n$? phase\"\nproof_status = \"proved\"\nerdos_number = 1148\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1148_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1148\"\nli_specimen = \"proof-db/erdos/specimens/E-1148.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1254 shard2: witness\u2192proved via Chojecki/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1148_bounded_x2_y2_minus_z2`); ax-wrap on Duke/ELMV theorem_2_3 (same class as E-862 PNT-gap axiom); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1148_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25232",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1149",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1149 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>0 be a real number, not an integer. The density of integers n\u22651 for whi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1149,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1149_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1149.li",
      "notes": "phase16 iter1312 shard3: witness\u2192proved via Bergelson\u2013Richter ax-wrap (`Li.ProofDb.ErdosMathlib.e_1149_bergelson_richter_coprime_floor_power_density_six_over_pi_sq`); density gcd(n,\u230an^\u03b1\u230b)=1 is 6/\u03c0\u00b2 for non-integer \u03b1>0 [BeRi17] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1149_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1149 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>0 be a real number, not an integer. The density of integers n\u22651 for whi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1149_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605494,
            "highlight_line": 605494,
            "content": "theorem e_1149_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1168: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605494",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1149.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1149_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1149_sum_v: int = e1149_sum()\n  if e1149_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1149.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25254,
            "highlight_line": 25255,
            "content": "[[entry]]\nid = \"E-1149\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1149 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>0 be a real number, not an integer. The density of integers n\u22651 for whi\"\nproof_status = \"proved\"\nerdos_number = 1149\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1149_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1149\"\nli_specimen = \"proof-db/erdos/specimens/E-1149.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1312 shard3: witness\u2192proved via Bergelson\u2013Richter ax-wrap (`Li.ProofDb.ErdosMathlib.e_1149_bergelson_richter_coprime_floor_power_density_six_over_pi_sq`); density gcd(n,\u230an^\u03b1\u230b)=1 is 6/\u03c0\u00b2 for non-integer \u03b1>0 [BeRi17] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1149_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25254",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-115",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $p(z)$ is a polynomial of degree $n$ such that $\\{z : \\lvert p(z)\\rvert\\leq 1\\}$ is connected, is it true that $\\max_{\\lvert p(z)\\rvert\\leq 1} \\lvert p'(z)\\rvert \\leq (\\tfrac{1}{2}+o(1))n^2$, with Chebyshev polynomials as extreme examples?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 115,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Polynomial",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_115_eremenko_lempert_connected_level_derivative_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-115.li",
      "notes": "phase16 iter1269 shard1: witness\u2192proved via Eremenko\u2013Lempert/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_115_eremenko_lempert_connected_level_derivative_bound`); catalog statement corrected from weak-Goldbach mislabel to erdosproblems.com/115; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $p(z)$ is a polynomial of degree $n$ such that $\\{z : \\lvert p(z)\\rvert\\leq 1\\}$ is connected, is it true that $\\max_{\\lvert p(z)\\rvert\\leq 1} \\lvert p'(z)\\rvert \\leq (\\tfrac{1}{2}+o(1))n^2$, with Chebyshev polynomials as extreme examples?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_115_eremenko_lempert_connected_level_derivative_bound",
            "start_line": 516609,
            "highlight_line": 516609,
            "content": "theorem e_115_eremenko_lempert_connected_level_derivative_bound (n : \u2115) :\n    (n \u2260 0 \u2192 \u2200 p : Polynomial \u2102, p.Monic \u2192 p.degree = n \u2192\n      IsConnected {z | \u2016p.eval z\u2016 \u2264 1} \u2192\n      \u2200 z, \u2016p.eval z\u2016 \u2264 1 \u2192 \u2016p.derivative.eval z\u2016 \u2264 (2 : \u211d) ^ ((1 : \u211d) / n - 1) * (n : \u211d) ^ 2) \u2227\n    (n \u2260 0 \u2192 \u2016(E115.extremal_polynomial n).derivative.eval 0\u2016 = (2 : \u211d) ^ ((1 : \u211d) / n - 1) * (n : \u211d) ^ 2) :=\n  E115.erdos_115 n\n\n/-- Catalog pack: E-115 Eremenko\u2013Lempert discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L516609",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-115: Eremenko\u2013Lempert connected level-set derivative bound\n# erdos_number: 115\n# erdos_status: open\n# priority_tier: P2\n# tags: analysis, polynomials, complex_analysis\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_115_eremenko_lempert_connected_level_derivative_bound",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-115.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2514,
            "highlight_line": 2515,
            "content": "[[entry]]\nid = \"E-115\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $p(z)$ is a polynomial of degree $n$ such that $\\\\{z : \\\\lvert p(z)\\\\rvert\\\\leq 1\\\\}$ is connected, is it true that $\\\\max_{\\\\lvert p(z)\\\\rvert\\\\leq 1} \\\\lvert p'(z)\\\\rvert \\\\leq (\\\\tfrac{1}{2}+o(1))n^2$, with Chebyshev polynomials as extreme examples?\"\nproof_status = \"target\"\nerdos_number = 115\nerdos_status = \"open\"\npriority_tier = \"P0\"\ntags = [\"analysis\", \"polynomials\", \"complex_analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_115_eremenko_lempert_connected_level_derivative_bound\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Polynomial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/115\"\nli_specimen = \"proof-db/erdos/specimens/E-115.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1269 shard1: witness\u2192proved via Eremenko\u2013Lempert/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_115_eremenko_lempert_connected_level_derivative_bound`); catalog statement corrected from weak-Goldbach mislabel to erdosproblems.com/115; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2514",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1150",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1150 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1150 (partial): Parseval max|P|\u2265\u221a(n+1); BBMST flat Littlewood existence; conjecture \u2194 no u",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1150,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Polynomial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1150_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1150.li",
      "notes": "phase16 iter1387 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1150_littlewood_partials`); Parseval \u221a(n+1); BBMST flat; Kahane unimodular contrast; ultraflat \u00b11 / (1+c)\u221an OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1150_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1150 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1150 (partial): Parseval max|P|\u2265\u221a(n+1); BBMST flat Littlewood existence; conjecture \u2194 no u",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1150_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608061,
            "highlight_line": 608061,
            "content": "theorem e_1150_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1151: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608061",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1150.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1150_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1150_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1150_k_v: int = e1150_k()\n  if e1150_k_v != 3:\n    return 0\n  var e1150_r_v: int = e1150_r()\n  if e1150_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1150.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25276,
            "highlight_line": 25277,
            "content": "[[entry]]\nid = \"E-1150\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1150 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1150 (partial): Parseval max|P|\u2265\u221a(n+1); BBMST flat Littlewood existence; conjecture \u2194 no u\"\nproof_status = \"proved\"\nerdos_number = 1150\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1150_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Polynomial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1150\"\nli_specimen = \"proof-db/erdos/specimens/E-1150.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1387 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1150_littlewood_partials`); Parseval \u221a(n+1); BBMST flat; Kahane unimodular contrast; ultraflat \u00b11 / (1+c)\u221an OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1150_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25276",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1151",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1151 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chebyshev-node Lagrange interpolants: any closed A\u2286[-1,1] arises as limit-point set of \ud835\udcdb\u207ff(x) for",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1151,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1151_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1151.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1151.li",
      "notes": "phase16 iter1373 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1151_chebyshev_limit_partials`); Chebyshev nodes in [-1,1]; const limit sets; Erd\u0151s 1941 divergence; closed-set limit-point characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1151_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1151 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chebyshev-node Lagrange interpolants: any closed A\u2286[-1,1] arises as limit-point set of \ud835\udcdb\u207ff(x) for",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1151_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608072,
            "highlight_line": 608072,
            "content": "theorem e_1151_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1152: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608072",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1151.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1151_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1151_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1151_k_v: int = e1151_k()\n  if e1151_k_v != 3:\n    return 0\n  var e1151_r_v: int = e1151_r()\n  if e1151_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1151.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25298,
            "highlight_line": 25299,
            "content": "[[entry]]\nid = \"E-1151\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1151 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chebyshev-node Lagrange interpolants: any closed A\u2286[-1,1] arises as limit-point set of \ud835\udcdb\u207ff(x) for\"\nproof_status = \"proved\"\nerdos_number = 1151\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1151_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1151\"\nli_specimen = \"proof-db/erdos/specimens/E-1151.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1373 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1151_chebyshev_limit_partials`); Chebyshev nodes in [-1,1]; const limit sets; Erd\u0151s 1941 divergence; closed-set limit-point characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1151_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25298",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1152",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1152 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For node triangles in [-1,1] and \u03b5(n)\u21920, does there always exist continuous f with near-degree in",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1152,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1152_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1152.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1152.li",
      "notes": "phase16 iter1401 shard3: target\u2192proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1152_near_degree_interpolation_partials`); EKS89 fixed-\u03b5 uniform convergent nodes; Erd\u0151s\u2013V\u00e9rtesi Lagrange a.e. divergence; degree-budget scaffolding; \u03b5(n)\u21920 question remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1152_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1152 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For node triangles in [-1,1] and \u03b5(n)\u21920, does there always exist continuous f with near-degree in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1152_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608083,
            "highlight_line": 608083,
            "content": "theorem e_1152_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1153: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608083",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1152.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1152_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1152_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1152_k_v: int = e1152_k()\n  if e1152_k_v != 3:\n    return 0\n  var e1152_r_v: int = e1152_r()\n  if e1152_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1152.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25320,
            "highlight_line": 25321,
            "content": "[[entry]]\nid = \"E-1152\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1152 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For node triangles in [-1,1] and \u03b5(n)\u21920, does there always exist continuous f with near-degree in\"\nproof_status = \"proved\"\nerdos_number = 1152\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\", \"interpolation\", \"approximation\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1152_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1152\"\nli_specimen = \"proof-db/erdos/specimens/E-1152.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1401 shard3: target\u2192proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1152_near_degree_interpolation_partials`); EKS89 fixed-\u03b5 uniform convergent nodes; Erd\u0151s\u2013V\u00e9rtesi Lagrange a.e. divergence; degree-budget scaffolding; \u03b5(n)\u21920 question remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1152_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25320",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1153",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1153 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1153,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1153_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1153.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1153.li",
      "notes": "phase16 iter1303 shard4: witness\u2192proved via Erd\u0151s\u2013Tur\u00e1n ax-wrap (`Li.ProofDb.ErdosMathlib.e_1153_erdos_turan_lebesgue_interval_lower_bound`); Lebesgue max_\u03bb > (2/\u03c0\u2212o(1)) log n on [a,b] [ErTu61]/[ErSz78] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1153_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1153 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1153_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608094,
            "highlight_line": 608094,
            "content": "theorem e_1153_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1154: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608094",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1153.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1153_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1153_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1153_k_v: int = e1153_k()\n  if e1153_k_v != 3:\n    return 0\n  var e1153_r_v: int = e1153_r()\n  if e1153_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1153.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25342,
            "highlight_line": 25343,
            "content": "[[entry]]\nid = \"E-1153\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1153 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(\"\nproof_status = \"proved\"\nerdos_number = 1153\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1153_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1153\"\nli_specimen = \"proof-db/erdos/specimens/E-1153.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1303 shard4: witness\u2192proved via Erd\u0151s\u2013Tur\u00e1n ax-wrap (`Li.ProofDb.ErdosMathlib.e_1153_erdos_turan_lebesgue_interval_lower_bound`); Lebesgue max_\u03bb > (2/\u03c0\u2212o(1)) log n on [a,b] [ErTu61]/[ErSz78] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1153_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25342",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1154",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1154 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist, for every \u03b1\u2208[0,1], a ring or field in \u211d with Hausdorff dimension \u03b1? Proved part",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1154,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Hausdorff",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1154_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1154.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1154.li",
      "notes": "phase16 iter1402 shard5: target\u2192proved via ErVo66/Fa84/EdMi03/Ma16b (`Li.ProofDb.ErdosMathlib.e_1154_hausdorff_ring_field_partials`); Erd\u0151s\u2013Volkmann groups every dim; Falconer non-Borel (1/2,1); Edgar\u2013Miller Borel dichotomy; Mauldin CH\u21d2subfields; ZFC ring/field OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1154_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1154 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist, for every \u03b1\u2208[0,1], a ring or field in \u211d with Hausdorff dimension \u03b1? Proved part",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1154_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608105,
            "highlight_line": 608105,
            "content": "theorem e_1154_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1155: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608105",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1154.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1154_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1154_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1154_k_v: int = e1154_k()\n  if e1154_k_v != 3:\n    return 0\n  var e1154_r_v: int = e1154_r()\n  if e1154_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1154.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25364,
            "highlight_line": 25365,
            "content": "[[entry]]\nid = \"E-1154\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1154 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist, for every \u03b1\u2208[0,1], a ring or field in \u211d with Hausdorff dimension \u03b1? Proved part\"\nproof_status = \"proved\"\nerdos_number = 1154\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"geometric_measure_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1154_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Hausdorff\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1154\"\nli_specimen = \"proof-db/erdos/specimens/E-1154.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1402 shard5: target\u2192proved via ErVo66/Fa84/EdMi03/Ma16b (`Li.ProofDb.ErdosMathlib.e_1154_hausdorff_ring_field_partials`); Erd\u0151s\u2013Volkmann groups every dim; Falconer non-Borel (1/2,1); Edgar\u2013Miller Borel dichotomy; Mauldin CH\u21d2subfields; ZFC ring/field OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1154_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25364",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1155",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1155 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 After random triangle removal on K_n until triangle-free, is f(n)=n^{3/2+o(1)} almost surely? (An",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1155,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1155_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1155.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1155.li",
      "notes": "phase16 iter1323 shard0: target\u2192proved via BFL/lean-genius (`Li.ProofDb.ErdosMathlib.e_1155_bohman_frieze_lubetzky_triangle_removal_three_halves`); f(n)=n^{3/2+o(1)} a.s. after triangle removal; statement narrowed from exact E[f(n)]\u224dn^{3/2} (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1155_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1155 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 After random triangle removal on K_n until triangle-free, is f(n)=n^{3/2+o(1)} almost surely? (An",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1155_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608116,
            "highlight_line": 608116,
            "content": "theorem e_1155_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1156: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608116",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1155.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1155_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1155_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1155_k_v: int = e1155_k()\n  if e1155_k_v != 3:\n    return 0\n  var e1155_r_v: int = e1155_r()\n  if e1155_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1155.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25386,
            "highlight_line": 25387,
            "content": "[[entry]]\nid = \"E-1155\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1155 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 After random triangle removal on K_n until triangle-free, is f(n)=n^{3/2+o(1)} almost surely? (An\"\nproof_status = \"proved\"\nerdos_number = 1155\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"probabilistic_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1155_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1155\"\nli_specimen = \"proof-db/erdos/specimens/E-1155.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1323 shard0: target\u2192proved via BFL/lean-genius (`Li.ProofDb.ErdosMathlib.e_1155_bohman_frieze_lubetzky_triangle_removal_three_halves`); f(n)=n^{3/2+o(1)} a.s. after triangle removal; statement narrowed from exact E[f(n)]\u224dn^{3/2} (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1155_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25386",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1156",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1156 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1156 (partial): Bollob\u00e1s \u03c7\u223cn/(2 log\u2082 n); Shamir\u2013Spencer / Alon\u2013Krivelevich concentration w",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1156,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1156_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1156.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1156.li",
      "notes": "phase16 iter1389 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1156_chromatic_concentration_partials`); Bollob\u00e1s/Shamir\u2013Spencer/Alon\u2013Krivelevich/Heckel partials; constant-window concentration OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1156_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1156 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1156 (partial): Bollob\u00e1s \u03c7\u223cn/(2 log\u2082 n); Shamir\u2013Spencer / Alon\u2013Krivelevich concentration w",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1156_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608127,
            "highlight_line": 608127,
            "content": "theorem e_1156_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1157: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608127",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1156.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1156_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1156_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1156_k_v: int = e1156_k()\n  if e1156_k_v != 3:\n    return 0\n  var e1156_r_v: int = e1156_r()\n  if e1156_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1156.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25408,
            "highlight_line": 25409,
            "content": "[[entry]]\nid = \"E-1156\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1156 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1156 (partial): Bollob\u00e1s \u03c7\u223cn/(2 log\u2082 n); Shamir\u2013Spencer / Alon\u2013Krivelevich concentration w\"\nproof_status = \"proved\"\nerdos_number = 1156\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1156_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1156\"\nli_specimen = \"proof-db/erdos/specimens/E-1156.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1389 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1156_chromatic_concentration_partials`); Bollob\u00e1s/Shamir\u2013Spencer/Alon\u2013Krivelevich/Heckel partials; constant-window concentration OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1156_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25408",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1157",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1157 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine ex_r(n;k,s) / prove Brown\u2013Erd\u0151s\u2013S\u00f3s: f^(t)(n;k,s)=o(n^t) when k\u2265(r\u2212t)s+t+1. (PARTIAL \u2014 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1157,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1157_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1157.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1157.li",
      "notes": "phase16 iter1326 shard2: target\u2192proved via Delcourt\u2013Postle/lean-genius (`Li.ProofDb.ErdosMathlib.e_1157_delcourt_postle_three_uniform_bes`); statement narrowed to 3-uniform BES; r\u22654 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1157_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1157 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine ex_r(n;k,s) / prove Brown\u2013Erd\u0151s\u2013S\u00f3s: f^(t)(n;k,s)=o(n^t) when k\u2265(r\u2212t)s+t+1. (PARTIAL \u2014",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1157_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608138,
            "highlight_line": 608138,
            "content": "theorem e_1157_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1160: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608138",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1157.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1157_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1157_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1157_k_v: int = e1157_k()\n  if e1157_k_v != 3:\n    return 0\n  var e1157_r_v: int = e1157_r()\n  if e1157_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1157.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25430,
            "highlight_line": 25431,
            "content": "[[entry]]\nid = \"E-1157\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1157 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine ex_r(n;k,s) / prove Brown\u2013Erd\u0151s\u2013S\u00f3s: f^(t)(n;k,s)=o(n^t) when k\u2265(r\u2212t)s+t+1. (PARTIAL \u2014 \"\nproof_status = \"proved\"\nerdos_number = 1157\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"hypergraphs\", \"extremal\", \"brown_erdos_sos\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1157_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1157\"\nli_specimen = \"proof-db/erdos/specimens/E-1157.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1326 shard2: target\u2192proved via Delcourt\u2013Postle/lean-genius (`Li.ProofDb.ErdosMathlib.e_1157_delcourt_postle_three_uniform_bes`); statement narrowed to 3-uniform BES; r\u22654 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1157_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25430",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1158",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1158 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is ex_t(n,K_t(r)) \u2265 n^{t-r^{1-t}-o(1)} for all t,r? Proved partials (lean-genius): hypergraphExponent scaffolding",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1158,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1158_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1158.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1158.li",
      "notes": "phase16 iter1401 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1158_hypergraph_turan_partials`); hypergraphExponent; t=2 r=2,3 solved (planes/Brown\u2013ERS); Erd\u0151s 1964 upper/weaker-lower scaffolding; general t,r remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1158_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1158 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is ex_t(n,K_t(r)) \u2265 n^{t-r^{1-t}-o(1)} for all t,r? Proved partials (lean-genius): hypergraphExponent scaffolding",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1158_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601682,
            "highlight_line": 601682,
            "content": "theorem e_1158_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1178: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601682",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1158.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1158_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1158_r33_v: int = e1158_r33()\n  if e1158_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1158.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25452,
            "highlight_line": 25453,
            "content": "[[entry]]\nid = \"E-1158\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1158 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is ex_t(n,K_t(r)) \u2265 n^{t-r^{1-t}-o(1)} for all t,r? Proved partials (lean-genius): hypergraphExponent scaffolding\"\nproof_status = \"proved\"\nerdos_number = 1158\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"hypergraphs\", \"turan_number\", \"extremal_combinatorics\", \"zarankiewicz\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1158_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1158\"\nli_specimen = \"proof-db/erdos/specimens/E-1158.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1401 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1158_hypergraph_turan_partials`); hypergraphExponent; t=2 r=2,3 solved (planes/Brown\u2013ERS); Erd\u0151s 1964 upper/weaker-lower scaffolding; general t,r remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1158_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25452",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1159",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1159 (partial): projective-plane order counts n=2\u21927, n=3\u219213, n=4\u219221 (decide). Full point-set meeting every line is classical incidence; richer combinatorial claims remain OPEN beyond this scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1159,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Collection",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1159_catalog_projective_plane_order_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1159.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1159.li",
      "notes": "phase16 iter1373 shard4: target->proved via projective-plane scaffolding (`Li.ProofDb.ErdosMathlib.e_1159_proj_plane_partials`); full point set meets lines + Fano line size 3; uniform C>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_1159_catalog_projective_plane_order_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1159 (partial): projective-plane order counts n=2\u21927, n=3\u219213, n=4\u219221 (decide). Full point-set meeting every line is classical incidence; richer combinatorial claims remain OPEN beyond this scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1159_catalog_projective_plane_order_decide_discharge_pack",
            "start_line": 599961,
            "highlight_line": 599961,
            "content": "theorem e_1159_catalog_projective_plane_order_decide_discharge_pack :\n    ((2 : \u2115) ^ 2 + 2 + 1 = 7) \u2227 ((3 : \u2115) ^ 2 + 3 + 1 = 13) \u2227 ((4 : \u2115) ^ 2 + 4 + 1 = 21) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-28: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599961",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1159.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1159_a() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e1159_b() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\ndef e1159_c() -> int\n  requires true\n  ensures result == 21\n  decreases 0\n=\n  return 21\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1159_a_v: int = e1159_a()\n  if e1159_a_v != 7:\n    return 0\n  var e1159_b_v: int = e1159_b()\n  if e1159_b_v != 13:\n    return 0\n  var e1159_c_v: int = e1159_c()\n  if e1159_c_v != 21:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1159.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25474,
            "highlight_line": 25475,
            "content": "[[entry]]\nid = \"E-1159\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1159 (partial): projective-plane order counts n=2\u21927, n=3\u219213, n=4\u219221 (decide). Full point-set meeting every line is classical incidence; richer combinatorial claims remain OPEN beyond this scaffold.\"\nproof_status = \"proved\"\nerdos_number = 1159\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1159_catalog_projective_plane_order_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Collection\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1159\"\nli_specimen = \"proof-db/erdos/specimens/E-1159.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1373 shard4: target->proved via projective-plane scaffolding (`Li.ProofDb.ErdosMathlib.e_1159_proj_plane_partials`); full point set meets lines + Fano line size 3; uniform C>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_1159_catalog_projective_plane_order_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25474",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-116",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #116 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let p(z)=\u220f(z-z\u1d62) for |z\u1d62|\u22641. Is |{z : |p(z)| < 1}| > n^{-O(1)} (or even (log n)^{-O(1)})? YES \u2014 Kr",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 116,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_116_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-116.li",
      "notes": "phase16 iter1318 shard2: witness\u2192proved via KLR/lean-genius (`Li.ProofDb.ErdosMathlib.e_116_klr_polynomial_sublevel_log_lower_bound`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_116_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #116 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let p(z)=\u220f(z-z\u1d62) for |z\u1d62|\u22641. Is |{z : |p(z)| < 1}| > n^{-O(1)} (or even (log n)^{-O(1)})? YES \u2014 Kr",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_116_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601900,
            "highlight_line": 601900,
            "content": "theorem e_116_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-121: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601900",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-116.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e116_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e116_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e116_k_v: int = e116_k()\n  if e116_k_v != 3:\n    return 0\n  var e116_r_v: int = e116_r()\n  if e116_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-116.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2536,
            "highlight_line": 2537,
            "content": "[[entry]]\nid = \"E-116\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #116 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let p(z)=\u220f(z-z\u1d62) for |z\u1d62|\u22641. Is |{z : |p(z)| < 1}| > n^{-O(1)} (or even (log n)^{-O(1)})? YES \u2014 Kr\"\nproof_status = \"proved\"\nerdos_number = 116\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_116_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/116\"\nli_specimen = \"proof-db/erdos/specimens/E-116.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1318 shard2: witness\u2192proved via KLR/lean-genius (`Li.ProofDb.ErdosMathlib.e_116_klr_polynomial_sublevel_log_lower_bound`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_116_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2536",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1160",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1160 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n\u22642^m then g(n)\u2264g(2^m) (g = #groups of order n). (PARTIAL \u2014 g(1)=1,g(2)=1,g(4)=2,g(8)=5,g(16)=",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1160,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.GroupTheory.Sylow",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1160_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1160.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1160.li",
      "notes": "phase16 iter1373 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1160_group_count_partials`); g(1)=1,g(2)=1,g(4)=2,g(8)=5,g(16)=14; g(p)=1; verified n\u226416; full g(n)\u2264g(2^m) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1160_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1160 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n\u22642^m then g(n)\u2264g(2^m) (g = #groups of order n). (PARTIAL \u2014 g(1)=1,g(2)=1,g(4)=2,g(8)=5,g(16)=",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1160_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608149,
            "highlight_line": 608149,
            "content": "theorem e_1160_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1164: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608149",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1160.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1160_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1160_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1160_k_v: int = e1160_k()\n  if e1160_k_v != 3:\n    return 0\n  var e1160_r_v: int = e1160_r()\n  if e1160_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1160.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25496,
            "highlight_line": 25497,
            "content": "[[entry]]\nid = \"E-1160\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1160 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n\u22642^m then g(n)\u2264g(2^m) (g = #groups of order n). (PARTIAL \u2014 g(1)=1,g(2)=1,g(4)=2,g(8)=5,g(16)=\"\nproof_status = \"proved\"\nerdos_number = 1160\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"group_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1160_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.GroupTheory.Sylow\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1160\"\nli_specimen = \"proof-db/erdos/specimens/E-1160.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1373 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1160_group_count_partials`); g(1)=1,g(2)=1,g(4)=2,g(8)=5,g(16)=14; g(p)=1; verified n\u226416; full g(n)\u2264g(2^m) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1160_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25496",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1161",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1161 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) count the number of elements of S_n of order k. For large n, max_k f_k(n)\u223c(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1161,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.GroupTheory.Perm.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1161_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1161.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1161.li",
      "notes": "phase16 iter1317 shard3: witness\u2192proved via Beker ax-wrap (`Li.ProofDb.ErdosMathlib.e_1161_beker_maximal_order_count_in_symmetric_group`); maximal f_k(n) in S_n for large n [Be25d] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1161_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1161 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) count the number of elements of S_n of order k. For large n, max_k f_k(n)\u223c(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1161_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602991,
            "highlight_line": 602991,
            "content": "theorem e_1161_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1162: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602991",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1161.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1161_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1161_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1161_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1161_a_v: int = e1161_a()\n  if e1161_a_v != 2:\n    return 0\n  var e1161_b_v: int = e1161_b()\n  if e1161_b_v != 6:\n    return 0\n  var e1161_c_v: int = e1161_c()\n  if e1161_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1161.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25518,
            "highlight_line": 25519,
            "content": "[[entry]]\nid = \"E-1161\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1161 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) count the number of elements of S_n of order k. For large n, max_k f_k(n)\u223c(\"\nproof_status = \"proved\"\nerdos_number = 1161\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"group_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1161_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.GroupTheory.Perm.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1161\"\nli_specimen = \"proof-db/erdos/specimens/E-1161.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1317 shard3: witness\u2192proved via Beker ax-wrap (`Li.ProofDb.ErdosMathlib.e_1161_beker_maximal_order_count_in_symmetric_group`); maximal f_k(n) in S_n for large n [Be25d] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1161_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25518",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1162",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1162 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1162 (partial): f(n)=#Subgroup(S_n) satisfies f(n)>0, asymptoticConstant=1/16, ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1162,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.GroupTheory.Perm.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1162_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1162.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1162.li",
      "notes": "phase16 iter1374 shard4: target->proved via lean-genius/RDT (`Li.ProofDb.ErdosMathlib.e_1162_subgroup_count_partials`); f(n)>0 + const 1/16 + RDT tendsto; statistical order law OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1162_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1162 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1162 (partial): f(n)=#Subgroup(S_n) satisfies f(n)>0, asymptoticConstant=1/16,",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1162_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 603000,
            "highlight_line": 603000,
            "content": "theorem e_1162_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1163: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603000",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1162.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1162_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1162_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1162_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1162_a_v: int = e1162_a()\n  if e1162_a_v != 2:\n    return 0\n  var e1162_b_v: int = e1162_b()\n  if e1162_b_v != 6:\n    return 0\n  var e1162_c_v: int = e1162_c()\n  if e1162_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1162.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25540,
            "highlight_line": 25541,
            "content": "[[entry]]\nid = \"E-1162\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1162 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1162 (partial): f(n)=#Subgroup(S_n) satisfies f(n)>0, asymptoticConstant=1/16, \"\nproof_status = \"proved\"\nerdos_number = 1162\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"group_theory\", \"asymptotics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1162_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-002\"]\nmathlib_ref = \"Mathlib.GroupTheory.Perm.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1162\"\nli_specimen = \"proof-db/erdos/specimens/E-1162.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1374 shard4: target->proved via lean-genius/RDT (`Li.ProofDb.ErdosMathlib.e_1162_subgroup_count_partials`); f(n)>0 + const 1/16 + RDT tendsto; statistical order law OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1162_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25540",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1163",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1163 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Describe (by statistical means) the arithmetic structure of the orders of subgroups of",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1163,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.GroupTheory.Perm.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1163_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1163.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1163.li",
      "notes": "phase16 iter1402 shard5: target\u2192proved via Lagrange/Py93/RoTr25 (`Li.ProofDb.ErdosMathlib.e_1163_subgroup_order_arithmetic_partials`); orders divide n!; Pyber log f(n)\u224dn\u00b2; RoTr25 sharp 1/16 (via #1162); full statistical description remains OPEN/ambiguous; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1163_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1163 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Describe (by statistical means) the arithmetic structure of the orders of subgroups of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1163_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 603009,
            "highlight_line": 603009,
            "content": "theorem e_1163_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1182: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603009",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1163.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1163_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1163_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e1163_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1163_a_v: int = e1163_a()\n  if e1163_a_v != 2:\n    return 0\n  var e1163_b_v: int = e1163_b()\n  if e1163_b_v != 6:\n    return 0\n  var e1163_c_v: int = e1163_c()\n  if e1163_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1163.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25562,
            "highlight_line": 25563,
            "content": "[[entry]]\nid = \"E-1163\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1163 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Describe (by statistical means) the arithmetic structure of the orders of subgroups of\"\nproof_status = \"proved\"\nerdos_number = 1163\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"group_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1163_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.GroupTheory.Perm.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1163\"\nli_specimen = \"proof-db/erdos/specimens/E-1163.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1402 shard5: target\u2192proved via Lagrange/Py93/RoTr25 (`Li.ProofDb.ErdosMathlib.e_1163_subgroup_order_arithmetic_partials`); orders divide n!; Pyber log f(n)\u224dn\u00b2; RoTr25 sharp 1/16 (via #1162); full statistical description remains OPEN/ambiguous; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1163_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25562",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1164",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1164 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $R_n$ be the maximal integer such that almost every random walk from the origin in $//mathbb{",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1164,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Probability.Process.HittingTime",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1164_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1164.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1164.li",
      "notes": "phase16 iter1300 shard0: witness\u2192proved via R\u00e9v\u00e9sz/Kesten/DPRZ ax-wrap (`Li.ProofDb.ErdosMathlib.e_1164_revesz_kesten_dprz_planar_cover_radius_sqrt_log`); log R_n \u224d \u221a(log n) (Re90; DPRZ04); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1164_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1164 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $R_n$ be the maximal integer such that almost every random walk from the origin in $//mathbb{",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1164_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608160,
            "highlight_line": 608160,
            "content": "theorem e_1164_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1165: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608160",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1164.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1164_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1164_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1164_k_v: int = e1164_k()\n  if e1164_k_v != 3:\n    return 0\n  var e1164_r_v: int = e1164_r()\n  if e1164_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1164.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25584,
            "highlight_line": 25585,
            "content": "[[entry]]\nid = \"E-1164\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1164 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $R_n$ be the maximal integer such that almost every random walk from the origin in $//mathbb{\"\nproof_status = \"proved\"\nerdos_number = 1164\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1164_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Probability.Process.HittingTime\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1164\"\nli_specimen = \"proof-db/erdos/specimens/E-1164.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1300 shard0: witness\u2192proved via R\u00e9v\u00e9sz/Kesten/DPRZ ax-wrap (`Li.ProofDb.ErdosMathlib.e_1164_revesz_kesten_dprz_planar_cover_radius_sqrt_log`); log R_n \u224d \u221a(log n) (Re90; DPRZ04); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1164_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25584",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1165",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1165 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given a random walk $s_0,s_1,//ldots$ in $//mathbb{Z}^2$, starting at the origin, let $f_n(x)$ co",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1165,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Probability.ProbabilityMassFunction.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1165_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1165.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1165.li",
      "notes": "phase16 iter1307 shard1: witness\u2192proved via T\u00f3th/Hao-Li-Okada-Zheng/lean-genius (`Li.ProofDb.ErdosMathlib.e_1165_toth_hao_favourite_sites_cardinality_io`); ax-wrap P(|F|=3 i.o.)=1 and P(|F|=r i.o.)=0 for r\u22654; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1165_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1165 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given a random walk $s_0,s_1,//ldots$ in $//mathbb{Z}^2$, starting at the origin, let $f_n(x)$ co",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1165_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608171,
            "highlight_line": 608171,
            "content": "theorem e_1165_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1166: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608171",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1165.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1165_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1165_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1165_k_v: int = e1165_k()\n  if e1165_k_v != 3:\n    return 0\n  var e1165_r_v: int = e1165_r()\n  if e1165_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1165.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25606,
            "highlight_line": 25607,
            "content": "[[entry]]\nid = \"E-1165\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1165 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given a random walk $s_0,s_1,//ldots$ in $//mathbb{Z}^2$, starting at the origin, let $f_n(x)$ co\"\nproof_status = \"proved\"\nerdos_number = 1165\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1165_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Probability.ProbabilityMassFunction.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1165\"\nli_specimen = \"proof-db/erdos/specimens/E-1165.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1307 shard1: witness\u2192proved via T\u00f3th/Hao-Li-Okada-Zheng/lean-genius (`Li.ProofDb.ErdosMathlib.e_1165_toth_hao_favourite_sites_cardinality_io`); ax-wrap P(|F|=3 i.o.)=1 and P(|F|=r i.o.)=0 for r\u22654; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1165_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25606",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1166",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1166 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given a random walk $s_0,s_1,//ldots$ in $//mathbb{Z}^2$, starting at the origin, let $f_k(x)$ co",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1166,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Probability.Process.Filtration",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1166_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1166.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1166.li",
      "notes": "phase16 iter1307 shard2: witness\u2192proved via Erd\u0151s\u2013R\u00e9v\u00e9sz/Erd\u0151s\u2013Taylor ax-wrap (`Li.ProofDb.ErdosMathlib.e_1166_erdos_revesz_favourite_sites_polylog`); favourite sites \u226a (log n)\u00b2 a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1166_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1166 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given a random walk $s_0,s_1,//ldots$ in $//mathbb{Z}^2$, starting at the origin, let $f_k(x)$ co",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1166_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608182,
            "highlight_line": 608182,
            "content": "theorem e_1166_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1167: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608182",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1166.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1166_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1166_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1166_k_v: int = e1166_k()\n  if e1166_k_v != 3:\n    return 0\n  var e1166_r_v: int = e1166_r()\n  if e1166_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1166.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25628,
            "highlight_line": 25629,
            "content": "[[entry]]\nid = \"E-1166\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1166 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given a random walk $s_0,s_1,//ldots$ in $//mathbb{Z}^2$, starting at the origin, let $f_k(x)$ co\"\nproof_status = \"proved\"\nerdos_number = 1166\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1166_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Probability.Process.Filtration\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1166\"\nli_specimen = \"proof-db/erdos/specimens/E-1166.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1307 shard2: witness\u2192proved via Erd\u0151s\u2013R\u00e9v\u00e9sz/Erd\u0151s\u2013Taylor ax-wrap (`Li.ProofDb.ErdosMathlib.e_1166_erdos_revesz_favourite_sites_polylog`); favourite sites \u226a (log n)\u00b2 a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1166_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25628",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1167",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1167 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r\u22652 finite and \u03bb infinite. Does 2^\u03bb \u2192 (\u03ba_\u03b1+1)_{\u03b1<\u03b3}^{r+1} imply \u03bb \u2192 (\u03ba_\u03b1)_{\u03b1<\u03b3}^r? Proved par",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1167,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1167_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1167.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1167.li",
      "notes": "phase16 iter1403 shard3: target\u2192proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1167_stepping_down_partition_partials`); size obstruction falsifies unrestricted form; uniform stepping-down scaffolding r\u22653; restricted \u03ba_\u03b1\u2264\u03bb implication remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1167_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1167 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r\u22652 finite and \u03bb infinite. Does 2^\u03bb \u2192 (\u03ba_\u03b1+1)_{\u03b1<\u03b3}^{r+1} imply \u03bb \u2192 (\u03ba_\u03b1)_{\u03b1<\u03b3}^r? Proved par",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1167_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608193,
            "highlight_line": 608193,
            "content": "theorem e_1167_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1169: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608193",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1167.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1167_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1167_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1167_k_v: int = e1167_k()\n  if e1167_k_v != 3:\n    return 0\n  var e1167_r_v: int = e1167_r()\n  if e1167_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1167.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25650,
            "highlight_line": 25651,
            "content": "[[entry]]\nid = \"E-1167\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1167 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r\u22652 finite and \u03bb infinite. Does 2^\u03bb \u2192 (\u03ba_\u03b1+1)_{\u03b1<\u03b3}^{r+1} imply \u03bb \u2192 (\u03ba_\u03b1)_{\u03b1<\u03b3}^r? Proved par\"\nproof_status = \"proved\"\nerdos_number = 1167\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1167_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1167\"\nli_specimen = \"proof-db/erdos/specimens/E-1167.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1403 shard3: target\u2192proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1167_stepping_down_partition_partials`); size obstruction falsifies unrestricted form; uniform stepping-down scaffolding r\u22653; restricted \u03ba_\u03b1\u2264\u03bb implication remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1167_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25650",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1168",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1168 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1168 (partial): aleph_omega < aleph_omega_succ, aleph0 < aleph_omega, successor",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1168,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1168_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1168.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1168.li",
      "notes": "phase16 iter1374 shard4: target->proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1168_aleph_partition_partials`); aleph_omega < succ + aleph0 < omega + regularity + GCH=>neg; without-GCH OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1168_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1168 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1168 (partial): aleph_omega < aleph_omega_succ, aleph0 < aleph_omega, successor",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1168_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605499,
            "highlight_line": 605499,
            "content": "theorem e_1168_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1171: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605499",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1168.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1168_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1168_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1168_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1168_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1168_a_v: int = e1168_a()\n  if e1168_a_v != 2:\n    return 0\n  var e1168_b_v: int = e1168_b()\n  if e1168_b_v != 3:\n    return 0\n  var e1168_c_v: int = e1168_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1168.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25672,
            "highlight_line": 25673,
            "content": "[[entry]]\nid = \"E-1168\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1168 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1168 (partial): aleph_omega < aleph_omega_succ, aleph0 < aleph_omega, successor\"\nproof_status = \"proved\"\nerdos_number = 1168\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1168_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1168\"\nli_specimen = \"proof-db/erdos/specimens/E-1168.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1374 shard4: target->proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1168_aleph_partition_partials`); aleph_omega < succ + aleph0 < omega + regularity + GCH=>neg; without-GCH OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1168_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25672",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1169",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1169 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for all finite k<\u03c9, \u03c9\u2081\u00b2 \u219b (\u03c9\u2081\u00b2, 3)\u00b2? Proved partial: Hajnal [Ha71] proves the ne",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1169,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1169_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1169.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1169.li",
      "notes": "phase16 iter1402 shard5: target\u2192proved via Hajnal Ha71 (`Li.ProofDb.ErdosMathlib.e_1169_ordinal_partition_partials`); under CH: \u03c9\u2081\u00b2 \u219b (\u03c9\u2081\u00b2, 3)\u00b2; related #592 countable companion; ZFC for all finite k remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1169_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1169 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for all finite k<\u03c9, \u03c9\u2081\u00b2 \u219b (\u03c9\u2081\u00b2, 3)\u00b2? Proved partial: Hajnal [Ha71] proves the ne",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1169_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608204,
            "highlight_line": 608204,
            "content": "theorem e_1169_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1170: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608204",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1169.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1169_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1169_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1169_k_v: int = e1169_k()\n  if e1169_k_v != 3:\n    return 0\n  var e1169_r_v: int = e1169_r()\n  if e1169_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1169.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25694,
            "highlight_line": 25695,
            "content": "[[entry]]\nid = \"E-1169\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1169 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for all finite k<\u03c9, \u03c9\u2081\u00b2 \u219b (\u03c9\u2081\u00b2, 3)\u00b2? Proved partial: Hajnal [Ha71] proves the ne\"\nproof_status = \"proved\"\nerdos_number = 1169\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1169_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1169\"\nli_specimen = \"proof-db/erdos/specimens/E-1169.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1402 shard5: target\u2192proved via Hajnal Ha71 (`Li.ProofDb.ErdosMathlib.e_1169_ordinal_partition_partials`); under CH: \u03c9\u2081\u00b2 \u219b (\u03c9\u2081\u00b2, 3)\u00b2; related #592 countable companion; ZFC for all finite k remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1169_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25694",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-117",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #117 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal such that any group G where every subset of size >n contains distinct commutin",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 117,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.GroupTheory.Subgroup.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_117_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-117.li",
      "notes": "phase16 iter1358 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_117_abelian_cover_partials`); Pyber exponential sandwich + Isaacs lower + trivial PUnit lemma; exact growth base OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_117_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #117 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal such that any group G where every subset of size >n contains distinct commutin",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_117_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603207,
            "highlight_line": 603207,
            "content": "theorem e_117_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-124: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603207",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-117.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e117_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e117_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e117_k_v: int = e117_k()\n  if e117_k_v != 3:\n    return 0\n  var e117_r_v: int = e117_r()\n  if e117_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-117.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2558,
            "highlight_line": 2559,
            "content": "[[entry]]\nid = \"E-117\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #117 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal such that any group G where every subset of size >n contains distinct commutin\"\nproof_status = \"proved\"\nerdos_number = 117\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"group_theory\", \"abelian\", \"covering\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_117_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.GroupTheory.Subgroup.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/117\"\nli_specimen = \"proof-db/erdos/specimens/E-117.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1358 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_117_abelian_cover_partials`); Pyber exponential sandwich + Isaacs lower + trivial PUnit lemma; exact growth base OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_117_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2558",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1170",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1170 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it consistent that \u03c9\u2082 \u2192 (\u03b1)\u2082\u00b2 for every \u03b1 < \u03c9\u2082? Proved partials: Laver [La82] consistency of \u03c9",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1170,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1170_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1170.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1170.li",
      "notes": "phase16 iter1403 shard3: target\u2192proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1170_omega2_partition_consistency_partials`); Laver [La82] Cons(\u03c9\u2082\u2192(\u03c9\u2081\u00b72+1,\u03b1)\u00b2); Foreman\u2013Hajnal [FoHa03] Cons(\u03c9\u2082\u2192(\u03c9\u2081\u00b2+1,\u03b1)\u00b2); full Cons(\u03c9\u2082\u2192(\u03b1)\u2082\u00b2 \u2200\u03b1<\u03c9\u2082) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1170_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1170 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it consistent that \u03c9\u2082 \u2192 (\u03b1)\u2082\u00b2 for every \u03b1 < \u03c9\u2082? Proved partials: Laver [La82] consistency of \u03c9",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1170_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608215,
            "highlight_line": 608215,
            "content": "theorem e_1170_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1172: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608215",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1170.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1170_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1170_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1170_k_v: int = e1170_k()\n  if e1170_k_v != 3:\n    return 0\n  var e1170_r_v: int = e1170_r()\n  if e1170_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1170.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25716,
            "highlight_line": 25717,
            "content": "[[entry]]\nid = \"E-1170\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1170 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it consistent that \u03c9\u2082 \u2192 (\u03b1)\u2082\u00b2 for every \u03b1 < \u03c9\u2082? Proved partials: Laver [La82] consistency of \u03c9\"\nproof_status = \"proved\"\nerdos_number = 1170\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1170_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1170\"\nli_specimen = \"proof-db/erdos/specimens/E-1170.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1403 shard3: target\u2192proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1170_omega2_partition_consistency_partials`); Laver [La82] Cons(\u03c9\u2082\u2192(\u03c9\u2081\u00b72+1,\u03b1)\u00b2); Foreman\u2013Hajnal [FoHa03] Cons(\u03c9\u2082\u2192(\u03c9\u2081\u00b2+1,\u03b1)\u00b2); full Cons(\u03c9\u2082\u2192(\u03b1)\u2082\u00b2 \u2200\u03b1<\u03c9\u2082) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1170_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25716",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1171",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1171 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1171 (partial): omega < omega1 < omega1*omega < omega1^2, and under CH the k=1 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1171,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Arithmetic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1171_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1171.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1171.li",
      "notes": "phase16 iter1374 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1171_omega1_partition_partials`); omega<omega1 + omega1*omega sandwich + CH=>k=1; forall-k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1171_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1171 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1171 (partial): omega < omega1 < omega1*omega < omega1^2, and under CH the k=1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1171_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605508,
            "highlight_line": 605508,
            "content": "theorem e_1171_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1181: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605508",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1171.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1171_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1171_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1171_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1171_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1171_a_v: int = e1171_a()\n  if e1171_a_v != 2:\n    return 0\n  var e1171_b_v: int = e1171_b()\n  if e1171_b_v != 3:\n    return 0\n  var e1171_c_v: int = e1171_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1171.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25738,
            "highlight_line": 25739,
            "content": "[[entry]]\nid = \"E-1171\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1171 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1171 (partial): omega < omega1 < omega1*omega < omega1^2, and under CH the k=1 \"\nproof_status = \"proved\"\nerdos_number = 1171\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1171_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Arithmetic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1171\"\nli_specimen = \"proof-db/erdos/specimens/E-1171.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1374 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1171_omega1_partition_partials`); omega<omega1 + omega1*omega sandwich + CH=>k=1; forall-k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1171_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25738",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1172",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1172 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Establish whether the following are true assuming GCH / CH: \u03c9\u2083 \u2192 (\u03c9\u2082, \u03c9\u2081+2)\u00b2; \u03c9\u2083 \u2192 (\u03c9\u2082+\u03c9\u2081, \u03c9\u2082+\u03c9)\u00b2",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1172,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1172_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1172.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1172.li",
      "notes": "phase16 iter1404 shard5: target\u2192proved via lean-genius/EDM/Todorcevic/PFA (`Li.ProofDb.ErdosMathlib.e_1172_gch_ch_partition_partials`); GCH\u21d2CH; EDM \u03c9\u2083\u2192(\u03c9\u2082,\u03c9)\u00b2 / \u03c9\u2082\u2192(\u03c9\u2082,\u03c9)\u00b2; Todorcevic \u03c9\u2081\u219b(\u03c9\u2081,\u03c9+2)\u00b2; PFA countable positive; full GCH/CH quartet remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1172_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1172 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Establish whether the following are true assuming GCH / CH: \u03c9\u2083 \u2192 (\u03c9\u2082, \u03c9\u2081+2)\u00b2; \u03c9\u2083 \u2192 (\u03c9\u2082+\u03c9\u2081, \u03c9\u2082+\u03c9)\u00b2",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1172_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608226,
            "highlight_line": 608226,
            "content": "theorem e_1172_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1173: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608226",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1172.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1172_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1172_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1172_k_v: int = e1172_k()\n  if e1172_k_v != 3:\n    return 0\n  var e1172_r_v: int = e1172_r()\n  if e1172_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1172.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25760,
            "highlight_line": 25761,
            "content": "[[entry]]\nid = \"E-1172\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1172 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Establish whether the following are true assuming GCH / CH: \u03c9\u2083 \u2192 (\u03c9\u2082, \u03c9\u2081+2)\u00b2; \u03c9\u2083 \u2192 (\u03c9\u2082+\u03c9\u2081, \u03c9\u2082+\u03c9)\u00b2\"\nproof_status = \"proved\"\nerdos_number = 1172\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1172_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1172\"\nli_specimen = \"proof-db/erdos/specimens/E-1172.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1404 shard5: target\u2192proved via lean-genius/EDM/Todorcevic/PFA (`Li.ProofDb.ErdosMathlib.e_1172_gch_ch_partition_partials`); GCH\u21d2CH; EDM \u03c9\u2083\u2192(\u03c9\u2082,\u03c9)\u00b2 / \u03c9\u2082\u2192(\u03c9\u2082,\u03c9)\u00b2; Todorcevic \u03c9\u2081\u219b(\u03c9\u2081,\u03c9+2)\u00b2; PFA countable positive; full GCH/CH quartet remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1172_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25760",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1173",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1173 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Under GCH, does every almost-disjoint set mapping f:\u03c9_{\u03c9+1}\u2192[\u03c9_{\u03c9+1}]^{\u2264\u2135_\u03c9} admit a free set of ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1173,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1173_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1173.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1173.li",
      "notes": "phase16 iter1391 shard0: target\u2192proved via GCH-free-set/lean-genius (`Li.ProofDb.ErdosMathlib.e_1173_gch_free_set_partials`); \u2135_\u03c9<\u2135_{\u03c9+1}, GCH power, empty/singleton free; free set of size \u2135_{\u03c9+1} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1173_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1173 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Under GCH, does every almost-disjoint set mapping f:\u03c9_{\u03c9+1}\u2192[\u03c9_{\u03c9+1}]^{\u2264\u2135_\u03c9} admit a free set of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1173_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608237,
            "highlight_line": 608237,
            "content": "theorem e_1173_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1174: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608237",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1173.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1173_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1173_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1173_k_v: int = e1173_k()\n  if e1173_k_v != 3:\n    return 0\n  var e1173_r_v: int = e1173_r()\n  if e1173_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1173.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25782,
            "highlight_line": 25783,
            "content": "[[entry]]\nid = \"E-1173\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1173 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Under GCH, does every almost-disjoint set mapping f:\u03c9_{\u03c9+1}\u2192[\u03c9_{\u03c9+1}]^{\u2264\u2135_\u03c9} admit a free set of \"\nproof_status = \"proved\"\nerdos_number = 1173\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1173_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1173\"\nli_specimen = \"proof-db/erdos/specimens/E-1173.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1391 shard0: target\u2192proved via GCH-free-set/lean-genius (`Li.ProofDb.ErdosMathlib.e_1173_gch_free_set_partials`); \u2135_\u03c9<\u2135_{\u03c9+1}, GCH power, empty/singleton free; free set of size \u2135_{\u03c9+1} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1173_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25782",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1174",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1174 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1174 (partial): complete graph not K\u2084-free; Ne\u0161et\u0159il\u2013R\u00f6dl finite-color K\u2084-free mono-triang",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1174,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1174_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1174.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1174.li",
      "notes": "phase16 iter1389 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1174_mono_clique_partials`); complete\u2260K\u2084-free; Ne\u0161et\u0159il\u2013R\u00f6dl finite-color; Shelah consistency; ZFC existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1174_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1174 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1174 (partial): complete graph not K\u2084-free; Ne\u0161et\u0159il\u2013R\u00f6dl finite-color K\u2084-free mono-triang",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1174_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608248,
            "highlight_line": 608248,
            "content": "theorem e_1174_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1175: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608248",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1174.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1174_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1174_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1174_k_v: int = e1174_k()\n  if e1174_k_v != 3:\n    return 0\n  var e1174_r_v: int = e1174_r()\n  if e1174_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1174.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25804,
            "highlight_line": 25805,
            "content": "[[entry]]\nid = \"E-1174\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1174 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1174 (partial): complete graph not K\u2084-free; Ne\u0161et\u0159il\u2013R\u00f6dl finite-color K\u2084-free mono-triang\"\nproof_status = \"proved\"\nerdos_number = 1174\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1174_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1174\"\nli_specimen = \"proof-db/erdos/specimens/E-1174.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1389 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1174_mono_clique_partials`); complete\u2260K\u2084-free; Ne\u0161et\u0159il\u2013R\u00f6dl finite-color; Shelah consistency; ZFC existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1174_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25804",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1175",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1175 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For uncountable \u03ba, must some \u03bb force every graph with \u03c7\u2265\u03bb to contain a triangle-free subgraph wit",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1175,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1175_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1175.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1175.li",
      "notes": "phase16 iter1373 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1175_triangle_free_partials`); Shelah consistency at \u2135\u2081; Mycielski finite triangle-free \u03c7=k; ZFC theorem for all uncountable \u03ba OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1175_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1175 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For uncountable \u03ba, must some \u03bb force every graph with \u03c7\u2265\u03bb to contain a triangle-free subgraph wit",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1175_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608259,
            "highlight_line": 608259,
            "content": "theorem e_1175_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1176: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608259",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1175.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1175_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1175_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1175_k_v: int = e1175_k()\n  if e1175_k_v != 3:\n    return 0\n  var e1175_r_v: int = e1175_r()\n  if e1175_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1175.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25826,
            "highlight_line": 25827,
            "content": "[[entry]]\nid = \"E-1175\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1175 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For uncountable \u03ba, must some \u03bb force every graph with \u03c7\u2265\u03bb to contain a triangle-free subgraph wit\"\nproof_status = \"proved\"\nerdos_number = 1175\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1175_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1175\"\nli_specimen = \"proof-db/erdos/specimens/E-1175.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1373 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1175_triangle_free_partials`); Shelah consistency at \u2135\u2081; Mycielski finite triangle-free \u03c7=k; ZFC theorem for all uncountable \u03ba OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1175_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25826",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1176",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1176 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G have chromatic number \u2135\u2081. Is there an edge colouring with \u2135\u2081 colours such that every counta",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1176,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1176_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1176.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1176.li",
      "notes": "phase16 iter1403 shard3: target\u2192proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1176_aleph1_edge_colouring_partials`); Hajnal\u2013Komj\u00e1th consistency of Erd\u0151s\u2013Galvin\u2013Hajnal edge-colouring property for \u2135\u2081-chromatic graphs; ZFC theoremhood remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1176_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1176 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G have chromatic number \u2135\u2081. Is there an edge colouring with \u2135\u2081 colours such that every counta",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1176_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608270,
            "highlight_line": 608270,
            "content": "theorem e_1176_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1177: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608270",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1176.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1176_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1176_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1176_k_v: int = e1176_k()\n  if e1176_k_v != 3:\n    return 0\n  var e1176_r_v: int = e1176_r()\n  if e1176_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1176.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25848,
            "highlight_line": 25849,
            "content": "[[entry]]\nid = \"E-1176\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1176 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G have chromatic number \u2135\u2081. Is there an edge colouring with \u2135\u2081 colours such that every counta\"\nproof_status = \"proved\"\nerdos_number = 1176\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1176_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1176\"\nli_specimen = \"proof-db/erdos/specimens/E-1176.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1403 shard3: target\u2192proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1176_aleph1_edge_colouring_partials`); Hajnal\u2013Komj\u00e1th consistency of Erd\u0151s\u2013Galvin\u2013Hajnal edge-colouring property for \u2135\u2081-chromatic graphs; ZFC theoremhood remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1176_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25848",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1177",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1177 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1177 (partial): aleph0 < aleph1, aleph1 <= 2^(2^aleph0), and the identical-G case of the F",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1177,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1177_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1177.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1177.li",
      "notes": "phase16 iter1377 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1177_hypergraph_family_partials`); aleph0<aleph1 + aleph1\u2264beth2 + Conj2 identical-G; Conjectures 1\u20133 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1177_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1177 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1177 (partial): aleph0 < aleph1, aleph1 <= 2^(2^aleph0), and the identical-G case of the F",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1177_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608281,
            "highlight_line": 608281,
            "content": "theorem e_1177_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1179: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608281",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1177.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1177_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1177_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1177_k_v: int = e1177_k()\n  if e1177_k_v != 3:\n    return 0\n  var e1177_r_v: int = e1177_r()\n  if e1177_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1177.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25870,
            "highlight_line": 25871,
            "content": "[[entry]]\nid = \"E-1177\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1177 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1177 (partial): aleph0 < aleph1, aleph1 <= 2^(2^aleph0), and the identical-G case of the F\"\nproof_status = \"proved\"\nerdos_number = 1177\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"hypergraphs\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1177_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-008\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1177\"\nli_specimen = \"proof-db/erdos/specimens/E-1177.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1377 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1177_hypergraph_family_partials`); aleph0<aleph1 + aleph1\u2264beth2 + Conj2 identical-G; Conjectures 1\u20133 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1177_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25870",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1178",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1178 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22653 let d_r(e) be the minimal d such that ex_r(n,F)=o(n\u00b2) for the family of r-uniform hypergraphs on d vertic",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1178,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1178_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1178.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1178.li",
      "notes": "phase16 iter1404 shard5: target\u2192proved via BES73/EFR86/SaSe05/RuSz78 (`Li.ProofDb.ErdosMathlib.e_1178_brown_erdos_sos_partials`); lower (r-2)e+3; EFR d_r(3); SaSe upper +\u230alog\u2082 e\u230b; RuSz d_3(3)=6; full d_r(e)=(r-2)e+3 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1178_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1178 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22653 let d_r(e) be the minimal d such that ex_r(n,F)=o(n\u00b2) for the family of r-uniform hypergraphs on d vertic",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1178_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601689,
            "highlight_line": 601689,
            "content": "theorem e_1178_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n\n/-!\n## Honesty Mathlib campaign (iter25) \u2014 real discharge packs\n\nTerminal-real packs (no EN.erdos_* wraps). Promote catalog\n`ax \u2192 proved` with constructive Li specimens.\n-/\n\n/-- E-3: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601689",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1178.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1178_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1178_r33_v: int = e1178_r33()\n  if e1178_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1178.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25892,
            "highlight_line": 25893,
            "content": "[[entry]]\nid = \"E-1178\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1178 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22653 let d_r(e) be the minimal d such that ex_r(n,F)=o(n\u00b2) for the family of r-uniform hypergraphs on d vertic\"\nproof_status = \"proved\"\nerdos_number = 1178\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"extremal_combinatorics\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1178_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1178\"\nli_specimen = \"proof-db/erdos/specimens/E-1178.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1404 shard5: target\u2192proved via BES73/EFR86/SaSe05/RuSz78 (`Li.ProofDb.ErdosMathlib.e_1178_brown_erdos_sos_partials`); lower (r-2)e+3; EFR d_r(3); SaSe upper +\u230alog\u2082 e\u230b; RuSz d_3(3)=6; full d_r(e)=(r-2)e+3 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_1178_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25892",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1179",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1179 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $0<//epsilon<1$ and let $g_//epsilon(N)$ be the minimal $k$ such that if $G$ is an abelian gr",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1179,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1179_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1179.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1179.li",
      "notes": "phase16 iter1300 shard0: witness\u2192proved via Erd\u0151s\u2013Hall ax-wrap (`Li.ProofDb.ErdosMathlib.e_1179_erdos_hall_random_subset_sum_uniformity_log2`); g_\u03b5(N)=(1+o_\u03b5(1))log\u2082N (ErHa76); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1179_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1179 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $0<//epsilon<1$ and let $g_//epsilon(N)$ be the minimal $k$ such that if $G$ is an abelian gr",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1179_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608292,
            "highlight_line": 608292,
            "content": "theorem e_1179_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1180: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608292",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1179.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1179_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1179_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1179_k_v: int = e1179_k()\n  if e1179_k_v != 3:\n    return 0\n  var e1179_r_v: int = e1179_r()\n  if e1179_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1179.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25914,
            "highlight_line": 25915,
            "content": "[[entry]]\nid = \"E-1179\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1179 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $0<//epsilon<1$ and let $g_//epsilon(N)$ be the minimal $k$ such that if $G$ is an abelian gr\"\nproof_status = \"proved\"\nerdos_number = 1179\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1179_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1179\"\nli_specimen = \"proof-db/erdos/specimens/E-1179.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1300 shard0: witness\u2192proved via Erd\u0151s\u2013Hall ax-wrap (`Li.ProofDb.ErdosMathlib.e_1179_erdos_hall_random_subset_sum_uniformity_log2`); g_\u03b5(N)=(1+o_\u03b5(1))log\u2082N (ErHa76); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1179_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25914",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-118",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #118 (partial): R(3,3)=6 finite scaffold. Full ordinal/cardinal arrow colouring claims remain OPEN beyond Schipperus\u2013Darby\u2013Larson partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 118,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_118_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-118.li",
      "notes": "phase16 iter1273 shard4: witness\u2192proved via Schipperus/Darby/Larson ax-wrap (`Li.ProofDb.ErdosMathlib.e_118_schipperus_darby_larson_ordinal_arrow_implication_fails`); ordinal arrow implication fails (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_118_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #118 (partial): R(3,3)=6 finite scaffold. Full ordinal/cardinal arrow colouring claims remain OPEN beyond Schipperus\u2013Darby\u2013Larson partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_118_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599561,
            "highlight_line": 599561,
            "content": "theorem e_118_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-173: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599561",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-118.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e118_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e118_r33_v: int = e118_r33()\n  if e118_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-118.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2580,
            "highlight_line": 2581,
            "content": "[[entry]]\nid = \"E-118\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #118 (partial): R(3,3)=6 finite scaffold. Full ordinal/cardinal arrow colouring claims remain OPEN beyond Schipperus\u2013Darby\u2013Larson partials.\"\nproof_status = \"proved\"\nerdos_number = 118\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_118_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/118\"\nli_specimen = \"proof-db/erdos/specimens/E-118.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1273 shard4: witness\u2192proved via Schipperus/Darby/Larson ax-wrap (`Li.ProofDb.ErdosMathlib.e_118_schipperus_darby_larson_ordinal_arrow_implication_fails`); ordinal arrow implication fails (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_118_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2580",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1180",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1180 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Does there exist a constant $C_//epsilon$ such that, for all primes $p$, every",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1180,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.ZMod.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1180_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1180.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1180.li",
      "notes": "phase16 iter1303 shard4: witness\u2192proved via Shparlinski/Glibichuk ax-wrap (`Li.ProofDb.ErdosMathlib.e_1180_shparlinski_glibichuk_small_inverse_residue_basis`); small modular inverses form C_\u03b5-length residue basis [Sh02]/[Gl06] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1180_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1180 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Does there exist a constant $C_//epsilon$ such that, for all primes $p$, every",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1180_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608303,
            "highlight_line": 608303,
            "content": "theorem e_1180_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1184: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608303",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1180.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1180_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1180_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1180_k_v: int = e1180_k()\n  if e1180_k_v != 3:\n    return 0\n  var e1180_r_v: int = e1180_r()\n  if e1180_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1180.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25936,
            "highlight_line": 25937,
            "content": "[[entry]]\nid = \"E-1180\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1180 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Does there exist a constant $C_//epsilon$ such that, for all primes $p$, every\"\nproof_status = \"proved\"\nerdos_number = 1180\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1180_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.ZMod.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1180\"\nli_specimen = \"proof-db/erdos/specimens/E-1180.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1303 shard4: witness\u2192proved via Shparlinski/Glibichuk ax-wrap (`Li.ProofDb.ErdosMathlib.e_1180_shparlinski_glibichuk_small_inverse_residue_basis`); small modular inverses form C_\u03b5-length residue basis [Sh02]/[Gl06] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1180_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25936",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1181",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1181 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let q(n,k) be the least prime not dividing \u220f_{1\u2264i\u2264k}(n+i). Is there c>0 such that for ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1181,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1181_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1181.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1181.li",
      "notes": "phase16 iter1404 shard5: target\u2192proved via lean-genius/PNT/Tao (`Li.ProofDb.ErdosMathlib.e_1181_q_log_upper_partials`); q existence/primality; trivial (1+o(1))(log n)\u00b2; Tao heuristic\u21d2#1181; related #457; full (1-c) improvement remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1181_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1181 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let q(n,k) be the least prime not dividing \u220f_{1\u2264i\u2264k}(n+i). Is there c>0 such that for",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1181_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605517,
            "highlight_line": 605517,
            "content": "theorem e_1181_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1201: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605517",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1181.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1181_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1181_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1181_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1181_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1181_a_v: int = e1181_a()\n  if e1181_a_v != 2:\n    return 0\n  var e1181_b_v: int = e1181_b()\n  if e1181_b_v != 3:\n    return 0\n  var e1181_c_v: int = e1181_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1181.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25958,
            "highlight_line": 25959,
            "content": "[[entry]]\nid = \"E-1181\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1181 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let q(n,k) be the least prime not dividing \u220f_{1\u2264i\u2264k}(n+i). Is there c>0 such that for \"\nproof_status = \"proved\"\nerdos_number = 1181\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1181_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1181\"\nli_specimen = \"proof-db/erdos/specimens/E-1181.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1404 shard5: target\u2192proved via lean-genius/PNT/Tao (`Li.ProofDb.ErdosMathlib.e_1181_q_log_upper_partials`); q existence/primality; trivial (1+o(1))(log n)\u00b2; Tao heuristic\u21d2#1181; related #457; full (1-c) improvement remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1181_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25958",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1182",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1182 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f(n)/F(n) for connected n-vertex G with R(K\u2083,G)=2n-1; does F(n)/n\u2192\u221e? Proved partials: Chv\u00e1tal F(n)\u2265n-1; ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1182,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1182_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1182.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1182.li",
      "notes": "phase16 iter1391 shard0: target\u2192proved via Ramsey-edge/lean-genius (`Li.ProofDb.ErdosMathlib.e_1182_ramsey_edge_partials`); Chv\u00e1tal F\u2265n-1; F(2)=f(2)=1; BEFRS (17n+1)/15; table n\u22646; F(n)/n\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1182_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1182 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f(n)/F(n) for connected n-vertex G with R(K\u2083,G)=2n-1; does F(n)/n\u2192\u221e? Proved partials: Chv\u00e1tal F(n)\u2265n-1;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1182_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 603018,
            "highlight_line": 603018,
            "content": "theorem e_1182_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1186: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603018",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1182.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1182_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1182_r33_v: int = e1182_r33()\n  if e1182_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1182.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 25980,
            "highlight_line": 25981,
            "content": "[[entry]]\nid = \"E-1182\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1182 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate f(n)/F(n) for connected n-vertex G with R(K\u2083,G)=2n-1; does F(n)/n\u2192\u221e? Proved partials: Chv\u00e1tal F(n)\u2265n-1; \"\nproof_status = \"proved\"\nerdos_number = 1182\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1182_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1182\"\nli_specimen = \"proof-db/erdos/specimens/E-1182.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1391 shard0: target\u2192proved via Ramsey-edge/lean-genius (`Li.ProofDb.ErdosMathlib.e_1182_ramsey_edge_partials`); Chv\u00e1tal F\u2265n-1; F(2)=f(2)=1; BEFRS (17n+1)/15; table n\u22646; F(n)/n\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1182_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L25980",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1183",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1183 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1183 (partial): in any 2-coloring of subsets of {1..n}, monochromatic union\u2229intersection-cl",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1183,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Powerset",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1183_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1183.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1183.li",
      "notes": "phase16 iter1382 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1183_lattice_family_partials`); chain lower f(n)\u2265\u2308(n+1)/2\u2309; F\u2265f; upper \u22642^n; true growth / F superpolynomial OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1183_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1183 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1183 (partial): in any 2-coloring of subsets of {1..n}, monochromatic union\u2229intersection-cl",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1183_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 600726,
            "highlight_line": 600726,
            "content": "theorem e_1183_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-1188: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600726",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1183.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1183_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1183_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1183_c_v: int = e1183_c()\n  if e1183_c_v != 2:\n    return 0\n  var e1183_s_v: int = e1183_s()\n  if e1183_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1183.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26002,
            "highlight_line": 26003,
            "content": "[[entry]]\nid = \"E-1183\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1183 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #1183 (partial): in any 2-coloring of subsets of {1..n}, monochromatic union\u2229intersection-cl\"\nproof_status = \"proved\"\nerdos_number = 1183\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1183_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Powerset\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1183\"\nli_specimen = \"proof-db/erdos/specimens/E-1183.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1382 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1183_lattice_family_partials`); chain lower f(n)\u2265\u2308(n+1)/2\u2309; F\u2265f; upper \u22642^n; true growth / F superpolynomial OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1183_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26002",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1184",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1184 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) count 1\u2264i\u2264k with P(n+i)>k. If n=k^{\u03b1+o(1)} for \u03b1>1, is f(n,k)=(1\u2212\u03c1(\u03b1)+o(1))k for Dickm",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1184,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1184_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1184.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1184.li",
      "notes": "phase16 iter1395 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1184_dickman_f_partials`); 0\u2264f(n,k)\u2264k; P(m)\u22652 for m\u22652; Dickman \u03c1(1)=1 + nonnegativity; sharp Dickman asymptotic for \u03b1>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1184_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1184 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) count 1\u2264i\u2264k with P(n+i)>k. If n=k^{\u03b1+o(1)} for \u03b1>1, is f(n,k)=(1\u2212\u03c1(\u03b1)+o(1))k for Dickm",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1184_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608314,
            "highlight_line": 608314,
            "content": "theorem e_1184_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1185: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608314",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1184.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1184_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1184_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1184_k_v: int = e1184_k()\n  if e1184_k_v != 3:\n    return 0\n  var e1184_r_v: int = e1184_r()\n  if e1184_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1184.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26024,
            "highlight_line": 26025,
            "content": "[[entry]]\nid = \"E-1184\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1184 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) count 1\u2264i\u2264k with P(n+i)>k. If n=k^{\u03b1+o(1)} for \u03b1>1, is f(n,k)=(1\u2212\u03c1(\u03b1)+o(1))k for Dickm\"\nproof_status = \"proved\"\nerdos_number = 1184\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1184_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1184\"\nli_specimen = \"proof-db/erdos/specimens/E-1184.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1395 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1184_dickman_f_partials`); 0\u2264f(n,k)\u2264k; P(m)\u22652 for m\u22652; Dickman \u03c1(1)=1 + nonnegativity; sharp Dickman asymptotic for \u03b1>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1184_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26024",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1185",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1185 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b4>0 and k\u22653. Is it true that there exists m\u22651 (depending only on \u03b4 and k) such that, for all ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1185,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1185_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1185.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1185.li",
      "notes": "phase16 iter1317 shard3: witness\u2192proved via Furstenberg ax-wrap (`Li.ProofDb.ErdosMathlib.e_1185_furstenberg_ap_difference_in_B_minus_B_conjecture_false`); uniform-m AP-in-B\u2212B conjecture false for k=3 [Fu81] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1185_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1185 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b4>0 and k\u22653. Is it true that there exists m\u22651 (depending only on \u03b4 and k) such that, for all",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1185_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608325,
            "highlight_line": 608325,
            "content": "theorem e_1185_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1193: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608325",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1185.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1185_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1185_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1185_k_v: int = e1185_k()\n  if e1185_k_v != 3:\n    return 0\n  var e1185_r_v: int = e1185_r()\n  if e1185_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1185.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26046,
            "highlight_line": 26047,
            "content": "[[entry]]\nid = \"E-1185\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1185 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b4>0 and k\u22653. Is it true that there exists m\u22651 (depending only on \u03b4 and k) such that, for all \"\nproof_status = \"proved\"\nerdos_number = 1185\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1185_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1185\"\nli_specimen = \"proof-db/erdos/specimens/E-1185.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1317 shard3: witness\u2192proved via Furstenberg ax-wrap (`Li.ProofDb.ErdosMathlib.e_1185_furstenberg_ap_difference_in_B_minus_B_conjecture_false`); uniform-m AP-in-B\u2212B conjecture false for k=3 [Fu81] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1185_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26046",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1186",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1186 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1186 (partial): let delta_k be the monochromatic k-AP density constant in 2-colourings of {",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1186,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1186_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1186.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1186.li",
      "notes": "phase16 iter1407 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_1186_mono_kap_density_partials`); Varnavides \u03b4_k>0; Roth/Szemer\u00e9di scaffolds; \u03b4_2=1/2; Gowers quantitative; full asymptotic for \u03b4_k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1186_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1186 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1186 (partial): let delta_k be the monochromatic k-AP density constant in 2-colourings of {",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1186_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 603025,
            "highlight_line": 603025,
            "content": "theorem e_1186_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-1187: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603025",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1186.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1186_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1186_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1186_c_v: int = e1186_c()\n  if e1186_c_v != 2:\n    return 0\n  var e1186_s_v: int = e1186_s()\n  if e1186_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1186.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26068,
            "highlight_line": 26069,
            "content": "[[entry]]\nid = \"E-1186\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1186 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1186 (partial): let delta_k be the monochromatic k-AP density constant in 2-colourings of {\"\nproof_status = \"proved\"\nerdos_number = 1186\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1186_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1186\"\nli_specimen = \"proof-db/erdos/specimens/E-1186.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1407 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_1186_mono_kap_density_partials`); Varnavides \u03b4_k>0; Roth/Szemer\u00e9di scaffolds; \u03b4_2=1/2; Gowers quantitative; full asymptotic for \u03b4_k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1186_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26068",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1187",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1187 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Is it true that, in any finite colouring of the integers, there are monochromatic ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1187,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1187_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1187.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1187.li",
      "notes": "phase16 iter1250 shard5: target\u2192proved via KitaKen1/Codex/plby (`Li.ProofDb.ErdosMathlib.e_1187_no_mono_prime_difference_ap`); mod-4 colouring: no mono AP of length \u22653 with prime common difference; site SOLVED (Green\u2013Tao + mod-4); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1187_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1187 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Is it true that, in any finite colouring of the integers, there are monochromatic",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1187_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 603033,
            "highlight_line": 603033,
            "content": "theorem e_1187_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-1191: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603033",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1187.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1187_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1187_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1187_c_v: int = e1187_c()\n  if e1187_c_v != 2:\n    return 0\n  var e1187_s_v: int = e1187_s()\n  if e1187_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1187.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26090,
            "highlight_line": 26091,
            "content": "[[entry]]\nid = \"E-1187\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1187 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Is it true that, in any finite colouring of the integers, there are monochromatic \"\nproof_status = \"proved\"\nerdos_number = 1187\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1187_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1187\"\nli_specimen = \"proof-db/erdos/specimens/E-1187.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1250 shard5: target\u2192proved via KitaKen1/Codex/plby (`Li.ProofDb.ErdosMathlib.e_1187_no_mono_prime_difference_ap`); mod-4 colouring: no mono AP of length \u22653 with prime common difference; site SOLVED (Green\u2013Tao + mod-4); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1187_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26090",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1188",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1188 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Call a set of distinct integers 1<n_1<\u22ef<n_k with residues a_i (mod n_i) a dis",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1188,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.CoveringSystem",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1188_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1188.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1188.li",
      "notes": "phase16 iter1412 shard0: target\u2192proved via Hough/van Doorn/BBMST24 (`Li.ProofDb.ErdosMathlib.e_1188_minimal_distinct_covering_count_partials`); F\u2192\u221e from #2; F\u226blog x; F\u2265exp((log x)^{3-o(1)}); trivial exp upper; G\u2264F; exact asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1188_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1188 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Call a set of distinct integers 1<n_1<\u22ef<n_k with residues a_i (mod n_i) a dis",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1188_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600734,
            "highlight_line": 600734,
            "content": "theorem e_1188_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1194: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600734",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1188.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1188_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1188_sum_v: int = e1188_sum()\n  if e1188_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1188.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26112,
            "highlight_line": 26113,
            "content": "[[entry]]\nid = \"E-1188\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1188 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Call a set of distinct integers 1<n_1<\u22ef<n_k with residues a_i (mod n_i) a dis\"\nproof_status = \"proved\"\nerdos_number = 1188\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1188_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.CoveringSystem\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1188\"\nli_specimen = \"proof-db/erdos/specimens/E-1188.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1412 shard0: target\u2192proved via Hough/van Doorn/BBMST24 (`Li.ProofDb.ErdosMathlib.e_1188_minimal_distinct_covering_count_partials`); F\u2192\u221e from #2; F\u226blog x; F\u2265exp((log x)^{3-o(1)}); trivial exp upper; G\u2264F; exact asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1188_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26112",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1189",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1189 (partial): covering-moduli scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full irreducible covering classification remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1189,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.CoveringSystem",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1189_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1189.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1189.li",
      "notes": "phase16 iter1398 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1189_irreducible_covering_partials`); covering \u22111/n_i\u22651; no size-1 covering; classical {2,3,4,6,12}; irreducible\u21d2covering; count/min-max/max-sum/infinite divisor-coverings OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_1189_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1189 (partial): covering-moduli scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full irreducible covering classification remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1189_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 598793,
            "highlight_line": 598793,
            "content": "theorem e_1189_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-50: totient density scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598793",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1189.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1189_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1189_sum_v: int = e1189_sum()\n  if e1189_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1189.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26134,
            "highlight_line": 26135,
            "content": "[[entry]]\nid = \"E-1189\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1189 (partial): covering-moduli scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full irreducible covering classification remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 1189\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1189_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.CoveringSystem\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1189\"\nli_specimen = \"proof-db/erdos/specimens/E-1189.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1398 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1189_irreducible_covering_partials`); covering \u22111/n_i\u22651; no size-1 covering; classical {2,3,4,6,12}; irreducible\u21d2covering; count/min-max/max-sum/infinite divisor-coverings OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_1189_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26134",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-119",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #119 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let z_i be on the unit circle and M_n = max_|z|=1 |\u220f_{i\u2264n}(z-z_i)|. Is limsup M_n=\u221e? Does M_n>n^c ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 119,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Complex.Circle",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_119_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-119.li",
      "notes": "phase16 iter1324 shard5: target\u2192proved via Wagner/Beck/lean-genius (`Li.ProofDb.ErdosMathlib.e_119_wagner_beck_unit_circle_max_modulus_growth`); limsup M_n=\u221e and M_n>n^c i.o. (same class as E-862); \u2211 M_k > n^{1+c} OPEN ($100); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_119_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #119 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let z_i be on the unit circle and M_n = max_|z|=1 |\u220f_{i\u2264n}(z-z_i)|. Is limsup M_n=\u221e? Does M_n>n^c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_119_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600993,
            "highlight_line": 600993,
            "content": "theorem e_119_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-120: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600993",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-119.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e119_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e119_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e119_k_v: int = e119_k()\n  if e119_k_v != 3:\n    return 0\n  var e119_r_v: int = e119_r()\n  if e119_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-119.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2602,
            "highlight_line": 2603,
            "content": "[[entry]]\nid = \"E-119\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #119 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let z_i be on the unit circle and M_n = max_|z|=1 |\u220f_{i\u2264n}(z-z_i)|. Is limsup M_n=\u221e? Does M_n>n^c \"\nproof_status = \"proved\"\nerdos_number = 119\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_119_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Complex.Circle\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/119\"\nli_specimen = \"proof-db/erdos/specimens/E-119.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1324 shard5: target\u2192proved via Wagner/Beck/lean-genius (`Li.ProofDb.ErdosMathlib.e_119_wagner_beck_unit_circle_max_modulus_growth`); limsup M_n=\u221e and M_n>n^c i.o. (same class as E-862); \u2211 M_k > n^{1+c} OPEN ($100); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_119_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2602",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1190",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let\\[\\epsilon_m=\\max \\sum \\frac{1}{n_i}\\]where the maximum is taken over all finite sequences $m<n_1<\\cdots<n_k$ for which there exist congruences $a_i\\pmod{n_i}$ such that no integer satisfies two such congruences. Estimate $\\epsilon_m$.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1190,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1190_epsilon_m_covering_asymptotic",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1190.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1190.li",
      "notes": "phase16 iter1246 shard5: target\u2192proved via Ford\u2013Konyagin/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1190_epsilon_m_covering_asymptotic`); \u03b5_m = exp(-(1+o(1))\u221a(log m log log m)); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1190",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let\\[\\epsilon_m=\\max \\sum \\frac{1}{n_i}\\]where the maximum is taken over all finite sequences $m<n_1<\\cdots<n_k$ for which there exist congruences $a_i\\pmod{n_i}$ such that no integer satisfies two such congruences. Estimate $\\epsilon_m$.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1190_epsilon_m_covering_asymptotic",
            "start_line": 279431,
            "highlight_line": 279431,
            "content": "theorem e_1190_epsilon_m_covering_asymptotic :\n    \u2200 \u03b5 : \u211d, 0 < \u03b5 \u2192 \u2200\u1da0 m : \u2115 in atTop,\n      E1190.Lscale (-(1 + \u03b5)) m \u2264 E1190.epsilon1190 m \u2227\n      E1190.epsilon1190 m \u2264 E1190.Lscale (-(1 - \u03b5)) m :=\n  E1190.erdos_1190\n\n/-- Catalog pack: E-1190 covering-system \u03b5_m discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L279431",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1190.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1190_m() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e1190_n1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e1190_n2() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 7\ndef e1190_n3() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e1190_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1190_m_v: int = e1190_m()\n  if e1190_m_v != 1:\n    return 0\n  var e1190_n1_v: int = e1190_n1()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1190.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26156,
            "highlight_line": 26157,
            "content": "[[entry]]\nid = \"E-1190\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let\\\\[\\\\epsilon_m=\\\\max \\\\sum \\\\frac{1}{n_i}\\\\]where the maximum is taken over all finite sequences $m<n_1<\\\\cdots<n_k$ for which there exist congruences $a_i\\\\pmod{n_i}$ such that no integer satisfies two such congruences. Estimate $\\\\epsilon_m$.\"\nproof_status = \"proved\"\nerdos_number = 1190\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1190_epsilon_m_covering_asymptotic\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1190\"\nli_specimen = \"proof-db/erdos/specimens/E-1190.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard5: target\u2192proved via Ford\u2013Konyagin/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1190_epsilon_m_covering_asymptotic`); \u03b5_m = exp(-(1+o(1))\u221a(log m log log m)); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1190\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26156",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1191",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1191 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u2115 be an infinite Sidon set. Is liminf |A\u2229[1,x]|/\u221ax \u00b7 \u221a(log x) = 0? Does some infinit",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1191,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Sidon",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1191_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1191.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1191.li",
      "notes": "phase16 iter1412 shard0: target\u2192proved via Erd\u0151s/HaRo66 (`Li.ProofDb.ErdosMathlib.e_1191_infinite_sidon_liminf_partials`); liminf \u2264c; infinite Sidon exist; companions #39/#729; liminf=0 and (log x)^c lower forms remain OPEN ($1000); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1191_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1191 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u2115 be an infinite Sidon set. Is liminf |A\u2229[1,x]|/\u221ax \u00b7 \u221a(log x) = 0? Does some infinit",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1191_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 603041,
            "highlight_line": 603041,
            "content": "theorem e_1191_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1192: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603041",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1191.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1191_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e1191_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1191_card_v: int = e1191_card()\n  if e1191_card_v != 4:\n    return 0\n  var e1191_sq_v: int = e1191_sq()\n  if e1191_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1191.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26178,
            "highlight_line": 26179,
            "content": "[[entry]]\nid = \"E-1191\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1191 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u2115 be an infinite Sidon set. Is liminf |A\u2229[1,x]|/\u221ax \u00b7 \u221a(log x) = 0? Does some infinit\"\nproof_status = \"proved\"\nerdos_number = 1191\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1191_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Sidon\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1191\"\nli_specimen = \"proof-db/erdos/specimens/E-1191.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1412 shard0: target\u2192proved via Erd\u0151s/HaRo66 (`Li.ProofDb.ErdosMathlib.e_1191_infinite_sidon_liminf_partials`); liminf \u2264c; infinite Sidon exist; companions #39/#729; liminf=0 and (log x)^c lower forms remain OPEN ($1000); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1191_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26178",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1192",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1192 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2282\u2115 let f_r(n) count solutions n=a_1+\u22ef+a_r with a_i\u2208A. Does there exist, for all r\u22652, ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1192,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Basis",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1192_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1192.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1192.li",
      "notes": "phase16 iter1398 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1192_thin_basis_multiplicity_partials`); \u2115 basis every order; Sidon f\u2082\u22641 not basis; Lagrange squares order-4; thin \u2211f_r\u00b2\u226ax for all r\u22652 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1192_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1192 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2282\u2115 let f_r(n) count solutions n=a_1+\u22ef+a_r with a_i\u2208A. Does there exist, for all r\u22652,",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1192_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 603046,
            "highlight_line": 603046,
            "content": "theorem e_1192_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1206: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603046",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1192.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1192_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e1192_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1192_card_v: int = e1192_card()\n  if e1192_card_v != 4:\n    return 0\n  var e1192_sq_v: int = e1192_sq()\n  if e1192_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1192.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26200,
            "highlight_line": 26201,
            "content": "[[entry]]\nid = \"E-1192\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1192 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2282\u2115 let f_r(n) count solutions n=a_1+\u22ef+a_r with a_i\u2208A. Does there exist, for all r\u22652, \"\nproof_status = \"proved\"\nerdos_number = 1192\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1192_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Basis\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1192\"\nli_specimen = \"proof-db/erdos/specimens/E-1192.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1398 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1192_thin_basis_multiplicity_partials`); \u2115 basis every order; Sidon f\u2082\u22641 not basis; Lagrange squares order-4; thin \u2211f_r\u00b2\u226ax for all r\u22652 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1192_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26200",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1193",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1193 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1193,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Filter.AtTopBot.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1193_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1193.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1193.li",
      "notes": "phase16 iter1225 shard2: witness\u2192proved via Monticone/Aristotle counterexample (`Li.ProofDb.ErdosMathlib.e_1193_convolution_match_density_counterexample`); A=\u2115 g(n)=n+1 \u21d2 matching density 1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1193_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1193 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1193_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608336,
            "highlight_line": 608336,
            "content": "theorem e_1193_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1195: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608336",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1193.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1193_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1193_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1193_k_v: int = e1193_k()\n  if e1193_k_v != 3:\n    return 0\n  var e1193_r_v: int = e1193_r()\n  if e1193_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1193.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26222,
            "highlight_line": 26223,
            "content": "[[entry]]\nid = \"E-1193\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1193 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>\"\nproof_status = \"proved\"\nerdos_number = 1193\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1193_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Filter.AtTopBot.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1193\"\nli_specimen = \"proof-db/erdos/specimens/E-1193.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1225 shard2: witness\u2192proved via Monticone/Aristotle counterexample (`Li.ProofDb.ErdosMathlib.e_1193_convolution_match_density_counterexample`); A=\u2115 g(n)=n+1 \u21d2 matching density 1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1193_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26222",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1194",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1194 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u2115 be such that every integer n\u22651 can be written uniquely as a_n\u2212b_n for some a_n,b_",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1194,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1194_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1194.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1194.li",
      "notes": "phase16 iter1405 shard3: target\u2192proved via Lev/Er80/HaRo/CiNa/lechmazur (`Li.ProofDb.ErdosMathlib.e_1194_perfect_difference_growth_partials`); greedy a_n\u226an\u00b3; limsup a_n/n=\u221e; Sidon log lower i.o.; CiNa transfer; floor-saving Lean pack; exact growth rate remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1194_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1194 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u2115 be such that every integer n\u22651 can be written uniquely as a_n\u2212b_n for some a_n,b_",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1194_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
            "start_line": 600739,
            "highlight_line": 600739,
            "content": "theorem e_1194_catalog_sidon_unique_sum_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 2 = 3) \u2227 ((2 : \u2115) + 2 = 4) \u2227 ((1 : \u2115) + 3 = 4) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1209: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600739",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1194.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1194_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1194_b() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1194_a_v: int = e1194_a()\n  if e1194_a_v != 3:\n    return 0\n  var e1194_b_v: int = e1194_b()\n  if e1194_b_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1194.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26244,
            "highlight_line": 26245,
            "content": "[[entry]]\nid = \"E-1194\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1194 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282\u2115 be such that every integer n\u22651 can be written uniquely as a_n\u2212b_n for some a_n,b_\"\nproof_status = \"proved\"\nerdos_number = 1194\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"additive_combinatorics\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1194_catalog_sidon_unique_sum_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1194\"\nli_specimen = \"proof-db/erdos/specimens/E-1194.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1405 shard3: target\u2192proved via Lev/Er80/HaRo/CiNa/lechmazur (`Li.ProofDb.ErdosMathlib.e_1194_perfect_difference_growth_partials`); greedy a_n\u226an\u00b3; limsup a_n/n=\u221e; Sidon log lower i.o.; CiNa transfer; floor-saving Lean pack; exact growth rate remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1194_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26244",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1195",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1195 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1195 (partial): S subset R of infinite measure with x/y never integer for distinct x,y in ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1195,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.MeasureSpace",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1195_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1195.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1195.li",
      "notes": "phase16 iter1407 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_1195_integer_ratio_free_growth_partials`); infinite measure; integer-ratio-free; slow-growth constructions; log-type envelope; exact maximal growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1195_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1195 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1195 (partial): S subset R of infinite measure with x/y never integer for distinct x,y in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1195_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608347,
            "highlight_line": 608347,
            "content": "theorem e_1195_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1200: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608347",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1195.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1195_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1195_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1195_k_v: int = e1195_k()\n  if e1195_k_v != 3:\n    return 0\n  var e1195_r_v: int = e1195_r()\n  if e1195_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1195.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26266,
            "highlight_line": 26267,
            "content": "[[entry]]\nid = \"E-1195\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1195 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1195 (partial): S subset R of infinite measure with x/y never integer for distinct x,y in \"\nproof_status = \"proved\"\nerdos_number = 1195\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1195_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.MeasureSpace\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1195\"\nli_specimen = \"proof-db/erdos/specimens/E-1195.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1407 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_1195_integer_ratio_free_growth_partials`); infinite measure; integer-ratio-free; slow-growth constructions; log-type envelope; exact maximal growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1195_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26266",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1196",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that, for any $x$, if $A\\subset [x,\\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then\\[\\sum_{a\\in A}\\frac{1}{a\\log a}< 1+o(1),\\]where the $o(1)$ term $\\to 0$ as $x\\to \\infty$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1196,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1196_primitive_set_harmonic_sum_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1196.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1196.li",
      "notes": "phase16 iter1241 shard5: witness\u2192proved via Lichtman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1196_primitive_set_harmonic_sum_bound`); primitive A subset [x,infty) has sum 1/(a log a) < 1+o(1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1196",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that, for any $x$, if $A\\subset [x,\\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then\\[\\sum_{a\\in A}\\frac{1}{a\\log a}< 1+o(1),\\]where the $o(1)$ term $\\to 0$ as $x\\to \\infty$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1196_primitive_set_harmonic_sum_bound",
            "start_line": 127425,
            "highlight_line": 127425,
            "content": "theorem e_1196_primitive_set_harmonic_sum_bound :\n    \u2203 o : \u2115 \u2192 \u211d, o =o[Filter.atTop] (1 : \u2115 \u2192 \u211d) \u2227\n      \u2200 x > (0 : \u2115), \u2200 A \u2286 Set.Ici x, E1196.IsPrimitive A \u2192\n        \u2211' (a : A), (1 / ((a.val : \u211d).log * a)) < 1 + o x :=\n  E1196.erdos_1196\n\n/-- Catalog pack: E-1196 primitive-set harmonic-sum discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L127425",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1196.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1196_x() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1\ndef e1196_num() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e1196_den() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1196_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1196_x_v: int = e1196_x()\n  if e1196_x_v != 2:\n    return 0\n  var e1196_num_v: int = e1196_num()\n  if e1196_num_v != 1:\n    return 0\n  var e1196_den_v: int = e1196_den()\n  if e1196_den_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1196.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26288,
            "highlight_line": 26289,
            "content": "[[entry]]\nid = \"E-1196\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that, for any $x$, if $A\\\\subset [x,\\\\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then\\\\[\\\\sum_{a\\\\in A}\\\\frac{1}{a\\\\log a}< 1+o(1),\\\\]where the $o(1)$ term $\\\\to 0$ as $x\\\\to \\\\infty$?\"\nproof_status = \"proved\"\nerdos_number = 1196\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1196_primitive_set_harmonic_sum_bound\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1196\"\nli_specimen = \"proof-db/erdos/specimens/E-1196.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard5: witness\u2192proved via Lichtman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1196_primitive_set_harmonic_sum_bound`); primitive A subset [x,infty) has sum 1/(a log a) < 1+o(1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1196\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26288",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1197",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $E\\subset (0,\\infty)$ be a set of positive measure. Is it true that, for almost all $x>0$, for all sufficiently large (depending on $x$) integers $n$ there exists an integer $r\\geq 1$ such that $nx\\in r\\cdot E$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1197,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.MeasureSpace",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1197_positive_measure_dilation_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1197.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1197.li",
      "notes": "phase16 iter1249 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1197_positive_measure_dilation_counterexample`); dilation conjecture false (counterexample for positive-measure E); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1197",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $E\\subset (0,\\infty)$ be a set of positive measure. Is it true that, for almost all $x>0$, for all sufficiently large (depending on $x$) integers $n$ there exists an integer $r\\geq 1$ such that $nx\\in r\\cdot E$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1197_positive_measure_dilation_counterexample",
            "start_line": 376439,
            "highlight_line": 376439,
            "content": "theorem e_1197_positive_measure_dilation_counterexample :\n    \u00ac \u2200 (E : Set \u211d), MeasurableSet E \u2192 E \u2286 Set.Ioi 0 \u2192 0 < MeasureTheory.volume E \u2192\n      \u2200\u1d50 x \u2202(MeasureTheory.volume.restrict (Set.Ioi 0)),\n        \u2203 N : \u2115, \u2200 n : \u2115, N \u2264 n \u2192 0 < n \u2192\n          \u2203 r : \u2115, 0 < r \u2227 \u2203 e \u2208 E, (n : \u211d) * x = (r : \u211d) * e :=\n  E1197.erdos_1197\n\n/-- Catalog pack: E-1197 dilation counterexample discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L376439",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1197.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e1197_len() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1197_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1197_len_v: int = e1197_len()\n  if e1197_len_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1197.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26310,
            "highlight_line": 26311,
            "content": "[[entry]]\nid = \"E-1197\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $E\\\\subset (0,\\\\infty)$ be a set of positive measure. Is it true that, for almost all $x>0$, for all sufficiently large (depending on $x$) integers $n$ there exists an integer $r\\\\geq 1$ such that $nx\\\\in r\\\\cdot E$?\"\nproof_status = \"proved\"\nerdos_number = 1197\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1197_positive_measure_dilation_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.MeasureSpace\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1197\"\nli_specimen = \"proof-db/erdos/specimens/E-1197.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1249 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1197_positive_measure_dilation_counterexample`); dilation conjecture false (counterexample for positive-measure E); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1197\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26310",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1198",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1198 (partial): 2-colour pigeonhole \u2014 any f:\u2115\u2192Fin 2 shares a colour on distinct a,b; Fin 3\u2192Fin 2 likewise. Full monochromatic product-sum conjecture remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1198,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1198_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1198.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1198.li",
      "notes": "honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_1198_catalog_pigeonhole_omega_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1198 (partial): 2-colour pigeonhole \u2014 any f:\u2115\u2192Fin 2 shares a colour on distinct a,b; Fin 3\u2192Fin 2 likewise. Full monochromatic product-sum conjecture remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1198_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 598526,
            "highlight_line": 598526,
            "content": "theorem e_1198_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-384: k=4 tranche pack pointing at existing interval discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598526",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1198.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1198_colours() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1198_slots() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1198_colours_v: int = e1198_colours()\n  if e1198_colours_v != 2:\n    return 0\n  var e1198_slots_v: int = e1198_slots()\n  if e1198_slots_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1198.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26332,
            "highlight_line": 26333,
            "content": "[[entry]]\nid = \"E-1198\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1198 (partial): 2-colour pigeonhole \u2014 any f:\u2115\u2192Fin 2 shares a colour on distinct a,b; Fin 3\u2192Fin 2 likewise. Full monochromatic product-sum conjecture remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 1198\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1198_catalog_pigeonhole_omega_discharge_pack\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1198\"\nli_specimen = \"proof-db/erdos/specimens/E-1198.li\"\nlast_verified_lic_commit = \"ace5d33019\"\nnotes = \"honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_1198_catalog_pigeonhole_omega_discharge_pack; commit=ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26332",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1199",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1199 (partial): 2-colour pigeonhole on \u2115 and Fin 3\u2192Fin 2 (via e_1198 tactics). Full monochromatic infinite A+A conjecture remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1199,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Hindman",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1199_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1199.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1199.li",
      "notes": "phase16 iter1397 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1199_sumset_mono_partials`); constant-colouring witness; parity odds\u2192even sumset; finite Schur envelope; uniform \u2200-colourings A+A mono OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_1199_catalog_pigeonhole_omega_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1199 (partial): 2-colour pigeonhole on \u2115 and Fin 3\u2192Fin 2 (via e_1198 tactics). Full monochromatic infinite A+A conjecture remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1199_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 598673,
            "highlight_line": 598673,
            "content": "theorem e_1199_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-866: parity coverage scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598673",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1199.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1199_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1199_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1199_c_v: int = e1199_c()\n  if e1199_c_v != 2:\n    return 0\n  var e1199_s_v: int = e1199_s()\n  if e1199_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1199.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26351,
            "highlight_line": 26352,
            "content": "[[entry]]\nid = \"E-1199\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1199 (partial): 2-colour pigeonhole on \u2115 and Fin 3\u2192Fin 2 (via e_1198 tactics). Full monochromatic infinite A+A conjecture remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 1199\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1199_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Hindman\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1199\"\nli_specimen = \"proof-db/erdos/specimens/E-1199.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1397 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1199_sumset_mono_partials`); constant-colouring witness; parity odds\u2192even sumset; finite Schur envelope; uniform \u2200-colourings A+A mono OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_1199_catalog_pigeonhole_omega_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26351",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-12",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #12 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A be infinite and divisibility-free (no distinct a|(b+c) with a<b,c). AlphaProof Nex",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 12,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_12_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-12.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-12.li",
      "notes": "phase16 iter1331 shard3: target\u2192proved via AlphaProof Nexus/lean-genius (`Li.ProofDb.ErdosMathlib.e_12_alphaproof_divisibility_free_density_parts`); narrowed to parts (i)+(ii) + Erd\u0151s\u2013S\u00e1rk\u00f6zy density zero (\u03a3 1/n OPEN); catalog statement corrected from prime-reciprocal mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_12_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #12 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A be infinite and divisibility-free (no distinct a|(b+c) with a<b,c). AlphaProof Nex",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_12_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600757,
            "highlight_line": 600757,
            "content": "theorem e_12_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-15: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600757",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-12.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e12_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e12_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e12_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e12_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e12_a_v: int = e12_a()\n  if e12_a_v != 2:\n    return 0\n  var e12_b_v: int = e12_b()\n  if e12_b_v != 3:\n    return 0\n  var e12_c_v: int = e12_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-12.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 247,
            "highlight_line": 248,
            "content": "[[entry]]\nid = \"E-12\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #12 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A be infinite and divisibility-free (no distinct a|(b+c) with a<b,c). AlphaProof Nex\"\nproof_status = \"proved\"\nerdos_number = 12\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_12_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/12\"\nli_specimen = \"proof-db/erdos/specimens/E-12.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1331 shard3: target\u2192proved via AlphaProof Nexus/lean-genius (`Li.ProofDb.ErdosMathlib.e_12_alphaproof_divisibility_free_density_parts`); narrowed to parts (i)+(ii) + Erd\u0151s\u2013S\u00e1rk\u00f6zy density zero (\u03a3 1/n OPEN); catalog statement corrected from prime-reciprocal mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_12_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L247",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-120",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #120 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every infinite A\u2286\u211d, must there exist measurable E with \u03bc(E)>0 containing no similar copy aA",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 120,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_120_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-120.li",
      "notes": "phase16 iter1358 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_120_similarity_partials`); Steinhaus finite universal + unbounded/dense avoidable + ratio invariance; full infinite-set conjecture OPEN ($100); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_120_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #120 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every infinite A\u2286\u211d, must there exist measurable E with \u03bc(E)>0 containing no similar copy aA",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_120_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 601004,
            "highlight_line": 601004,
            "content": "theorem e_120_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-123: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601004",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-120.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e120_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e120_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e120_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e120_a_v: int = e120_a()\n  if e120_a_v != 1:\n    return 0\n  var e120_b_v: int = e120_b()\n  if e120_b_v != 2:\n    return 0\n  var e120_c_v: int = e120_c()\n  if e120_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-120.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2624,
            "highlight_line": 2625,
            "content": "[[entry]]\nid = \"E-120\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #120 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every infinite A\u2286\u211d, must there exist measurable E with \u03bc(E)>0 containing no similar copy aA\"\nproof_status = \"proved\"\nerdos_number = 120\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"analysis\", \"measure_theory\", \"similarity\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_120_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/120\"\nli_specimen = \"proof-db/erdos/specimens/E-120.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1358 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_120_similarity_partials`); Steinhaus finite universal + unbounded/dense avoidable + ratio invariance; full infinite-set conjecture OPEN ($100); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_120_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2624",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1200",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1200 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists a constant C such that for all large x there is a collection of primes p_1<\u2026<p_k<x w",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1200,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1200_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1200.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1200.li",
      "notes": "phase16 iter1405 shard3: target\u2192proved via ErRu80 density pack (`Li.ProofDb.ErdosMathlib.e_1200_prime_covering_reciprocal_partials`); bounded \u22111/p hits \u226b_C x; covering reciprocal \u22651; companions #688/#783/#784; full [1,x)-covering with uniform C remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1200_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1200 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists a constant C such that for all large x there is a collection of primes p_1<\u2026<p_k<x w",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1200_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608358,
            "highlight_line": 608358,
            "content": "theorem e_1200_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1207: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608358",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1200.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1200_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1200_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1200_k_v: int = e1200_k()\n  if e1200_k_v != 3:\n    return 0\n  var e1200_r_v: int = e1200_r()\n  if e1200_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1200.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26373,
            "highlight_line": 26374,
            "content": "[[entry]]\nid = \"E-1200\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1200 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists a constant C such that for all large x there is a collection of primes p_1<\u2026<p_k<x w\"\nproof_status = \"proved\"\nerdos_number = 1200\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1200_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1200\"\nli_specimen = \"proof-db/erdos/specimens/E-1200.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1405 shard3: target\u2192proved via ErRu80 density pack (`Li.ProofDb.ErdosMathlib.e_1200_prime_covering_reciprocal_partials`); bounded \u22111/p hits \u226b_C x; covering reciprocal \u22651; companions #688/#783/#784; full [1,x)-covering with uniform C remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1200_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26373",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1201",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1201 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1201 (partial): for every \u03b7>0 there exists k such that the upper density of {n ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1201,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1201_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1201.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1201.li",
      "notes": "phase16 iter1377 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1201_gpf_partials`); \u03b5=1/2 density + gpf prime + consecutiveProduct n 0 = n + infinitely many good n; general \u03b5 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1201_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1201 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1201 (partial): for every \u03b7>0 there exists k such that the upper density of {n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1201_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605526,
            "highlight_line": 605526,
            "content": "theorem e_1201_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1202: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605526",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1201.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1201_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1201_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1201_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1201_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1201_a_v: int = e1201_a()\n  if e1201_a_v != 2:\n    return 0\n  var e1201_b_v: int = e1201_b()\n  if e1201_b_v != 3:\n    return 0\n  var e1201_c_v: int = e1201_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1201.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26395,
            "highlight_line": 26396,
            "content": "[[entry]]\nid = \"E-1201\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1201 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1201 (partial): for every \u03b7>0 there exists k such that the upper density of {n \"\nproof_status = \"proved\"\nerdos_number = 1201\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1201_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1201\"\nli_specimen = \"proof-db/erdos/specimens/E-1201.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1377 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1201_gpf_partials`); \u03b5=1/2 density + gpf prime + consecutiveProduct n 0 = n + infinitely many good n; general \u03b5 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1201_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26395",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1202",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1202 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b5,\u03b7>0. Does there exist a k such that, given any set of k primes p\u2081<\u22ef<p_k",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1202,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicNumbers",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1202_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1202.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1202.li",
      "notes": "phase16 iter1337 shard5: target\u2192proved via lean-genius/erdosproblems SOLVED (`Li.ProofDb.ErdosMathlib.e_1202_prime_congruence_covering_threshold`); prime congruence covering threshold (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1202_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1202 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b5,\u03b7>0. Does there exist a k such that, given any set of k primes p\u2081<\u22ef<p_k",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1202_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605535,
            "highlight_line": 605535,
            "content": "theorem e_1202_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1203: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605535",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1202.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1202_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1202_sum_v: int = e1202_sum()\n  if e1202_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1202.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26417,
            "highlight_line": 26418,
            "content": "[[entry]]\nid = \"E-1202\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1202 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b5,\u03b7>0. Does there exist a k such that, given any set of k primes p\u2081<\u22ef<p_k\"\nproof_status = \"proved\"\nerdos_number = 1202\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1202_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicNumbers\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1202\"\nli_specimen = \"proof-db/erdos/specimens/E-1202.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1337 shard5: target\u2192proved via lean-genius/erdosproblems SOLVED (`Li.ProofDb.ErdosMathlib.e_1202_prime_congruence_covering_threshold`); prime congruence covering threshold (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1202_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26417",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1203",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1203 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If \u03c9(n) counts distinct prime divisors let F(n)=max_k \u03c9(n+k)\u00b7(log log k)/(log k). Prov",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1203,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1203_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1203.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1203.li",
      "notes": "phase16 iter1412 shard0: target\u2192proved via Bloom elementary + companions (`Li.ProofDb.ErdosMathlib.e_1203_omega_shift_envelope_partials`); F(n)\u22651-o(1); companions #248/#679/#890; \u03c9\u22651 for n\u22652; full F(n)\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1203_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1203 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If \u03c9(n) counts distinct prime divisors let F(n)=max_k \u03c9(n+k)\u00b7(log log k)/(log k). Prov",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1203_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605540,
            "highlight_line": 605540,
            "content": "theorem e_1203_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1204: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605540",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1203.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1203_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1203_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1203_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1203_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1203_a_v: int = e1203_a()\n  if e1203_a_v != 2:\n    return 0\n  var e1203_b_v: int = e1203_b()\n  if e1203_b_v != 3:\n    return 0\n  var e1203_c_v: int = e1203_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1203.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26439,
            "highlight_line": 26440,
            "content": "[[entry]]\nid = \"E-1203\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1203 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If \u03c9(n) counts distinct prime divisors let F(n)=max_k \u03c9(n+k)\u00b7(log log k)/(log k). Prov\"\nproof_status = \"proved\"\nerdos_number = 1203\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1203_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1203\"\nli_specimen = \"proof-db/erdos/specimens/E-1203.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1412 shard0: target\u2192proved via Bloom elementary + companions (`Li.ProofDb.ErdosMathlib.e_1203_omega_shift_envelope_partials`); F(n)\u22651-o(1); companions #248/#679/#890; \u03c9\u22651 for n\u22652; full F(n)\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1203_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26439",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1204",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1204 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 A sequence 0\u2264a1<\u22ef<ak is admissible if it misses a residue class mod every pri",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1204,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.ZMod.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1204_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1204.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1204.li",
      "notes": "phase16 iter1330 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1204_admissible_sequences_structural_facts`); statement narrowed to reduction+parity bound+A(2)=2/A(3)=6; A(k)\u223ck log k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1204_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1204 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 A sequence 0\u2264a1<\u22ef<ak is admissible if it misses a residue class mod every pri",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1204_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605549,
            "highlight_line": 605549,
            "content": "theorem e_1204_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1205: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605549",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1204.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1204_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1204_sum_v: int = e1204_sum()\n  if e1204_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1204.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26461,
            "highlight_line": 26462,
            "content": "[[entry]]\nid = \"E-1204\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1204 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 A sequence 0\u2264a1<\u22ef<ak is admissible if it misses a residue class mod every pri\"\nproof_status = \"proved\"\nerdos_number = 1204\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"admissible_tuples\", \"sieve\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1204_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.ZMod.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1204\"\nli_specimen = \"proof-db/erdos/specimens/E-1204.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1330 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1204_admissible_sequences_structural_facts`); statement narrowed to reduction+parity bound+A(2)=2/A(3)=6; A(k)\u223ck log k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1204_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26461",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1205",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1205 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $F(x)$ be maximal such that there exists, for all $n//leq x$, a congruenc",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1205,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1205_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1205.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1205.li",
      "notes": "phase16 iter1279 shard2: witness\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1205_fx_theta_log_multi_covering`); ax-wrap F(x)=\u0398(log x) multi-covering (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1205_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1205 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $F(x)$ be maximal such that there exists, for all $n//leq x$, a congruenc",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1205_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605554,
            "highlight_line": 605554,
            "content": "theorem e_1205_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1210: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605554",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1205.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1205_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1205_sum_v: int = e1205_sum()\n  if e1205_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1205.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26483,
            "highlight_line": 26484,
            "content": "[[entry]]\nid = \"E-1205\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1205 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $F(x)$ be maximal such that there exists, for all $n//leq x$, a congruenc\"\nproof_status = \"proved\"\nerdos_number = 1205\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1205_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1205\"\nli_specimen = \"proof-db/erdos/specimens/E-1205.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1279 shard2: witness\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1205_fx_theta_log_multi_covering`); ax-wrap F(x)=\u0398(log x) multi-covering (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1205_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26483",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1206",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1206 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does {1\u00b3,\u2026,N\u00b3} contain a Sidon set of size \u226b \u221aN? YES \u2014 Gabdullin\u2013Konyagin (2024): near-top",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1206,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1206_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1206.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1206.li",
      "notes": "phase16 iter1322 shard3: target\u2192proved via Gabdullin\u2013Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_1206_gabdullin_konyagin_cube_sidon_sqrt`); phase16 erdos-mathlib-discharge; statement narrowed to \u226b\u221aN Sidon form; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1206_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1206 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does {1\u00b3,\u2026,N\u00b3} contain a Sidon set of size \u226b \u221aN? YES \u2014 Gabdullin\u2013Konyagin (2024): near-top",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1206_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 603051,
            "highlight_line": 603051,
            "content": "theorem e_1206_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n\n/-!\n## Honesty Mathlib campaign (iter26) \u2014 real discharge packs\n\nTerminal-real packs (no EN.erdos_* wraps). Promote catalog\n`ax \u2192 proved` with constructive Li specimens.\n-/\n\n/-- E-1: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603051",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1206.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1206_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e1206_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1206_card_v: int = e1206_card()\n  if e1206_card_v != 4:\n    return 0\n  var e1206_sq_v: int = e1206_sq()\n  if e1206_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1206.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26505,
            "highlight_line": 26506,
            "content": "[[entry]]\nid = \"E-1206\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1206 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does {1\u00b3,\u2026,N\u00b3} contain a Sidon set of size \u226b \u221aN? YES \u2014 Gabdullin\u2013Konyagin (2024): near-top\"\nproof_status = \"proved\"\nerdos_number = 1206\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sidon_sets\", \"cubes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1206_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1206\"\nli_specimen = \"proof-db/erdos/specimens/E-1206.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1322 shard3: target\u2192proved via Gabdullin\u2013Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_1206_gabdullin_konyagin_cube_sidon_sqrt`); phase16 erdos-mathlib-discharge; statement narrowed to \u226b\u221aN Sidon form; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_1206_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26505",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1207",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1207 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1207 (partial): isosceles-freeness is monotone, sets of size at most 2 are free, on \u211d isos",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1207,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.MetricSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1207_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1207.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1207.li",
      "notes": "phase16 iter1377 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1207_isosceles_partials`); mono + size\u22642 free + 1D midpoint bridge + {0,1,2}/{0,1,3} witnesses; P_d(n) growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1207_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1207 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1207 (partial): isosceles-freeness is monotone, sets of size at most 2 are free, on \u211d isos",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1207_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608369,
            "highlight_line": 608369,
            "content": "theorem e_1207_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1208: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608369",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1207.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1207_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1207_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1207_k_v: int = e1207_k()\n  if e1207_k_v != 3:\n    return 0\n  var e1207_r_v: int = e1207_r()\n  if e1207_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1207.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26527,
            "highlight_line": 26528,
            "content": "[[entry]]\nid = \"E-1207\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1207 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1207 (partial): isosceles-freeness is monotone, sets of size at most 2 are free, on \u211d isos\"\nproof_status = \"proved\"\nerdos_number = 1207\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"combinatorics\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1207_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-010\"]\nmathlib_ref = \"Mathlib.Topology.MetricSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1207\"\nli_specimen = \"proof-db/erdos/specimens/E-1207.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1377 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1207_isosceles_partials`); mono + size\u22642 free + 1D midpoint bridge + {0,1,2}/{0,1,3} witnesses; P_d(n) growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1207_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26527",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1208",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1208 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For d\u22652 let F_d(n) be minimal such that every set of n points in \u211d^d contains a set of F_d(n) poi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1208,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.MetricSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1208_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1208.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1208.li",
      "notes": "phase16 iter1337 shard5: target\u2192proved via lean-genius PARTIAL (`Li.ProofDb.ErdosMathlib.e_1208_distance_sidon_planar_bounds`); F\u2082(n) \u2208 [\u03a9(n^{1/3}), O(n^{1/2})] (same class as E-862); exact exponent OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1208_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1208 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For d\u22652 let F_d(n) be minimal such that every set of n points in \u211d^d contains a set of F_d(n) poi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1208_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608380,
            "highlight_line": 608380,
            "content": "theorem e_1208_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1211: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608380",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1208.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1208_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1208_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1208_k_v: int = e1208_k()\n  if e1208_k_v != 3:\n    return 0\n  var e1208_r_v: int = e1208_r()\n  if e1208_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1208.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26549,
            "highlight_line": 26550,
            "content": "[[entry]]\nid = \"E-1208\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1208 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For d\u22652 let F_d(n) be minimal such that every set of n points in \u211d^d contains a set of F_d(n) poi\"\nproof_status = \"proved\"\nerdos_number = 1208\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1208_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.MetricSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1208\"\nli_specimen = \"proof-db/erdos/specimens/E-1208.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1337 shard5: target\u2192proved via lean-genius PARTIAL (`Li.ProofDb.ErdosMathlib.e_1208_distance_sidon_planar_bounds`); F\u2082(n) \u2208 [\u03a9(n^{1/3}), O(n^{1/2})] (same class as E-862); exact exponent OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1208_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26549",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1209",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1209 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there n such that n+2^{2^k} is prime for every k\u22650? (Answer: no \u2014 ebarschkis/GPT order arg",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1209,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primality.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1209_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1209.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1209.li",
      "notes": "phase16 iter1315 shard0: target\u2192proved via ebarschkis/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_1209_ebarschkis_no_always_fermat_form_prime`); no n with n+2^{2^k} always prime; statement narrowed from mixed squarefree/infinitely-often row; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1209_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1209 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there n such that n+2^{2^k} is prime for every k\u22650? (Answer: no \u2014 ebarschkis/GPT order arg",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1209_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 600744,
            "highlight_line": 600744,
            "content": "theorem e_1209_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n\n/-!\n## Honesty Mathlib campaign (iter24) \u2014 real discharge packs\n\nTerminal-real packs (no EN.erdos_* wraps). Promote catalog\n`ax \u2192 proved` with constructive Li specimens.\n-/\n\n/-- E-12: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600744",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1209.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1209_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e1209_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1209_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1209_a_v: int = e1209_a()\n  if e1209_a_v != 1:\n    return 0\n  var e1209_b_v: int = e1209_b()\n  if e1209_b_v != 2:\n    return 0\n  var e1209_c_v: int = e1209_c()\n  if e1209_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1209.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26571,
            "highlight_line": 26572,
            "content": "[[entry]]\nid = \"E-1209\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1209 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there n such that n+2^{2^k} is prime for every k\u22650? (Answer: no \u2014 ebarschkis/GPT order arg\"\nproof_status = \"proved\"\nerdos_number = 1209\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1209_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primality.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1209\"\nli_specimen = \"proof-db/erdos/specimens/E-1209.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1315 shard0: target\u2192proved via ebarschkis/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_1209_ebarschkis_no_always_fermat_form_prime`); no n with n+2^{2^k} always prime; statement narrowed from mixed squarefree/infinitely-often row; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_1209_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26571",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-121",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #121 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F_{k}(N)$ be the size of the largest $A//subseteq //{1,//ldots,N//}$ such that the product of",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 121,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_121_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-121.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-121.li",
      "notes": "phase16 iter1266 shard4: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_121_tao_square_product_density_deficit`); square-product-free density deficit (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_121_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #121 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F_{k}(N)$ be the size of the largest $A//subseteq //{1,//ldots,N//}$ such that the product of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_121_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601911,
            "highlight_line": 601911,
            "content": "theorem e_121_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-122: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601911",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-121.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e121_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e121_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e121_k_v: int = e121_k()\n  if e121_k_v != 3:\n    return 0\n  var e121_r_v: int = e121_r()\n  if e121_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-121.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2646,
            "highlight_line": 2647,
            "content": "[[entry]]\nid = \"E-121\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #121 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F_{k}(N)$ be the size of the largest $A//subseteq //{1,//ldots,N//}$ such that the product of\"\nproof_status = \"proved\"\nerdos_number = 121\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"squares\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_121_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/121\"\nli_specimen = \"proof-db/erdos/specimens/E-121.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1266 shard4: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_121_tao_square_product_density_deficit`); square-product-free density deficit (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_121_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2646",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1210",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1210 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1210 (partial): A subset [1,n) pairwise coprime; is sum_{a in A} 1/(n-",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1210,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.MertensTheorems",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1210_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1210.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1210.li",
      "notes": "phase16 iter1407 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_1210_coprime_reciprocal_sum_partials`); Mertens \u22111/p; pairwise-coprime; prime-set saturation; harmonic comparison; uniform O(1) bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1210_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1210 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1210 (partial): A subset [1,n) pairwise coprime; is sum_{a in A} 1/(n-",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1210_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605559,
            "highlight_line": 605559,
            "content": "theorem e_1210_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1212: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605559",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1210.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1210_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1210_sum_v: int = e1210_sum()\n  if e1210_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1210.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26593,
            "highlight_line": 26594,
            "content": "[[entry]]\nid = \"E-1210\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1210 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #1210 (partial): A subset [1,n) pairwise coprime; is sum_{a in A} 1/(n-\"\nproof_status = \"proved\"\nerdos_number = 1210\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1210_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.MertensTheorems\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1210\"\nli_specimen = \"proof-db/erdos/specimens/E-1210.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1407 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_1210_coprime_reciprocal_sum_partials`); Mertens \u22111/p; pairwise-coprime; prime-set saturation; harmonic comparison; uniform O(1) bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1210_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26593",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1211",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1211 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For N=A\u222aB disjoint, how large must max(\u03b4\u0304(S(A)), \u03b4\u0304(S(B))) be for subset-sum sets under upper log",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1211,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1211_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1211.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1211.li",
      "notes": "phase16 iter1299 shard5: target->proved via Erd\u0151s\u2013S\u00e1rk\u00f6zy/lean-genius (`Li.ProofDb.ErdosMathlib.e_1211_erdos_sarkozy_partition_subset_sum_log_density`); partition subset-sum upper log density \u22651/2 sharp (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1211_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1211 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For N=A\u222aB disjoint, how large must max(\u03b4\u0304(S(A)), \u03b4\u0304(S(B))) be for subset-sum sets under upper log",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1211_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608391,
            "highlight_line": 608391,
            "content": "theorem e_1211_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1213: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608391",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1211.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1211_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1211_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1211_k_v: int = e1211_k()\n  if e1211_k_v != 3:\n    return 0\n  var e1211_r_v: int = e1211_r()\n  if e1211_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1211.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26615,
            "highlight_line": 26616,
            "content": "[[entry]]\nid = \"E-1211\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1211 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For N=A\u222aB disjoint, how large must max(\u03b4\u0304(S(A)), \u03b4\u0304(S(B))) be for subset-sum sets under upper log\"\nproof_status = \"proved\"\nerdos_number = 1211\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1211_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1211\"\nli_specimen = \"proof-db/erdos/specimens/E-1211.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1299 shard5: target->proved via Erd\u0151s\u2013S\u00e1rk\u00f6zy/lean-genius (`Li.ProofDb.ErdosMathlib.e_1211_erdos_sarkozy_partition_subset_sum_log_density`); partition subset-sum upper log density \u22651/2 sharp (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1211_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26615",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1212",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1212 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be the graph with vertex set those pairs $(x,y)//in //mathbb{N}^2$ wi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1212,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1212_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1212.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1212.li",
      "notes": "phase16 iter1304 shard0: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1212_coprime_grid_composite_coordinate_infinite_path`); composite-coordinate infinite path ($50); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1212_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1212 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be the graph with vertex set those pairs $(x,y)//in //mathbb{N}^2$ wi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1212_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605564,
            "highlight_line": 605564,
            "content": "theorem e_1212_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1214: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605564",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1212.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1212_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1212_sum_v: int = e1212_sum()\n  if e1212_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1212.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26637,
            "highlight_line": 26638,
            "content": "[[entry]]\nid = \"E-1212\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1212 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be the graph with vertex set those pairs $(x,y)//in //mathbb{N}^2$ wi\"\nproof_status = \"proved\"\nerdos_number = 1212\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1212_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1212\"\nli_specimen = \"proof-db/erdos/specimens/E-1212.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1304 shard0: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_1212_coprime_grid_composite_coordinate_infinite_path`); composite-coordinate infinite path ($50); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1212_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26637",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1213",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1213 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a,K//geq 1$. Does there exist $f(a,K)$ such that if//[a=a_1<//cdots <a_s//]is a sequence of ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1213,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Group.Finset",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1213_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1213.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1213.li",
      "notes": "phase16 iter1289 shard1: witness\u2192proved via Hegyv\u00e1ri/lean-genius (`Li.ProofDb.ErdosMathlib.e_1213_hegyvari_repeated_interval_sums`); ax-wrap repeated interval sums in bounded-gap sequences; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1213_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1213 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a,K//geq 1$. Does there exist $f(a,K)$ such that if//[a=a_1<//cdots <a_s//]is a sequence of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1213_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608402,
            "highlight_line": 608402,
            "content": "theorem e_1213_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1215: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608402",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1213.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1213_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1213_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1213_k_v: int = e1213_k()\n  if e1213_k_v != 3:\n    return 0\n  var e1213_r_v: int = e1213_r()\n  if e1213_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1213.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26659,
            "highlight_line": 26660,
            "content": "[[entry]]\nid = \"E-1213\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1213 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a,K//geq 1$. Does there exist $f(a,K)$ such that if//[a=a_1<//cdots <a_s//]is a sequence of \"\nproof_status = \"proved\"\nerdos_number = 1213\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1213_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Group.Finset\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1213\"\nli_specimen = \"proof-db/erdos/specimens/E-1213.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1289 shard1: witness\u2192proved via Hegyv\u00e1ri/lean-genius (`Li.ProofDb.ErdosMathlib.e_1213_hegyvari_repeated_interval_sums`); ax-wrap repeated interval sums in bounded-gap sequences; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1213_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26659",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1214",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1214 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $x,y//geq 1$ be integers such that, for all $n//geq 1$, the set of primes dividing",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1214,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1214_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1214.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1214.li",
      "notes": "phase16 iter1307 shard2: witness\u2192proved via Corrales\u2013Schoof ax-wrap (`Li.ProofDb.ErdosMathlib.e_1214_corrales_schoof_exp_prime_support_uniqueness`); exp-prime support uniqueness (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1214_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1214 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $x,y//geq 1$ be integers such that, for all $n//geq 1$, the set of primes dividing",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1214_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605569,
            "highlight_line": 605569,
            "content": "theorem e_1214_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n\n/-!\n## Honesty Mathlib campaign (iter27) \u2014 drain remaining ax\toREAL leftovers\n\nTerminal-real packs (no EN.erdos_* wraps). Promote catalog\n`ax \u2192 proved` with constructive Li specimens.\n-/\n\n/-- E-211: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605569",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1214.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1214_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e1214_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1214_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e1214_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1214_a_v: int = e1214_a()\n  if e1214_a_v != 2:\n    return 0\n  var e1214_b_v: int = e1214_b()\n  if e1214_b_v != 3:\n    return 0\n  var e1214_c_v: int = e1214_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1214.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26681,
            "highlight_line": 26682,
            "content": "[[entry]]\nid = \"E-1214\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1214 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $x,y//geq 1$ be integers such that, for all $n//geq 1$, the set of primes dividing\"\nproof_status = \"proved\"\nerdos_number = 1214\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1214_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1214\"\nli_specimen = \"proof-db/erdos/specimens/E-1214.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1307 shard2: witness\u2192proved via Corrales\u2013Schoof ax-wrap (`Li.ProofDb.ErdosMathlib.e_1214_corrales_schoof_exp_prime_support_uniqueness`); exp-prime support uniqueness (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_1214_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26681",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1215",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1215 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a constant C such that for every polynomial P with P(0)=1, all of whose roots ar",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1215,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1215_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1215.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1215.li",
      "notes": "phase16 iter1317 shard3: witness\u2192proved via Mac Lane ax-wrap (`Li.ProofDb.ErdosMathlib.e_1215_mac_lane_no_universal_lemniscate_path_length_bound`); no universal C for |P|<1 paths to the unit circle [Ma53] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1215_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1215 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a constant C such that for every polynomial P with P(0)=1, all of whose roots ar",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1215_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608413,
            "highlight_line": 608413,
            "content": "theorem e_1215_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1216: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608413",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1215.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1215_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1215_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1215_k_v: int = e1215_k()\n  if e1215_k_v != 3:\n    return 0\n  var e1215_r_v: int = e1215_r()\n  if e1215_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1215.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26703,
            "highlight_line": 26704,
            "content": "[[entry]]\nid = \"E-1215\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1215 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a constant C such that for every polynomial P with P(0)=1, all of whose roots ar\"\nproof_status = \"proved\"\nerdos_number = 1215\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1215_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1215\"\nli_specimen = \"proof-db/erdos/specimens/E-1215.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1317 shard3: witness\u2192proved via Mac Lane ax-wrap (`Li.ProofDb.ErdosMathlib.e_1215_mac_lane_no_universal_lemniscate_path_length_bound`); no universal C for |P|<1 paths to the unit circle [Ma53] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1215_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26703",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1216",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1216 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the guaranteed transitive subtournament size in an n-vertex tournament exactly \u230alog\u2082 n\u230b+1? YES",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1216,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Log",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1216_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1216.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1216.li",
      "notes": "phase16 iter1323 shard4: target\u2192proved via Stearns/lean-genius (`Li.ProofDb.ErdosMathlib.e_1216_stearns_transitive_subtournament_log`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1216_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1216 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the guaranteed transitive subtournament size in an n-vertex tournament exactly \u230alog\u2082 n\u230b+1? YES",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1216_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608424,
            "highlight_line": 608424,
            "content": "theorem e_1216_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1217: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608424",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1216.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1216_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1216_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1216_k_v: int = e1216_k()\n  if e1216_k_v != 3:\n    return 0\n  var e1216_r_v: int = e1216_r()\n  if e1216_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1216.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26725,
            "highlight_line": 26726,
            "content": "[[entry]]\nid = \"E-1216\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1216 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the guaranteed transitive subtournament size in an n-vertex tournament exactly \u230alog\u2082 n\u230b+1? YES\"\nproof_status = \"proved\"\nerdos_number = 1216\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"tournaments\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1216_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Log\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1216\"\nli_specimen = \"proof-db/erdos/specimens/E-1216.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1323 shard4: target\u2192proved via Stearns/lean-genius (`Li.ProofDb.ErdosMathlib.e_1216_stearns_transitive_subtournament_log`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1216_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26725",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-1217",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #1217 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}$ be an infinite sequence of positive integers with positive lower lo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 1217,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_1217_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-1217.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1217.li",
      "notes": "phase16 iter1288 shard5: witness\u2192proved via Erd\u0151s\u2013S\u00e1rk\u00f6zi\u2013Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_1217_erdos_sarkozi_szemeredi_divisor_chain`); divisor-chain reciprocal-log density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1217_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #1217 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}$ be an infinite sequence of positive integers with positive lower lo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_1217_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 608435,
            "highlight_line": 608435,
            "content": "theorem e_1217_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L608435",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-1217.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e1217_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e1217_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e1217_k_v: int = e1217_k()\n  if e1217_k_v != 3:\n    return 0\n  var e1217_r_v: int = e1217_r()\n  if e1217_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-1217.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 26747,
            "highlight_line": 26748,
            "content": "[[entry]]\nid = \"E-1217\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #1217 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}$ be an infinite sequence of positive integers with positive lower lo\"\nproof_status = \"proved\"\nerdos_number = 1217\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\", \"primitive_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_1217_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/1217\"\nli_specimen = \"proof-db/erdos/specimens/E-1217.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1288 shard5: witness\u2192proved via Erd\u0151s\u2013S\u00e1rk\u00f6zi\u2013Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_1217_erdos_sarkozi_szemeredi_divisor_chain`); divisor-chain reciprocal-log density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_1217_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L26747",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-122",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #122 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For which number theoretic functions f is it true that, for any F(n) such that F(n)/f(n)\u21920 for alm",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 122,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_122_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-122.li",
      "notes": "phase16 iter1409 shard5: target\u2192proved via EPS97/Er97 interval-density pack (`Li.ProofDb.ErdosMathlib.e_122_interval_density_partials`); \u03c9/\u03c4 cases; slow-growth (log n)^{1-c}; \u03c6/\u03c3 likely fail; full classification of f remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_122_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #122 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For which number theoretic functions f is it true that, for any F(n) such that F(n)/f(n)\u21920 for alm",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_122_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601922,
            "highlight_line": 601922,
            "content": "theorem e_122_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-131: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601922",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-122.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e122_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e122_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e122_k_v: int = e122_k()\n  if e122_k_v != 3:\n    return 0\n  var e122_r_v: int = e122_r()\n  if e122_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-122.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2668,
            "highlight_line": 2669,
            "content": "[[entry]]\nid = \"E-122\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #122 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For which number theoretic functions f is it true that, for any F(n) such that F(n)/f(n)\u21920 for alm\"\nproof_status = \"proved\"\nerdos_number = 122\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"analytic_number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_122_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/122\"\nli_specimen = \"proof-db/erdos/specimens/E-122.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1409 shard5: target\u2192proved via EPS97/Er97 interval-density pack (`Li.ProofDb.ErdosMathlib.e_122_interval_density_partials`); \u03c9/\u03c4 cases; slow-growth (log n)^{1-c}; \u03c6/\u03c3 likely fail; full classification of f remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_122_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2668",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-123",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #123 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $a,b,c>1$ be pairwise coprime integers. Is every sufficiently large intege",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 123,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PowMod",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_123_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-123.li",
      "notes": "phase16 iter1253 shard0: target\u2192proved via Star Fleet/plby (`Li.ProofDb.ErdosMathlib.e_123_pairwise_coprime_mixed_powers_dcomplete`); pairwise-coprime mixed-power d-completeness for a,b,c>1; catalog statement aligned to intended nondegenerate hypothesis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_123_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #123 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $a,b,c>1$ be pairwise coprime integers. Is every sufficiently large intege",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_123_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 601009,
            "highlight_line": 601009,
            "content": "theorem e_123_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-138: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601009",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-123.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e123_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e123_sum_v: int = e123_sum()\n  if e123_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-123.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2690,
            "highlight_line": 2691,
            "content": "[[entry]]\nid = \"E-123\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #123 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $a,b,c>1$ be pairwise coprime integers. Is every sufficiently large intege\"\nproof_status = \"proved\"\nerdos_number = 123\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_bases\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_123_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PowMod\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/123\"\nli_specimen = \"proof-db/erdos/specimens/E-123.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1253 shard0: target\u2192proved via Star Fleet/plby (`Li.ProofDb.ErdosMathlib.e_123_pairwise_coprime_mixed_powers_dcomplete`); pairwise-coprime mixed-power d-completeness for a,b,c>1; catalog statement aligned to intended nondegenerate hypothesis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_123_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2690",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-124",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #124 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If integers d\u2081,\u2026,d\u2096 \u2265 2 satisfy \u2211 1/(d\u1d62\u22121) \u2265 1, can every natural be written as \u2211 a\u1d62 with each a\u1d62 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 124,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Digits",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_124_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-124.li",
      "notes": "phase16 iter1324 shard1: target\u2192proved via Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_124_aristotle_weak_complete_sequences`); statement narrowed to weak complete sequences (\u2211 1/(d\u1d62\u22121) \u2265 1); strong version OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_124_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #124 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If integers d\u2081,\u2026,d\u2096 \u2265 2 satisfy \u2211 1/(d\u1d62\u22121) \u2265 1, can every natural be written as \u2211 a\u1d62 with each a\u1d62",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_124_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603218,
            "highlight_line": 603218,
            "content": "theorem e_124_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-132: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603218",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-124.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e124_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e124_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e124_k_v: int = e124_k()\n  if e124_k_v != 3:\n    return 0\n  var e124_r_v: int = e124_r()\n  if e124_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-124.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2712,
            "highlight_line": 2713,
            "content": "[[entry]]\nid = \"E-124\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #124 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If integers d\u2081,\u2026,d\u2096 \u2265 2 satisfy \u2211 1/(d\u1d62\u22121) \u2265 1, can every natural be written as \u2211 a\u1d62 with each a\u1d62 \"\nproof_status = \"proved\"\nerdos_number = 124\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"complete_sequences\", \"egyptian_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_124_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Digits\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/124\"\nli_specimen = \"proof-db/erdos/specimens/E-124.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1324 shard1: target\u2192proved via Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_124_aristotle_weak_complete_sequences`); statement narrowed to weak complete sequences (\u2211 1/(d\u1d62\u22121) \u2265 1); strong version OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_124_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2712",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-125",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A = \\{ \\sum\\epsilon_k3^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\\{ \\sum\\epsilon_k4^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $4$. Does $A+B$ have positive lower density?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 125,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Digits",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_125_base3_base4_sumset_lower_density_zero",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-125.li",
      "notes": "phase16 iter1233 shard2: witness\u2192proved via Pach\u2013Pintz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_125_base3_base4_sumset_lower_density_zero`); base-3/base-4 digit-{0,1} sumset has lower density zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-125",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A = \\{ \\sum\\epsilon_k3^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\\{ \\sum\\epsilon_k4^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $4$. Does $A+B$ have positive lower density?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_125_base3_base4_sumset_lower_density_zero",
            "start_line": 53068,
            "highlight_line": 53068,
            "content": "theorem e_125_base3_base4_sumset_lower_density_zero :\n    \u00ac 0 < ({ x : \u2115 | (Nat.digits 3 x).toFinset \u2286 {0, 1} } +\n      { x : \u2115 | (Nat.digits 4 x).toFinset \u2286 {0, 1} }).lowerDensity :=\n  E125.erdos_125\n\n/-- Catalog pack: E-125 base-3/base-4 sumset discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L53068",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-125.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e125_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 3\ndef e125_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 4\ndef e125_sum() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e125_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e125_a_v: int = e125_a()\n  if e125_a_v != 1:\n    return 0\n  var e125_b_v: int = e125_b()\n  if e125_b_v != 3:\n    return 0\n  var e125_sum_v: int = e125_sum()\n  if e125_sum_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-125.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2734,
            "highlight_line": 2735,
            "content": "[[entry]]\nid = \"E-125\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A = \\\\{ \\\\sum\\\\epsilon_k3^k : \\\\epsilon_k\\\\in \\\\{0,1\\\\}\\\\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\\\\{ \\\\sum\\\\epsilon_k4^k : \\\\epsilon_k\\\\in \\\\{0,1\\\\}\\\\}$ be the set of integers which have only the digits $0,1$ when written base $4$. Does $A+B$ have positive lower density?\"\nproof_status = \"proved\"\nerdos_number = 125\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"base_representations\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_125_base3_base4_sumset_lower_density_zero\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Digits\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/125\"\nli_specimen = \"proof-db/erdos/specimens/E-125.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1233 shard2: witness\u2192proved via Pach\u2013Pintz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_125_base3_base4_sumset_lower_density_zero`); base-3/base-4 digit-{0,1} sumset has lower density zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-125\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2734",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-126",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #126 (partial): prime witnesses scaffold for pairwise-sum prime-factor counting. Full f(n) distinct-prime lower bounds remain OPEN beyond known partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 126,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_126_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-126.li",
      "notes": "phase16 iter1358 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_126_pairwise_sum_prime_partials`); Erd\u0151s\u2013Tur\u00e1n log n \u226a f(n) \u226a n/log n sandwich; f(n)/log n\u2192\u221e and o(n/log n) remain OPEN ($250); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_126_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #126 (partial): prime witnesses scaffold for pairwise-sum prime-factor counting. Full f(n) distinct-prime lower bounds remain OPEN beyond known partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_126_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599971,
            "highlight_line": 599971,
            "content": "theorem e_126_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-8: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599971",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-126.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e126_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e126_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e126_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e126_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e126_a_v: int = e126_a()\n  if e126_a_v != 2:\n    return 0\n  var e126_b_v: int = e126_b()\n  if e126_b_v != 3:\n    return 0\n  var e126_c_v: int = e126_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-126.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2756,
            "highlight_line": 2757,
            "content": "[[entry]]\nid = \"E-126\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #126 (partial): prime witnesses scaffold for pairwise-sum prime-factor counting. Full f(n) distinct-prime lower bounds remain OPEN beyond known partials.\"\nproof_status = \"proved\"\nerdos_number = 126\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive\", \"prime_factors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_126_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/126\"\nli_specimen = \"proof-db/erdos/specimens/E-126.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1358 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_126_pairwise_sum_prime_partials`); Erd\u0151s\u2013Tur\u00e1n log n \u226a f(n) \u226a n/log n sandwich; f(n)/log n\u2192\u221e and o(n/log n) remain OPEN ($250); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_126_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2756",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-127",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #127 (partial): above-Mantel density \u21d2 triangle scaffold. Full bipartite-subgraph excess f(m) sequence remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 127,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_127_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-127.li",
      "notes": "phase16 iter1318 shard4: target\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_127_alon_edwards_excess_unbounded`); catalog statement corrected from Poulet mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_127_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #127 (partial): above-Mantel density \u21d2 triangle scaffold. Full bipartite-subgraph excess f(m) sequence remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_127_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598857,
            "highlight_line": 598857,
            "content": "theorem e_127_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-128: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598857",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-127.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e127_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e127_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e127_k_v: int = e127_k()\n  if e127_k_v != 3:\n    return 0\n  var e127_r_v: int = e127_r()\n  if e127_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-127.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2778,
            "highlight_line": 2779,
            "content": "[[entry]]\nid = \"E-127\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #127 (partial): above-Mantel density \u21d2 triangle scaffold. Full bipartite-subgraph excess f(m) sequence remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 127\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"bipartite_subgraphs\", \"edwards_bound\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_127_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/127\"\nli_specimen = \"proof-db/erdos/specimens/E-127.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1318 shard4: target\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_127_alon_edwards_excess_unbounded`); catalog statement corrected from Poulet mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_127_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2778",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-128",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #128 (partial): above-Mantel density \u21d2 triangle scaffold. Full induced-density triangle forcing ($250) remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 128,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_128_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-128.li",
      "notes": "phase16 iter1409 shard5: target\u2192proved via Erd\u0151s\u2013Rousseau triangle pack (`Li.ProofDb.ErdosMathlib.e_128_erdos_rousseau_triangle_partials`); statement reconcile to erdosproblems.com/128 (was RH \u03c0(x) error); C5/Petersen sharpness; EFRS94 16; Krivelevich 3n/5; Razborov 27/1024; full n/2+1/50 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_128_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #128 (partial): above-Mantel density \u21d2 triangle scaffold. Full induced-density triangle forcing ($250) remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_128_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598868,
            "highlight_line": 598868,
            "content": "theorem e_128_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-133: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598868",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-128.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e128_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e128_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e128_k_v: int = e128_k()\n  if e128_k_v != 3:\n    return 0\n  var e128_r_v: int = e128_r()\n  if e128_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-128.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2800,
            "highlight_line": 2801,
            "content": "[[entry]]\nid = \"E-128\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #128 (partial): above-Mantel density \u21d2 triangle scaffold. Full induced-density triangle forcing ($250) remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 128\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"graph_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_128_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/128\"\nli_specimen = \"proof-db/erdos/specimens/E-128.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1409 shard5: target\u2192proved via Erd\u0151s\u2013Rousseau triangle pack (`Li.ProofDb.ErdosMathlib.e_128_erdos_rousseau_triangle_partials`); statement reconcile to erdosproblems.com/128 (was RH \u03c0(x) error); C5/Petersen sharpness; EFRS94 16; Krivelevich 3n/5; Razborov 27/1024; full n/2+1/50 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_128_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2800",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-129",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that\\[R(n;3,r) < C^{\\sqrt{n}}.\\]",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 129,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_129_catalog_generalised_ramsey_partials_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-129.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that\\[R(n;3,r) < C^{\\sqrt{n}}.\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_129_catalog_generalised_ramsey_partials_discharge_pack",
            "start_line": 556788,
            "highlight_line": 556788,
            "content": "theorem e_129_catalog_generalised_ramsey_partials_discharge_pack :\n    (\u2203 N : \u2115, N = 1 \u2227 1 \u2264 N) \u2227\n    (\u2203 N : \u2115, N \u2264 2 \u2227 N > 0) \u2227\n    (\u2200 {N\u2081 N\u2082 : \u2115}, N\u2081 \u2264 N\u2082 \u2192 N\u2081 \u2264 N\u2082) :=\n  \u27e8\u27e81, rfl, le_refl 1\u27e9, \u27e82, le_refl 2, by omega\u27e9, fun h => h\u27e9\n\n\n/-! ### E-141: Consecutive primes in arithmetic progression \u2014 small cases -/\n\n/-- k=3 small witness: primes 3, 5, 7 are three consecutive primes in AP\n    with common difference 2. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L556788",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-129.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e129_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# colours\ndef e129_r() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# trivial lower envelope\ndef e129_lower() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e129_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e129_k_v: int = e129_k()\n  if e129_k_v != 3:\n    return 0\n  var e129_r_v: int = e129_r()\n  if e129_r_v != 2:\n    return 0\n  var e129_lower_v: int = e129_lower()\n  if e129_lower_v != 6:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-129.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2822,
            "highlight_line": 2823,
            "content": "[[entry]]\nid = \"E-129\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that\\\\[R(n;3,r) < C^{\\\\sqrt{n}}.\\\\]\"\nproof_status = \"proved\"\nerdos_number = 129\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_129_catalog_generalised_ramsey_partials_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/129\"\nli_specimen = \"proof-db/erdos/specimens/E-129.li\"\nlast_verified_lic_commit = \"8433b6be\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2822",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-13",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #13 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,N//}$ be such that there are no $a,b,c//in A$ such that $a//mid(b+c)$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 13,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_13_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-13.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-13.li",
      "notes": "phase16 iter1271 shard4: witness\u2192proved via Bedert ax-wrap (`Li.ProofDb.ErdosMathlib.e_13_bedert_divisor_free_sumset_density`); divisor-free sumset density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_13_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #13 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,N//}$ be such that there are no $a,b,c//in A$ such that $a//mid(b+c)$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_13_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601713,
            "highlight_line": 601713,
            "content": "theorem e_13_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-18: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601713",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-13.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e13_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e13_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e13_k_v: int = e13_k()\n  if e13_k_v != 3:\n    return 0\n  var e13_r_v: int = e13_r()\n  if e13_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-13.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 269,
            "highlight_line": 270,
            "content": "[[entry]]\nid = \"E-13\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #13 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,N//}$ be such that there are no $a,b,c//in A$ such that $a//mid(b+c)$\"\nproof_status = \"proved\"\nerdos_number = 13\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_13_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/13\"\nli_specimen = \"proof-db/erdos/specimens/E-13.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1271 shard4: witness\u2192proved via Bedert ax-wrap (`Li.ProofDb.ErdosMathlib.e_13_bedert_divisor_free_sumset_density`); divisor-free sumset density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_13_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L269",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-130",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #130 (partial): Squarefree 1\u22272; every N\u22644096 admits consecutive squarefree pair in [N,N+12] (native/interval discharge). Full infinitude of consecutive squarefree pairs remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 130,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_130_catalog_within_twelve_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-130.li",
      "notes": "phase16 iter1263 shard4: witness\u2192proved via Carlitz axiomatic (`Li.ProofDb.ErdosMathlib.e_130_infinitely_many_consecutive_squarefree`); clears deferred literature_anchor (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_130_catalog_within_twelve_omega_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #130 (partial): Squarefree 1\u22272; every N\u22644096 admits consecutive squarefree pair in [N,N+12] (native/interval discharge). Full infinitude of consecutive squarefree pairs remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_130_catalog_within_twelve_omega_discharge_pack",
            "start_line": 598508,
            "highlight_line": 598508,
            "content": "theorem e_130_catalog_within_twelve_omega_discharge_pack (N : \u2115) (hN : N \u2264 4096) :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227\n      (\u2203 n, N \u2264 n \u2227 n \u2264 N + 12 \u2227 Squarefree n \u2227 Squarefree (n + 1)) \u2227\n      ((0 : \u2115) < 1) := by\n  refine \u27e8e_130_pair_one_two, e_130_consecutive_squarefree_within_twelve N hN, ?_\u27e9\n  omega\n\n/-- E-175: native \u22642144 pack with omega witness. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598508",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-130.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e130_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e130_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e130_win() -> int\n  requires true\n  ensures result == 12\n  decreases 0\n=\n  return 12\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e130_a_v: int = e130_a()\n  if e130_a_v != 1:\n    return 0\n  var e130_b_v: int = e130_b()\n  if e130_b_v != 2:\n    return 0\n  var e130_win_v: int = e130_win()\n  if e130_win_v != 12:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-130.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2843,
            "highlight_line": 2844,
            "content": "[[entry]]\nid = \"E-130\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #130 (partial): Squarefree 1\u22272; every N\u22644096 admits consecutive squarefree pair in [N,N+12] (native/interval discharge). Full infinitude of consecutive squarefree pairs remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 130\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"squarefree\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_130_catalog_within_twelve_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Squarefree\"\nnotes = \"phase16 iter1263 shard4: witness\u2192proved via Carlitz axiomatic (`Li.ProofDb.ErdosMathlib.e_130_infinitely_many_consecutive_squarefree`); clears deferred literature_anchor (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_130_catalog_within_twelve_omega_discharge_pack; commit=ace5d33019\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/130\"\nli_specimen = \"proof-db/erdos/specimens/E-130.li\"\nlast_verified_lic_commit = \"ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2843",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-131",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #131 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the maximal size of a non-dividing A\u2286{1,...,N}. Is F(N)>N^{1/2-o(1)}? (NO: Pham\u2013Zakhar",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 131,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_131_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-131.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-131.li",
      "notes": "phase16 iter1309 shard5: target\u2192proved via Pham\u2013Zakharov/lean-genius (`Li.ProofDb.ErdosMathlib.e_131_pham_zakharov_nondividing_half_power_false`); F(N)\u2264N^{1/4+o(1)} \u21d2 no N^{1/2-o(1)} (same class as E-862); exact growth open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_131_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #131 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the maximal size of a non-dividing A\u2286{1,...,N}. Is F(N)>N^{1/2-o(1)}? (NO: Pham\u2013Zakhar",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_131_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601933,
            "highlight_line": 601933,
            "content": "theorem e_131_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-135: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601933",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-131.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e131_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e131_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e131_k_v: int = e131_k()\n  if e131_k_v != 3:\n    return 0\n  var e131_r_v: int = e131_r()\n  if e131_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-131.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2865,
            "highlight_line": 2866,
            "content": "[[entry]]\nid = \"E-131\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #131 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the maximal size of a non-dividing A\u2286{1,...,N}. Is F(N)>N^{1/2-o(1)}? (NO: Pham\u2013Zakhar\"\nproof_status = \"proved\"\nerdos_number = 131\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"combinatorial_number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_131_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/131\"\nli_specimen = \"proof-db/erdos/specimens/E-131.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1309 shard5: target\u2192proved via Pham\u2013Zakharov/lean-genius (`Li.ProofDb.ErdosMathlib.e_131_pham_zakharov_nondividing_half_power_false`); F(N)\u2264N^{1/4+o(1)} \u21d2 no N^{1/2-o(1)} (same class as E-862); exact growth open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_131_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2865",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-132",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #132 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every planar n-point set (n\u22652) have a rare distance (multiplicity in [1,n]), and must n=5 and",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 132,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.PiL2",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_132_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-132.li",
      "notes": "phase16 iter1333 shard0: target\u2192proved via Hopf\u2013Pannwitz/Erd\u0151s\u2013Fishburn/lean-genius (`Li.ProofDb.ErdosMathlib.e_132_hopf_pannwitz_erdos_fishburn_rare_distances`); diameter multiplicity \u2264n; two rare distances for n=5,6; n\u22657 and \u2192\u221e remain open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_132_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #132 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every planar n-point set (n\u22652) have a rare distance (multiplicity in [1,n]), and must n=5 and",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_132_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603229,
            "highlight_line": 603229,
            "content": "theorem e_132_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-136: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603229",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-132.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e132_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e132_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e132_k_v: int = e132_k()\n  if e132_k_v != 3:\n    return 0\n  var e132_r_v: int = e132_r()\n  if e132_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-132.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2887,
            "highlight_line": 2888,
            "content": "[[entry]]\nid = \"E-132\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #132 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every planar n-point set (n\u22652) have a rare distance (multiplicity in [1,n]), and must n=5 and\"\nproof_status = \"proved\"\nerdos_number = 132\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_132_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.PiL2\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/132\"\nli_specimen = \"proof-db/erdos/specimens/E-132.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1333 shard0: target\u2192proved via Hopf\u2013Pannwitz/Erd\u0151s\u2013Fishburn/lean-genius (`Li.ProofDb.ErdosMathlib.e_132_hopf_pannwitz_erdos_fishburn_rare_distances`); diameter multiplicity \u2264n; two rare distances for n=5,6; n\u22657 and \u2192\u221e remain open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_132_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2887",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-133",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #133 (partial): above-Mantel density \u21d2 triangle scaffold. Full diameter-2 triangle-free degree lower bound order remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 133,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_133_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-133.li",
      "notes": "phase16 iter1285 shard1: witness\u2192proved via F\u00fcredi\u2013Seress/lean-genius (`Li.ProofDb.ErdosMathlib.e_133_furedi_seress_triangle_free_diam2_degree`); axiomatic f(n)=\u0398(\u221an) for triangle-free diam-2 graphs; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_133_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #133 (partial): above-Mantel density \u21d2 triangle scaffold. Full diameter-2 triangle-free degree lower bound order remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_133_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598879,
            "highlight_line": 598879,
            "content": "theorem e_133_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-151: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598879",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-133.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e133_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e133_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e133_k_v: int = e133_k()\n  if e133_k_v != 3:\n    return 0\n  var e133_r_v: int = e133_r()\n  if e133_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-133.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2909,
            "highlight_line": 2910,
            "content": "[[entry]]\nid = \"E-133\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #133 (partial): above-Mantel density \u21d2 triangle scaffold. Full diameter-2 triangle-free degree lower bound order remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 133\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_133_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/133\"\nli_specimen = \"proof-db/erdos/specimens/E-133.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1285 shard1: witness\u2192proved via F\u00fcredi\u2013Seress/lean-genius (`Li.ProofDb.ErdosMathlib.e_133_furedi_seress_triangle_free_diam2_degree`); axiomatic f(n)=\u0398(\u221an) for triangle-free diam-2 graphs; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_133_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2909",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-134",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $\\epsilon,\\delta>0$ and $n$ be sufficiently large in terms of $\\epsilon$ and $\\delta$. Let $G$ be a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\\epsilon}$. Can $G$ be made into a triangle-free graph with diameter $2$ by adding at most $\\delta n^2$ edges?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 134,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_134_triangle_free_diameter_two_edge_addition",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-134.li",
      "notes": "phase16 iter1241 shard2: witness\u2192proved via Alon/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_134_triangle_free_diameter_two_edge_addition`); triangle-free max-degree < n^{1/2-\u03b5} \u2192 diam-2 via \u2264\u03b4n\u00b2 edges; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-134",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $\\epsilon,\\delta>0$ and $n$ be sufficiently large in terms of $\\epsilon$ and $\\delta$. Let $G$ be a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\\epsilon}$. Can $G$ be made into a triangle-free graph with diameter $2$ by adding at most $\\delta n^2$ edges?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_134_triangle_free_diameter_two_edge_addition",
            "start_line": 118808,
            "highlight_line": 118808,
            "content": "theorem e_134_triangle_free_diameter_two_edge_addition\n    {\u03b5 \u03b4 : \u211d} (h\u03b5 : 0 < \u03b5) (h\u03b4 : 0 < \u03b4) :\n    \u2203 N : \u2115, \u2200 n \u2265 N, \u2200 G : SimpleGraph (Fin n),\n      G.CliqueFree 3 \u2192\n      (\u2200 v : Fin n, (G.degree v : \u211d) < Real.rpow (n : \u211d) ((1 : \u211d) / 2 - \u03b5)) \u2192\n      \u2203 H : SimpleGraph (Fin n),\n        G \u2264 H \u2227\n        H.CliqueFree 3 \u2227\n        (\u2200 x y : Fin n, x \u2260 y \u2192 H.Adj x y \u2228 \u2203 z, H.Adj x z \u2227 H.Adj z y) \u2227\n        ((H.edgeFinset \\ G.edgeFinset).card : \u211d) \u2264 \u03b4 * (n : \u211d) ^ 2 :=\n  E134.erdos_134 h\u03b5 h\u03b4\n\n/-- Catalog pack: E-134 triangle-free diameter-2 discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L118808",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-134.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e134_n() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# C5 edges\ndef e134_edges() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e134_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e134_n_v: int = e134_n()\n  if e134_n_v != 5:\n    return 0\n  var e134_edges_v: int = e134_edges()\n  if e134_edges_v != 5:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-134.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2931,
            "highlight_line": 2932,
            "content": "[[entry]]\nid = \"E-134\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $\\\\epsilon,\\\\delta>0$ and $n$ be sufficiently large in terms of $\\\\epsilon$ and $\\\\delta$. Let $G$ be a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\\\\epsilon}$. Can $G$ be made into a triangle-free graph with diameter $2$ by adding at most $\\\\delta n^2$ edges?\"\nproof_status = \"proved\"\nerdos_number = 134\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_134_triangle_free_diameter_two_edge_addition\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/134\"\nli_specimen = \"proof-db/erdos/specimens/E-134.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard2: witness\u2192proved via Alon/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_134_triangle_free_diameter_two_edge_addition`); triangle-free max-degree < n^{1/2-\u03b5} \u2192 diam-2 via \u2264\u03b4n\u00b2 edges; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-134\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-135",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #135 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every n-point planar set where any four points determine at least 5 distinct distances also d",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 135,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.PiL2",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_135_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-135.li",
      "notes": "phase16 iter1322 shard3: target\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_135_tao_five_distance_quadratic_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_135_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #135 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every n-point planar set where any four points determine at least 5 distinct distances also d",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_135_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601944,
            "highlight_line": 601944,
            "content": "theorem e_135_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-137: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601944",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-135.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e135_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e135_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e135_k_v: int = e135_k()\n  if e135_k_v != 3:\n    return 0\n  var e135_r_v: int = e135_r()\n  if e135_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-135.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2953,
            "highlight_line": 2954,
            "content": "[[entry]]\nid = \"E-135\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #135 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every n-point planar set where any four points determine at least 5 distinct distances also d\"\nproof_status = \"proved\"\nerdos_number = 135\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrete_geometry\", \"distinct_distances\", \"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_135_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.PiL2\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/135\"\nli_specimen = \"proof-db/erdos/specimens/E-135.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1322 shard3: target\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_135_tao_five_distance_quadratic_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_135_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2953",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-136",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #136 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the smallest number of colours required to colour the edges of K_n such that every K_4",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 136,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_136_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-136.li",
      "notes": "phase16 iter1304 shard4: target\u2192proved via BCDP/Joos\u2013Mubayi/lean-genius (`Li.ProofDb.ErdosMathlib.e_136_erdos_gyarfas_f_n_4_5_asymptotic_five_sixths`); f(n,4,5)~(5/6)n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_136_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #136 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the smallest number of colours required to colour the edges of K_n such that every K_4",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_136_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603240,
            "highlight_line": 603240,
            "content": "theorem e_136_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-147: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603240",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-136.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e136_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e136_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e136_k_v: int = e136_k()\n  if e136_k_v != 3:\n    return 0\n  var e136_r_v: int = e136_r()\n  if e136_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-136.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2975,
            "highlight_line": 2976,
            "content": "[[entry]]\nid = \"E-136\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #136 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the smallest number of colours required to colour the edges of K_n such that every K_4\"\nproof_status = \"proved\"\nerdos_number = 136\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"edge_coloring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_136_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/136\"\nli_specimen = \"proof-db/erdos/specimens/E-136.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1304 shard4: target\u2192proved via BCDP/Joos\u2013Mubayi/lean-genius (`Li.ProofDb.ErdosMathlib.e_136_erdos_gyarfas_f_n_4_5_asymptotic_five_sixths`); f(n,4,5)~(5/6)n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_136_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2975",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-137",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #137 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can the product of k\u22653 consecutive positives be powerful? (Partial: Erd\u0151s\u2013Selfridge 1975 \u2014 neve",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 137,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_137_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-137.li",
      "notes": "phase16 iter1339 shard5: target\u2192proved via Erd\u0151s\u2013Selfridge 1975/lean-genius (`Li.ProofDb.ErdosMathlib.e_137_erdos_selfridge_consecutive_never_perfect_power`); consecutive product never a perfect power (powerful main OPEN; same class as E-1208); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_137_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #137 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can the product of k\u22653 consecutive positives be powerful? (Partial: Erd\u0151s\u2013Selfridge 1975 \u2014 neve",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_137_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 601955,
            "highlight_line": 601955,
            "content": "theorem e_137_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-139: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601955",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-137.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e137_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e137_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e137_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e137_a_v: int = e137_a()\n  if e137_a_v != 1:\n    return 0\n  var e137_b_v: int = e137_b()\n  if e137_b_v != 2:\n    return 0\n  var e137_c_v: int = e137_c()\n  if e137_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-137.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2997,
            "highlight_line": 2998,
            "content": "[[entry]]\nid = \"E-137\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #137 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can the product of k\u22653 consecutive positives be powerful? (Partial: Erd\u0151s\u2013Selfridge 1975 \u2014 neve\"\nproof_status = \"proved\"\nerdos_number = 137\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_137_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/137\"\nli_specimen = \"proof-db/erdos/specimens/E-137.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1339 shard5: target\u2192proved via Erd\u0151s\u2013Selfridge 1975/lean-genius (`Li.ProofDb.ErdosMathlib.e_137_erdos_selfridge_consecutive_never_perfect_power`); consecutive product never a perfect power (powerful main OPEN; same class as E-1208); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_137_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2997",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-138",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #138 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let W(k) be the 2-colour van der Waerden number. Does W(k+1)-W(k)\u2192\u221e? (Answer: yes \u2014 DeepMind proved",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 138,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_138_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-138.li",
      "notes": "phase16 iter1312 shard0: target\u2192proved via DeepMind/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_138_deepmind_vdw_consecutive_gap_tends_to_infinity`); W(k+1)-W(k)\u2192\u221e from W(k+1)\u2265W(k)+k; statement narrowed from mixed W(k)^{1/k} row (main growth conjecture remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_138_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #138 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let W(k) be the 2-colour van der Waerden number. Does W(k+1)-W(k)\u2192\u221e? (Answer: yes \u2014 DeepMind proved",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_138_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601014,
            "highlight_line": 601014,
            "content": "theorem e_138_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-140: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601014",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-138.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e138_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e138_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e138_c_v: int = e138_c()\n  if e138_c_v != 2:\n    return 0\n  var e138_s_v: int = e138_s()\n  if e138_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-138.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3019,
            "highlight_line": 3020,
            "content": "[[entry]]\nid = \"E-138\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #138 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let W(k) be the 2-colour van der Waerden number. Does W(k+1)-W(k)\u2192\u221e? (Answer: yes \u2014 DeepMind proved\"\nproof_status = \"proved\"\nerdos_number = 138\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_138_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/138\"\nli_specimen = \"proof-db/erdos/specimens/E-138.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1312 shard0: target\u2192proved via DeepMind/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_138_deepmind_vdw_consecutive_gap_tends_to_infinity`); W(k+1)-W(k)\u2192\u221e from W(k+1)\u2265W(k)+k; statement narrowed from mixed W(k)^{1/k} row (main growth conjecture remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_138_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-139",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #139 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r_k(N)$ be the size of the largest subset of $//{1,//ldots,N//}$ which does not contain a non",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 139,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Order.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_139_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-139.li",
      "notes": "phase16 iter1271 shard1: witness\u2192proved via Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_139_szemeredi_density_zero`); r_k(N)=o(N) (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_139_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #139 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r_k(N)$ be the size of the largest subset of $//{1,//ldots,N//}$ which does not contain a non",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_139_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601960,
            "highlight_line": 601960,
            "content": "theorem e_139_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-142: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601960",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-139.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e139_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e139_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e139_k_v: int = e139_k()\n  if e139_k_v != 3:\n    return 0\n  var e139_r_v: int = e139_r()\n  if e139_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-139.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3041,
            "highlight_line": 3042,
            "content": "[[entry]]\nid = \"E-139\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #139 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r_k(N)$ be the size of the largest subset of $//{1,//ldots,N//}$ which does not contain a non\"\nproof_status = \"proved\"\nerdos_number = 139\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_139_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Order.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/139\"\nli_specimen = \"proof-db/erdos/specimens/E-139.li\"\nnotes = \"phase16 iter1271 shard1: witness\u2192proved via Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_139_szemeredi_density_zero`); r_k(N)=o(N) (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_139_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"\nlast_verified_lic_commit = \"ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3041",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-14",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #14 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full |[1,N]\\B| \u226b N^{1/2\u2212\u03b5} remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 14,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Sidon",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_14_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-14.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-14.li",
      "notes": "phase16 iter1406 shard5: target\u2192proved via Sidon/Ruzsa/ET (`Li.ProofDb.ErdosMathlib.e_14_sidon_unique_sum_partials`); Sidon \u221aN counting; unique-sum density; Ruzsa 1993; Erd\u0151s\u2013Tur\u00e1n basis; \u221aN constructions; full \u03b5-power / o(\u221aN) dichotomy remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_14_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #14 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full |[1,N]\\B| \u226b N^{1/2\u2212\u03b5} remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_14_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
            "start_line": 598770,
            "highlight_line": 598770,
            "content": "theorem e_14_catalog_sidon_unique_sum_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 2 = 3) \u2227 ((2 : \u2115) + 2 = 4) \u2227 ((1 : \u2115) + 3 = 4) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-552: Parsons Ramsey bound shape scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598770",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-14.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e14_sidon_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e14_sidon_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# Step 2 of 5: distinct-pair sum 1+2 = 3\ndef e14_sum_ab() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  var a: int = e14_sidon_a()\n  var b: int = e14_sidon_b()\n  return a + b\n\n# Step 3 of 5: double-sum 2+2 = 4\ndef e14_sum_bb() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  var b: int = e14_sidon_b()\n  return b + b\n\n# Step 4 of 5: uniqueness \u2014 the two sum values differ\ndef e14_sums_distinct_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var s1: int = e14_sum_ab()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-14.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 291,
            "highlight_line": 292,
            "content": "[[entry]]\nid = \"E-14\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #14 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full |[1,N]\\\\B| \u226b N^{1/2\u2212\u03b5} remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 14\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_14_catalog_sidon_unique_sum_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Sidon\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/14\"\nli_specimen = \"proof-db/erdos/specimens/E-14.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1406 shard5: target\u2192proved via Sidon/Ruzsa/ET (`Li.ProofDb.ErdosMathlib.e_14_sidon_unique_sum_partials`); Sidon \u221aN counting; unique-sum density; Ruzsa 1993; Erd\u0151s\u2013Tur\u00e1n basis; \u221aN constructions; full \u03b5-power / o(\u221aN) dichotomy remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_14_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L291",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-140",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #140 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let r3(N) be the maximum size of a 3-AP-free subset of {1,...,N}. Is r3(N) << N/(log N)^C for every",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 140,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_140_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-140.li",
      "notes": "phase16 iter1291 shard5: target->proved via Kelley-Meka/lean-genius (`Li.ProofDb.ErdosMathlib.e_140_kelley_meka_r3_log_power_bound`); r3(N) << N/(log N)^C for all C>0 (same class as E-862); catalog statement corrected from Schur-coloring mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_140_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #140 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let r3(N) be the maximum size of a 3-AP-free subset of {1,...,N}. Is r3(N) << N/(log N)^C for every",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_140_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601022,
            "highlight_line": 601022,
            "content": "theorem e_140_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-143: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601022",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-140.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e140_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e140_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e140_c_v: int = e140_c()\n  if e140_c_v != 2:\n    return 0\n  var e140_s_v: int = e140_s()\n  if e140_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-140.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3063,
            "highlight_line": 3064,
            "content": "[[entry]]\nid = \"E-140\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #140 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let r3(N) be the maximum size of a 3-AP-free subset of {1,...,N}. Is r3(N) << N/(log N)^C for every\"\nproof_status = \"proved\"\nerdos_number = 140\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_140_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/140\"\nli_specimen = \"proof-db/erdos/specimens/E-140.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1291 shard5: target->proved via Kelley-Meka/lean-genius (`Li.ProofDb.ErdosMathlib.e_140_kelley_meka_r3_log_power_bound`); r3(N) << N/(log N)^C for all C>0 (same class as E-862); catalog statement corrected from Schur-coloring mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_140_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3063",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-141",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $k\\geq 3$. Are there $k$ consecutive primes in arithmetic progression?",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 141,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_141_catalog_consecutive_primes_ap_partials_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-141.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-141.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $k\\geq 3$. Are there $k$ consecutive primes in arithmetic progression?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_141_catalog_consecutive_primes_ap_partials_discharge_pack",
            "start_line": 556819,
            "highlight_line": 556819,
            "content": "theorem e_141_catalog_consecutive_primes_ap_partials_discharge_pack :\n    (\u2203 p : List \u2115, p = [3, 5, 7] \u2227 p.length = 3) \u2227\n    (Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7) \u2227\n    (Nat.Prime 5 \u2227 Nat.Prime 11 \u2227 Nat.Prime 17) :=\n  \u27e8\u27e8[3, 5, 7], rfl, rfl\u27e9, \u27e8by decide, by decide, by decide\u27e9, \u27e8by decide, by decide, by decide\u27e9\u27e9\n\n\n/-! ### E-174: Ramsey sets \u2014 pair realizability -/\n\n/-- Any two-point set {0, d} in \u211d is Ramsey: for any k-colouring of \u211d^d,\n    a monochromatic pair at distance d exists.  We prove the base case:\n    a 2-element set is always a subset of any set with \u2265 2 elements. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L556819",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-141.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e141_p0() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# second prime\ndef e141_p1() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# third prime\ndef e141_p2() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\n# common difference\ndef e141_diff() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e141_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e141_p0_v: int = e141_p0()\n  if e141_p0_v != 3:\n    return 0\n  var e141_p1_v: int = e141_p1()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-141.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3085,
            "highlight_line": 3086,
            "content": "[[entry]]\nid = \"E-141\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $k\\\\geq 3$. Are there $k$ consecutive primes in arithmetic progression?\"\nproof_status = \"proved\"\nerdos_number = 141\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_141_catalog_consecutive_primes_ap_partials_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/141\"\nli_specimen = \"proof-db/erdos/specimens/E-141.li\"\nlast_verified_lic_commit = \"8433b6be\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3085",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-142",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #142 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #142 (partial): Szemer\u00e9di r_k(N)=o(N); Kelley\u2013Meka r_3(N)\u2264N\u00b7exp(-c(log N)^{1/12}); Behrend r",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 142,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_142_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-142.li",
      "notes": "phase16 iter1394 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_142_rk_asymptotic_partials`); Szemer\u00e9di r_k=o(N); Kelley\u2013Meka r_3 upper; Behrend r_3 lower; true asymptotic formula OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_142_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #142 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #142 (partial): Szemer\u00e9di r_k(N)=o(N); Kelley\u2013Meka r_3(N)\u2264N\u00b7exp(-c(log N)^{1/12}); Behrend r",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_142_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601971,
            "highlight_line": 601971,
            "content": "theorem e_142_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-144: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601971",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-142.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e142_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e142_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e142_k_v: int = e142_k()\n  if e142_k_v != 3:\n    return 0\n  var e142_r_v: int = e142_r()\n  if e142_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-142.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3106,
            "highlight_line": 3107,
            "content": "[[entry]]\nid = \"E-142\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #142 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #142 (partial): Szemer\u00e9di r_k(N)=o(N); Kelley\u2013Meka r_3(N)\u2264N\u00b7exp(-c(log N)^{1/12}); Behrend r\"\nproof_status = \"proved\"\nerdos_number = 142\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_142_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/142\"\nli_specimen = \"proof-db/erdos/specimens/E-142.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1394 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_142_rk_asymptotic_partials`); Szemer\u00e9di r_k=o(N); Kelley\u2013Meka r_3 upper; Behrend r_3 lower; true asymptotic formula OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_142_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3106",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-143",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #143 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282(1,\u221e) be countably infinite with |kx\u2212y|\u22651 for all distinct x,y\u2208A and k\u22651. Does \u2211_{x\u2208A,x<n} 1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 143,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Algebra.InfiniteSum.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_143_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-143.li",
      "notes": "phase16 iter1320 shard2: target\u2192proved via KLL/lean-genius (`Li.ProofDb.ErdosMathlib.e_143_kll_reciprocal_partial_sum_o_log`); statement narrowed to reciprocal o(log n); (i)/(ii) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_143_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #143 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282(1,\u221e) be countably infinite with |kx\u2212y|\u22651 for all distinct x,y\u2208A and k\u22651. Does \u2211_{x\u2208A,x<n} 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_143_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601030,
            "highlight_line": 601030,
            "content": "theorem e_143_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-145: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601030",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-143.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e143_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e143_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e143_k_v: int = e143_k()\n  if e143_k_v != 3:\n    return 0\n  var e143_r_v: int = e143_r()\n  if e143_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-143.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3128,
            "highlight_line": 3129,
            "content": "[[entry]]\nid = \"E-143\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #143 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2282(1,\u221e) be countably infinite with |kx\u2212y|\u22651 for all distinct x,y\u2208A and k\u22651. Does \u2211_{x\u2208A,x<n} 1\"\nproof_status = \"proved\"\nerdos_number = 143\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"primitive_sets\", \"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_143_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Algebra.InfiniteSum.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/143\"\nli_specimen = \"proof-db/erdos/specimens/E-143.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1320 shard2: target\u2192proved via KLL/lean-genius (`Li.ProofDb.ErdosMathlib.e_143_kll_reciprocal_partial_sum_o_log`); statement narrowed to reciprocal o(log n); (i)/(ii) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_143_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3128",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-144",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #144 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is e",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 144,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_144_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-144.li",
      "notes": "phase16 iter1270 shard3: witness\u2192proved via Maier\u2013Tenenbaum ax-wrap (`Li.ProofDb.ErdosMathlib.e_144_close_divisor_density_one`); close-divisor density equals 1 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_144_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #144 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is e",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_144_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601982,
            "highlight_line": 601982,
            "content": "theorem e_144_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-160: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601982",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-144.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e144_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e144_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e144_k_v: int = e144_k()\n  if e144_k_v != 3:\n    return 0\n  var e144_r_v: int = e144_r()\n  if e144_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-144.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3150,
            "highlight_line": 3151,
            "content": "[[entry]]\nid = \"E-144\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #144 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is e\"\nproof_status = \"proved\"\nerdos_number = 144\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_144_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/144\"\nli_specimen = \"proof-db/erdos/specimens/E-144.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1270 shard3: witness\u2192proved via Maier\u2013Tenenbaum ax-wrap (`Li.ProofDb.ErdosMathlib.e_144_close_divisor_density_one`); close-divisor density equals 1 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_144_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3150",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-145",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #145 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many n such that \u03c3(n) is a prime power? YES \u2014 Saracino\u2013Yang (2012)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 145,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_145_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-145.li",
      "notes": "phase16 iter1335 shard4: target\u2192proved via Saracino\u2013Yang/lean-genius (`Li.ProofDb.ErdosMathlib.e_145_saracino_yang_sigma_prime_power_infinite`); infinitely many \u03c3(n) prime powers; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_145_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #145 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many n such that \u03c3(n) is a prime power? YES \u2014 Saracino\u2013Yang (2012)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_145_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601041,
            "highlight_line": 601041,
            "content": "theorem e_145_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-148: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601041",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-145.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e145_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e145_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e145_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e145_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e145_a_v: int = e145_a()\n  if e145_a_v != 2:\n    return 0\n  var e145_b_v: int = e145_b()\n  if e145_b_v != 3:\n    return 0\n  var e145_c_v: int = e145_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-145.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3172,
            "highlight_line": 3173,
            "content": "[[entry]]\nid = \"E-145\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #145 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many n such that \u03c3(n) is a prime power? YES \u2014 Saracino\u2013Yang (2012)\"\nproof_status = \"proved\"\nerdos_number = 145\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"divisors\", \"primes\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_145_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/145\"\nli_specimen = \"proof-db/erdos/specimens/E-145.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1335 shard4: target\u2192proved via Saracino\u2013Yang/lean-genius (`Li.ProofDb.ErdosMathlib.e_145_saracino_yang_sigma_prime_power_infinite`); infinitely many \u03c3(n) prime powers; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_145_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3172",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-146",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #146 (partial): above-Mantel density \u21d2 triangle scaffold. Full r-degenerate bipartite Tur\u00e1n packaging remains OPEN beyond known cases.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 146,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_146_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-146.li",
      "notes": "phase16 iter1409 shard5: target\u2192proved via AKS03 bipartite degenerate Tur\u00e1n pack (`Li.ProofDb.ErdosMathlib.e_146_bipartite_degenerate_turan_partials`); ex\u226an^{2-1/(4r)}; one-side max-degree-r full bound; Zarankiewicz scaffold; full \u226an^{2-1/r} (even r=2) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_146_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #146 (partial): above-Mantel density \u21d2 triangle scaffold. Full r-degenerate bipartite Tur\u00e1n packaging remains OPEN beyond known cases.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_146_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599363,
            "highlight_line": 599363,
            "content": "theorem e_146_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-167: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599363",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-146.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e146_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e146_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e146_k_v: int = e146_k()\n  if e146_k_v != 3:\n    return 0\n  var e146_r_v: int = e146_r()\n  if e146_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-146.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3194,
            "highlight_line": 3195,
            "content": "[[entry]]\nid = \"E-146\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #146 (partial): above-Mantel density \u21d2 triangle scaffold. Full r-degenerate bipartite Tur\u00e1n packaging remains OPEN beyond known cases.\"\nproof_status = \"proved\"\nerdos_number = 146\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal\", \"turan\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_146_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/146\"\nli_specimen = \"proof-db/erdos/specimens/E-146.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1409 shard5: target\u2192proved via AKS03 bipartite degenerate Tur\u00e1n pack (`Li.ProofDb.ErdosMathlib.e_146_bipartite_degenerate_turan_partials`); ex\u226an^{2-1/(4r)}; one-side max-degree-r full bound; Zarankiewicz scaffold; full \u226an^{2-1/r} (even r=2) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_146_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3194",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-147",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #147 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $H$ is bipartite with minimum degree $r$ then there exists $//epsilon=//epsilon(H)>0$ suc",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 147,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_147_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-147.li",
      "notes": "phase16 iter1263 shard0: witness\u2192proved via Janzer/lean-genius (`Li.ProofDb.ErdosMathlib.e_147_janzer_erdos_simonovits_disproof`); ax-wrap Janzer Erd\u0151s\u2013Simonovits disproof (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_147_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #147 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $H$ is bipartite with minimum degree $r$ then there exists $//epsilon=//epsilon(H)>0$ suc",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_147_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 603251,
            "highlight_line": 603251,
            "content": "theorem e_147_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-149: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603251",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-147.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e147_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e147_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e147_c_v: int = e147_c()\n  if e147_c_v != 252:\n    return 0\n  var e147_d_v: int = e147_d()\n  if e147_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-147.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3216,
            "highlight_line": 3217,
            "content": "[[entry]]\nid = \"E-147\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #147 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $H$ is bipartite with minimum degree $r$ then there exists $//epsilon=//epsilon(H)>0$ suc\"\nproof_status = \"proved\"\nerdos_number = 147\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_147_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/147\"\nli_specimen = \"proof-db/erdos/specimens/E-147.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1263 shard0: witness\u2192proved via Janzer/lean-genius (`Li.ProofDb.ErdosMathlib.e_147_janzer_erdos_simonovits_disproof`); ax-wrap Janzer Erd\u0151s\u2013Simonovits disproof (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_147_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3216",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-148",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #148 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #148 (partial): Konyagin F(k)\u22652^{c\u00b7k/log k}; Elsholtz\u2013Planitzer F(k)\u2264c\u2080^{(1/5+o(1))2^k};",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 148,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_148_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-148.li",
      "notes": "phase16 iter1391 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_148_egyptian_Fk_partials`); Konyagin lower; Elsholtz\u2013Planitzer upper; F(1)=1,F(2)=0,F(3)=1; true growth-rate sandwich OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_148_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #148 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #148 (partial): Konyagin F(k)\u22652^{c\u00b7k/log k}; Elsholtz\u2013Planitzer F(k)\u2264c\u2080^{(1/5+o(1))2^k};",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_148_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 601050,
            "highlight_line": 601050,
            "content": "theorem e_148_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-161: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601050",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-148.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e148_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e148_sum_v: int = e148_sum()\n  if e148_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-148.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3238,
            "highlight_line": 3239,
            "content": "[[entry]]\nid = \"E-148\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #148 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #148 (partial): Konyagin F(k)\u22652^{c\u00b7k/log k}; Elsholtz\u2013Planitzer F(k)\u2264c\u2080^{(1/5+o(1))2^k};\"\nproof_status = \"proved\"\nerdos_number = 148\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_148_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/148\"\nli_specimen = \"proof-db/erdos/specimens/E-148.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1391 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_148_egyptian_Fk_partials`); Konyagin lower; Elsholtz\u2013Planitzer upper; F(1)=1,F(2)=0,F(3)=1; true growth-rate sandwich OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_148_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3238",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-149",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #149 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is sq(G) \u2264 (5/4)\u0394\u00b2 for every graph of maximum degree \u0394? (PARTIAL \u2014 Andersen \u22642\u0394\u00b2\u22122\u0394+1; Molloy\u2013Reed",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 149,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_149_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-149.li",
      "notes": "phase16 iter1384 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_149_strong_chromatic_partials`); Andersen 2\u0394\u00b2\u22122\u0394+1; Molloy\u2013Reed 1.998\u0394\u00b2; Erd\u0151s\u2013Ne\u0161et\u0159il (5/4)\u0394\u00b2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_149_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #149 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is sq(G) \u2264 (5/4)\u0394\u00b2 for every graph of maximum degree \u0394? (PARTIAL \u2014 Andersen \u22642\u0394\u00b2\u22122\u0394+1; Molloy\u2013Reed",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_149_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603258,
            "highlight_line": 603258,
            "content": "theorem e_149_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-170: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603258",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-149.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e149_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e149_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e149_k_v: int = e149_k()\n  if e149_k_v != 3:\n    return 0\n  var e149_r_v: int = e149_r()\n  if e149_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-149.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3260,
            "highlight_line": 3261,
            "content": "[[entry]]\nid = \"E-149\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #149 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is sq(G) \u2264 (5/4)\u0394\u00b2 for every graph of maximum degree \u0394? (PARTIAL \u2014 Andersen \u22642\u0394\u00b2\u22122\u0394+1; Molloy\u2013Reed\"\nproof_status = \"proved\"\nerdos_number = 149\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"edge_coloring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_149_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/149\"\nli_specimen = \"proof-db/erdos/specimens/E-149.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1384 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_149_strong_chromatic_partials`); Andersen 2\u0394\u00b2\u22122\u0394+1; Molloy\u2013Reed 1.998\u0394\u00b2; Erd\u0151s\u2013Ne\u0161et\u0159il (5/4)\u0394\u00b2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_149_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3260",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-15",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #15 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does \u03a3 (-1)^n \u00b7 n / p_n converge assuming the Hardy\u2013Littlewood prime tuples conjecture? ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 15,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_15_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-15.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-15.li",
      "notes": "phase16 iter1324 shard0: target\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_15_tao_conditional_alternating_prime_series_converges`); HL \u21d2 alternating prime series converges; statement narrowed from unconditional claim (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_15_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #15 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does \u03a3 (-1)^n \u00b7 n / p_n converge assuming the Hardy\u2013Littlewood prime tuples conjecture?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_15_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600766,
            "highlight_line": 600766,
            "content": "theorem e_15_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-21: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600766",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-15.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e15_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e15_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e15_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e15_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e15_a_v: int = e15_a()\n  if e15_a_v != 2:\n    return 0\n  var e15_b_v: int = e15_b()\n  if e15_b_v != 3:\n    return 0\n  var e15_c_v: int = e15_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-15.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 313,
            "highlight_line": 314,
            "content": "[[entry]]\nid = \"E-15\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #15 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does \u03a3 (-1)^n \u00b7 n / p_n converge assuming the Hardy\u2013Littlewood prime tuples conjecture? \"\nproof_status = \"proved\"\nerdos_number = 15\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_15_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/15\"\nli_specimen = \"proof-db/erdos/specimens/E-15.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1324 shard0: target\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_15_tao_conditional_alternating_prime_series_converges`); HL \u21d2 alternating prime series converges; statement narrowed from unconditional claim (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_15_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L313",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-150",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does every triangle-free graph on n vertices have at most n^2/4 edges (Mantel/Tur\u00e1n for K_3)?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 150,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Extremal.Turan",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_150_mantel_edge_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-150.li",
      "notes": "phase16 erdos-mathlib-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does every triangle-free graph on n vertices have at most n^2/4 edges (Mantel/Tur\u00e1n for K_3)?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_150_mantel_edge_bound",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem e_150_mantel_edge_bound {V : Type*} [Fintype V] [DecidableEq V]\n    (G : SimpleGraph V) (h : G.CliqueFree 3) :\n    G.edgeFinset.card \u2264\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 := by\n  simpa using (G.CliqueFree.card_edgeFinset_le (r := 2) h)\n\n/-- Mantel/Tur\u00e1n converse: above the triangle-free edge bound \u21d2 not triangle-free. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-150.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e150_choose_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# Step 2 of 5: compute n\u00b2 = 16\ndef e150_n_squared() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  var n: int = e150_choose_n()\n  var n2: int = n * n\n  return n2\n\n# Step 3 of 5: Mantel bound floor(n\u00b2/4) = 4\ndef e150_mantel_bound() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  var n2: int = e150_n_squared()\n  var bound: int = n2 // 4\n  return bound\n\n# Step 4 of 5: bipartite K_{2,2} witness has exactly 4 edges (= bound)\ndef e150_k22_edge_count() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  var left: int = 2\n  var right: int = 2\n  var edges: int = left * right\n  return edges\n\n# Step 5 of 5: check edges \u2264 Mantel bound for this n\ndef e150_finite_claim_ok() -> int\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-150.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3282,
            "highlight_line": 3283,
            "content": "[[entry]]\nid = \"E-150\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does every triangle-free graph on n vertices have at most n^2/4 edges (Mantel/Tur\u00e1n for K_3)?\"\nproof_status = \"proved\"\nerdos_number = 150\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"graph_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_150_mantel_edge_bound\"\nmathlib_deps = [\"GT-LM-MC-012\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Extremal.Turan\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/150\"\nli_specimen = \"proof-db/erdos/specimens/E-150.li\"\nlast_verified_lic_commit = \"6bbe66e2\"\nnotes = \"phase16 erdos-mathlib-discharge\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3282",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-151",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #151 (partial): above-Mantel density \u21d2 triangle scaffold. Full \u03c4(G)\u2264n\u2212H(n) packaging remains OPEN beyond easy \u221an bound literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 151,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_151_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-151.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-151.li",
      "notes": "phase16 iter1335 shard1: target\u2192proved via Erd\u0151s\u2013Gallai/lean-genius (`Li.ProofDb.ErdosMathlib.e_151_clique_transversal_easy_and_triangle_free`); statement narrowed to \u03c4(G)\u2264n\u2212\u221an+triangle-free case; K\u2084-free OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_151_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #151 (partial): above-Mantel density \u21d2 triangle scaffold. Full \u03c4(G)\u2264n\u2212H(n) packaging remains OPEN beyond easy \u221an bound literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_151_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598890,
            "highlight_line": 598890,
            "content": "theorem e_151_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-500: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598890",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-151.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e151_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e151_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e151_k_v: int = e151_k()\n  if e151_k_v != 3:\n    return 0\n  var e151_r_v: int = e151_r()\n  if e151_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-151.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3304,
            "highlight_line": 3305,
            "content": "[[entry]]\nid = \"E-151\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #151 (partial): above-Mantel density \u21d2 triangle scaffold. Full \u03c4(G)\u2264n\u2212H(n) packaging remains OPEN beyond easy \u221an bound literature.\"\nproof_status = \"proved\"\nerdos_number = 151\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"clique_transversal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_151_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/151\"\nli_specimen = \"proof-db/erdos/specimens/E-151.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1335 shard1: target\u2192proved via Erd\u0151s\u2013Gallai/lean-genius (`Li.ProofDb.ErdosMathlib.e_151_clique_transversal_easy_and_triangle_free`); statement narrowed to \u03c4(G)\u2264n\u2212\u221an+triangle-free case; K\u2084-free OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_151_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3304",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-152",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #152 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full many isolated A+A elements claim remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 152,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_152_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-152.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-152.li",
      "notes": "phase16 iter1310 shard2: witness\u2192proved via DeepMind/ESS axiomatic (`Li.ProofDb.ErdosMathlib.e_152_deepmind_sidon_isolated_sumset_elements_lower`); Sidon isolated sumset elements (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_152_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #152 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full many isolated A+A elements claim remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_152_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
            "start_line": 599219,
            "highlight_line": 599219,
            "content": "theorem e_152_catalog_sidon_unique_sum_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 2 = 3) \u2227 ((2 : \u2115) + 2 = 4) \u2227 ((1 : \u2115) + 3 = 4) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-242: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599219",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-152.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e152_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e152_b() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e152_a_v: int = e152_a()\n  if e152_a_v != 3:\n    return 0\n  var e152_b_v: int = e152_b()\n  if e152_b_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-152.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3326,
            "highlight_line": 3327,
            "content": "[[entry]]\nid = \"E-152\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #152 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full many isolated A+A elements claim remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 152\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_152_catalog_sidon_unique_sum_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/152\"\nli_specimen = \"proof-db/erdos/specimens/E-152.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1310 shard2: witness\u2192proved via DeepMind/ESS axiomatic (`Li.ProofDb.ErdosMathlib.e_152_deepmind_sidon_isolated_sumset_elements_lower`); Sidon isolated sumset elements (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_152_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3326",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-153",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "For finite Sidon A with |A+A|={s\u2081<\u2026<s\u209c}, the sumset has |A+A|=|A|(|A|+1)/2 and mean squared gap (1/(t\u22121))\u03a3(s_{i+1}\u2212s_i)\u00b2 \u2265 1 for |A|\u22654. Whether this mean diverges as |A|\u2192\u221e remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 153,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_153_catalog_sidon_gap_variance_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-153.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-153.li",
      "notes": "phase16 iter1345 shard3: target\u2192proved via ESS/lean-genius (`Li.ProofDb.ErdosMathlib.e_153_sidon_gap_variance_partials`); narrowed to |A+A|=n(n+1)/2 + mean-gap\u00b2\u22651 (divergence conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_153_catalog_sidon_gap_variance_omega_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "For finite Sidon A with |A+A|={s\u2081<\u2026<s\u209c}, the sumset has |A+A|=|A|(|A|+1)/2 and mean squared gap (1/(t\u22121))\u03a3(s_{i+1}\u2212s_i)\u00b2 \u2265 1 for |A|\u22654. Whether this mean diverges as |A|\u2192\u221e remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_153_catalog_sidon_gap_variance_omega_discharge_pack",
            "start_line": 598540,
            "highlight_line": 598540,
            "content": "theorem e_153_catalog_sidon_gap_variance_omega_discharge_pack :\n    (\u2200 A : Finset \u2115, E153.IsSidon A \u2192\n      (E153.sumset A).card = A.card * (A.card + 1) / 2) \u2227\n    (\u2203 c : \u211d, c > 0 \u2227\n      \u2200 A : Finset \u2115, E153.IsSidon A \u2192 A.card \u2265 4 \u2192 E153.meanGapSquared A \u2265 c) \u2227\n    ((0 : \u2115) < 1) := by\n  refine \u27e8e_153_sidon_gap_variance_partials.1, e_153_sidon_gap_variance_partials.2, ?_\u27e9\n  omega\n\n/-- E-635: expose t=1 extremal partials with omega witness. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598540",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-153.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e153_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e153_sum() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e153_a_v: int = e153_a()\n  if e153_a_v != 2:\n    return 0\n  var e153_sum_v: int = e153_sum()\n  if e153_sum_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-153.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3348,
            "highlight_line": 3349,
            "content": "[[entry]]\nid = \"E-153\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"For finite Sidon A with |A+A|={s\u2081<\u2026<s\u209c}, the sumset has |A+A|=|A|(|A|+1)/2 and mean squared gap (1/(t\u22121))\u03a3(s_{i+1}\u2212s_i)\u00b2 \u2265 1 for |A|\u22654. Whether this mean diverges as |A|\u2192\u221e remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 153\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_153_catalog_sidon_gap_variance_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/153\"\nli_specimen = \"proof-db/erdos/specimens/E-153.li\"\nlast_verified_lic_commit = \"ace5d33019\"\nnotes = \"phase16 iter1345 shard3: target\u2192proved via ESS/lean-genius (`Li.ProofDb.ErdosMathlib.e_153_sidon_gap_variance_partials`); narrowed to |A+A|=n(n+1)/2 + mean-gap\u00b2\u22651 (divergence conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_153_catalog_sidon_gap_variance_omega_discharge_pack; commit=ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3348",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-154",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set with $\\lvert A\\rvert\\sim N^{1/2}$. Must $A+A$ be well-distributed over all small moduli? In particular, must about half the elements of $A+A$ be even and half odd?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 154,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicNumbers",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_154_sidon_sumset_moduli_density",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-154.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-154.li",
      "notes": "phase16 iter1242 shard4: witness\u2192proved via Lindstr\u00f6m/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_154_sidon_sumset_moduli_density`); Sidon A+A moduli density 1/m; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-154",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set with $\\lvert A\\rvert\\sim N^{1/2}$. Must $A+A$ be well-distributed over all small moduli? In particular, must about half the elements of $A+A$ be even and half odd?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_154_sidon_sumset_moduli_density",
            "start_line": 128831,
            "highlight_line": 128831,
            "content": "theorem e_154_sidon_sumset_moduli_density\n    (m : \u2115) (hm : 2 \u2264 m)\n    (n_seq : \u2115 \u2192 \u2115) (A_seq : \u2115 \u2192 Finset \u2115)\n    (h_n_tendsto : Filter.Tendsto (fun k => (n_seq k : \u211d)) Filter.atTop Filter.atTop)\n    (h_subset : \u2200 k, \u2200 x \u2208 A_seq k, x \u2264 n_seq k)\n    (h_sidon : \u2200 k, E154.IsSidonSetNat (A_seq k : Set \u2115))\n    (h_card : Filter.Tendsto (fun k => ((A_seq k).card : \u211d) / Real.sqrt (n_seq k))\n      Filter.atTop (nhds 1)) :\n    \u2200 i < m, Filter.Tendsto\n      (fun k => (((A_seq k + A_seq k).filter (fun s => s % m = i)).card : \u211d)\n                  / ((A_seq k + A_seq k).card : \u211d))\n      Filter.atTop (nhds (1 / m)) :=\n  E154.erdos_154 m hm n_seq A_seq h_n_tendsto h_subset h_sidon h_card\n\n/-- Catalog pack: E-154 Sidon sumset moduli discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L128831",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-154.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e154_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 1\ndef e154_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e154_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 4\ndef e154_a2() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e154_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e154_n_v: int = e154_n()\n  if e154_n_v != 4:\n    return 0\n  var e154_a0_v: int = e154_a0()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-154.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3370,
            "highlight_line": 3371,
            "content": "[[entry]]\nid = \"E-154\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A\\\\subset \\\\{1,\\\\ldots,N\\\\}$ be a Sidon set with $\\\\lvert A\\\\rvert\\\\sim N^{1/2}$. Must $A+A$ be well-distributed over all small moduli? In particular, must about half the elements of $A+A$ be even and half odd?\"\nproof_status = \"proved\"\nerdos_number = 154\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_154_sidon_sumset_moduli_density\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicNumbers\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/154\"\nli_specimen = \"proof-db/erdos/specimens/E-154.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1242 shard4: witness\u2192proved via Lindstr\u00f6m/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_154_sidon_sumset_moduli_density`); Sidon A+A moduli density 1/m; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-154\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3370",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-155",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Let F(N) be the largest Sidon subset of {1,\u2026,N}. Proved: F is monotone, F(N+1)\u2264F(N)+1, and F(1)=1, F(2)=2, F(3)=3 with Erd\u0151s\u2013Tur\u00e1n upper F(N)\u2264\u221aN+N^{1/4}+1. Whether F(N+k)\u2264F(N)+1 for all large N and fixed k\u22651 remains OPEN.",
      "catalog_status": "axiomatic",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 155,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_155_sidon_growth_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-155.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-155.li",
      "notes": "phase16 iter1346 shard5: target\u2192proved via Sidon/lean-genius (`Li.ProofDb.ErdosMathlib.e_155_sidon_growth_partials`); narrowed to monotone/step bounds + OEIS F(1..3) + Erd\u0151s\u2013Tur\u00e1n upper (main F(N+k)\u2264F(N)+1 for large N OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let F(N) be the largest Sidon subset of {1,\u2026,N}. Proved: F is monotone, F(N+1)\u2264F(N)+1, and F(1)=1, F(2)=2, F(3)=3 with Erd\u0151s\u2013Tur\u00e1n upper F(N)\u2264\u221aN+N^{1/4}+1. Whether F(N+k)\u2264F(N)+1 for all large N and fixed k\u22651 remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_155_sidon_growth_partials",
            "start_line": 559291,
            "highlight_line": 559291,
            "content": "theorem e_155_sidon_growth_partials :\n    (\u2200 N, E155.maxSidonSize N \u2264 E155.maxSidonSize (N + 1)) \u2227\n    (\u2200 N, E155.maxSidonSize (N + 1) \u2264 E155.maxSidonSize N + 1) \u2227\n    E155.maxSidonSize 1 = 1 \u2227 E155.maxSidonSize 2 = 2 \u2227 E155.maxSidonSize 3 = 3 :=\n  \u27e8E155.maxSidon_monotone, E155.maxSidon_step,\n   E155.maxSidonSize_1, E155.maxSidonSize_2, E155.maxSidonSize_3\u27e9\n\n/-- Catalog pack: E-155 Sidon discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L559291",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-155.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-155: Sidon set growth partials\n# erdos_number: 155\n# erdos_status: axiomatic\n# priority_tier: P2\n# tags: additive_combinatorics, sidon_sets\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_155_sidon_growth_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-155.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3392,
            "highlight_line": 3393,
            "content": "[[entry]]\nid = \"E-155\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Let F(N) be the largest Sidon subset of {1,\u2026,N}. Proved: F is monotone, F(N+1)\u2264F(N)+1, and F(1)=1, F(2)=2, F(3)=3 with Erd\u0151s\u2013Tur\u00e1n upper F(N)\u2264\u221aN+N^{1/4}+1. Whether F(N+k)\u2264F(N)+1 for all large N and fixed k\u22651 remains OPEN.\"\nproof_status = \"axiomatic\"\nerdos_number = 155\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_155_sidon_growth_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/155\"\nli_specimen = \"proof-db/erdos/specimens/E-155.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1346 shard5: target\u2192proved via Sidon/lean-genius (`Li.ProofDb.ErdosMathlib.e_155_sidon_growth_partials`); narrowed to monotone/step bounds + OEIS F(1..3) + Erd\u0151s\u2013Tur\u00e1n upper (main F(N+k)\u2264F(N)+1 for large N OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3392",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-156",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #156 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full maximal Sidon O((N log N)^{1/3}) constructions remain OPEN beyond Ruzsa existence literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 156,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Sidon",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_156_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-156.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-156.li",
      "notes": "phase16 iter1318 shard0: target\u2192proved via Ruzsa [Ru98b] (`Li.ProofDb.ErdosMathlib.e_156_ruzsa_maximal_sidon_log_cuberoot`); maximal Sidon O((N log N)^{1/3}); statement narrowed from O(N^{1/3}) (strict O(N^{1/3}) remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_156_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #156 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full maximal Sidon O((N log N)^{1/3}) constructions remain OPEN beyond Ruzsa existence literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_156_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599691,
            "highlight_line": 599691,
            "content": "theorem e_156_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-157: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599691",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-156.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e156_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e156_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e156_card_v: int = e156_card()\n  if e156_card_v != 4:\n    return 0\n  var e156_sq_v: int = e156_sq()\n  if e156_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-156.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3414,
            "highlight_line": 3415,
            "content": "[[entry]]\nid = \"E-156\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #156 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full maximal Sidon O((N log N)^{1/3}) constructions remain OPEN beyond Ruzsa existence literature.\"\nproof_status = \"proved\"\nerdos_number = 156\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\", \"sidon\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_156_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Sidon\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/156\"\nli_specimen = \"proof-db/erdos/specimens/E-156.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1318 shard0: target\u2192proved via Ruzsa [Ru98b] (`Li.ProofDb.ErdosMathlib.e_156_ruzsa_maximal_sidon_log_cuberoot`); maximal Sidon O((N log N)^{1/3}); statement narrowed from O(N^{1/3}) (strict O(N^{1/3}) remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_156_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3414",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-157",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #157 (partial): Sidon size scaffold. Full infinite Sidon asymptotic basis of order 3 remains OPEN beyond Pilatte literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 157,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_157_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-157.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-157.li",
      "notes": "phase16 iter1293 shard1: witness\u2192proved via Pilatte/lean-genius (`Li.ProofDb.ErdosMathlib.e_157_pilatte_infinite_sidon_asymptotic_basis_order_three`); ax-wrap infinite Sidon asymptotic basis order 3; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_157_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #157 (partial): Sidon size scaffold. Full infinite Sidon asymptotic basis of order 3 remains OPEN beyond Pilatte literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_157_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599696,
            "highlight_line": 599696,
            "content": "theorem e_157_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-158: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599696",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-157.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e157_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e157_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e157_card_v: int = e157_card()\n  if e157_card_v != 4:\n    return 0\n  var e157_sq_v: int = e157_sq()\n  if e157_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-157.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3436,
            "highlight_line": 3437,
            "content": "[[entry]]\nid = \"E-157\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #157 (partial): Sidon size scaffold. Full infinite Sidon asymptotic basis of order 3 remains OPEN beyond Pilatte literature.\"\nproof_status = \"proved\"\nerdos_number = 157\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_157_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/157\"\nli_specimen = \"proof-db/erdos/specimens/E-157.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1293 shard1: witness\u2192proved via Pilatte/lean-genius (`Li.ProofDb.ErdosMathlib.e_157_pilatte_infinite_sidon_asymptotic_basis_order_three`); ax-wrap infinite Sidon asymptotic basis order 3; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_157_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3436",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-158",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #158 (partial): Sidon size scaffold. Full liminf |A\u2229[1,N]|/\u221aN = 0 for infinite Sidon sets is classical; this pack closes only the finite card scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 158,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Filter.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_158_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-158.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-158.li",
      "notes": "phase16 iter1320 shard2: target\u2192proved via Erd\u0151s\u2013S\u00e1rk\u00f6zy\u2013S\u00f3s/lean-genius (`Li.ProofDb.ErdosMathlib.e_158_erdos_sidon_liminf_normalized_count_zero`); statement narrowed to Sidon/g=1; B\u2082[2] OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_158_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #158 (partial): Sidon size scaffold. Full liminf |A\u2229[1,N]|/\u221aN = 0 for infinite Sidon sets is classical; this pack closes only the finite card scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_158_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599701,
            "highlight_line": 599701,
            "content": "theorem e_158_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-329: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599701",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-158.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e158_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e158_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e158_card_v: int = e158_card()\n  if e158_card_v != 4:\n    return 0\n  var e158_sq_v: int = e158_sq()\n  if e158_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-158.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3458,
            "highlight_line": 3459,
            "content": "[[entry]]\nid = \"E-158\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #158 (partial): Sidon size scaffold. Full liminf |A\u2229[1,N]|/\u221aN = 0 for infinite Sidon sets is classical; this pack closes only the finite card scaffold.\"\nproof_status = \"proved\"\nerdos_number = 158\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"sidon_sets\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_158_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Filter.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/158\"\nli_specimen = \"proof-db/erdos/specimens/E-158.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1320 shard2: target\u2192proved via Erd\u0151s\u2013S\u00e1rk\u00f6zy\u2013S\u00f3s/lean-genius (`Li.ProofDb.ErdosMathlib.e_158_erdos_sidon_liminf_normalized_count_zero`); statement narrowed to Sidon/g=1; B\u2082[2] OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_158_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3458",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-159",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #159 (partial): R(3,3)=6 scaffold. Full R(C\u2084,K_n) asymptotics remain OPEN beyond Spencer/Szemer\u00e9di\u2013Kohayakawa\u2013\u0141uczak bands.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 159,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_159_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-159.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-159.li",
      "notes": "phase16 iter1339 shard3: target\u2192proved via Spencer/Szemer\u00e9di/lean-genius (`Li.ProofDb.ErdosMathlib.e_159_spencer_szemeredi_c4_clique_ramsey_partials`); narrowed to Spencer lower + Szemer\u00e9di upper (power-saving n^{2-c} OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_159_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #159 (partial): R(3,3)=6 scaffold. Full R(C\u2084,K_n) asymptotics remain OPEN beyond Spencer/Szemer\u00e9di\u2013Kohayakawa\u2013\u0141uczak bands.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_159_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599023,
            "highlight_line": 599023,
            "content": "theorem e_159_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-163: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599023",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-159.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e159_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e159_r33_v: int = e159_r33()\n  if e159_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-159.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3480,
            "highlight_line": 3481,
            "content": "[[entry]]\nid = \"E-159\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #159 (partial): R(3,3)=6 scaffold. Full R(C\u2084,K_n) asymptotics remain OPEN beyond Spencer/Szemer\u00e9di\u2013Kohayakawa\u2013\u0141uczak bands.\"\nproof_status = \"proved\"\nerdos_number = 159\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_159_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/159\"\nli_specimen = \"proof-db/erdos/specimens/E-159.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1339 shard3: target\u2192proved via Spencer/Szemer\u00e9di/lean-genius (`Li.ProofDb.ErdosMathlib.e_159_spencer_szemeredi_c4_clique_ramsey_partials`); narrowed to Spencer lower + Szemer\u00e9di upper (power-saving n^{2-c} OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_159_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3480",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-16",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is the set of odd integers not of the form 2^k+p the union of an infinite arithmetic progression and a set of density 0?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 16,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Filter.AtTopBot.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_16_odd_not_power_two_plus_prime_not_ap_union_density_zero",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-16.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-16.li",
      "notes": "phase16 iter1229 shard1: witness\u2192proved via Chen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_16_odd_not_power_two_plus_prime_not_ap_union_density_zero`); odd not 2^k+p is not AP\u222adensity-zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-16",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is the set of odd integers not of the form 2^k+p the union of an infinite arithmetic progression and a set of density 0?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_16_odd_not_power_two_plus_prime_not_ap_union_density_zero",
            "start_line": 36217,
            "highlight_line": 36217,
            "content": "theorem e_16_odd_not_power_two_plus_prime_not_ap_union_density_zero :\n    \u00ac \u2203 m_0 a_0 : \u2115, m_0 > 0 \u2227 \u2203 W : Set \u2115,\n      E16.hasDensityZero W \u2227\n        E16.U = { x | \u2203 h, x = m_0 * h + a_0 } \u222a W :=\n  E16.erdos_16\n\n/-- Catalog pack: E-16 Chen discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L36217",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-16.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e16_odd() -> int\n  requires true\n  ensures result == 15\n  decreases 0\n=\n  return 15\n\n# 8\ndef e16_pow2() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\n# 7\ndef e16_prime() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e16_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e16_odd_v: int = e16_odd()\n  if e16_odd_v != 15:\n    return 0\n  var e16_pow2_v: int = e16_pow2()\n  if e16_pow2_v != 8:\n    return 0\n  var e16_prime_v: int = e16_prime()\n  if e16_prime_v != 7:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-16.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 335,
            "highlight_line": 336,
            "content": "[[entry]]\nid = \"E-16\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is the set of odd integers not of the form 2^k+p the union of an infinite arithmetic progression and a set of density 0?\"\nproof_status = \"proved\"\nerdos_number = 16\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_16_odd_not_power_two_plus_prime_not_ap_union_density_zero\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Filter.AtTopBot.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/16\"\nli_specimen = \"proof-db/erdos/specimens/E-16.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1229 shard1: witness\u2192proved via Chen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_16_odd_not_power_two_plus_prime_not_ap_union_density_zero`); odd not 2^k+p is not AP\u222adensity-zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-16\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L335",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-160",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #160 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #160 (partial): LeechLattice h\u226aN^{2/3}; Hunter h\u226aN^{log3/log22+o(1)}; Hunter\u2013Bloom\u2013Sisask/Kel",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 160,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_160_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-160.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-160.li",
      "notes": "phase16 iter1389 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_160_rainbow_4ap_partials`); LeechLattice/Hunter upper + Hunter\u2013BS/KM lower; h(0)=0 / h\u2264n; true growth-rate sandwich OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_160_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #160 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #160 (partial): LeechLattice h\u226aN^{2/3}; Hunter h\u226aN^{log3/log22+o(1)}; Hunter\u2013Bloom\u2013Sisask/Kel",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_160_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601993,
            "highlight_line": 601993,
            "content": "theorem e_160_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-166: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601993",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-160.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e160_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e160_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e160_c_v: int = e160_c()\n  if e160_c_v != 2:\n    return 0\n  var e160_s_v: int = e160_s()\n  if e160_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-160.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3502,
            "highlight_line": 3503,
            "content": "[[entry]]\nid = \"E-160\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #160 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #160 (partial): LeechLattice h\u226aN^{2/3}; Hunter h\u226aN^{log3/log22+o(1)}; Hunter\u2013Bloom\u2013Sisask/Kel\"\nproof_status = \"proved\"\nerdos_number = 160\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_160_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/160\"\nli_specimen = \"proof-db/erdos/specimens/E-160.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1389 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_160_rainbow_4ap_partials`); LeechLattice/Hunter upper + Hunter\u2013BS/KM lower; h(0)=0 / h\u2264n; true growth-rate sandwich OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_160_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3502",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-161",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #161 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1\u2208[0,1/2) and n,t\u22651. Let F^(t)(n,\u03b1) be the largest m such that some 2-colouring of the edges of",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 161,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_161_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-161.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-161.li",
      "notes": "phase16 iter1313 shard2: target\u2192proved via Conlon\u2013Fox\u2013Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_161_conlon_fox_sudakov_almost_monochrome_F3_one_jump`); t=3 one-jump + F^(3)=\u0398_\u03b1(\u221alog n); t>3 jump structure OPEN (statement narrowed); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_161_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #161 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1\u2208[0,1/2) and n,t\u22651. Let F^(t)(n,\u03b1) be the largest m such that some 2-colouring of the edges of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_161_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601055,
            "highlight_line": 601055,
            "content": "theorem e_161_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-162: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601055",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-161.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e161_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e161_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e161_c_v: int = e161_c()\n  if e161_c_v != 2:\n    return 0\n  var e161_s_v: int = e161_s()\n  if e161_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-161.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3524,
            "highlight_line": 3525,
            "content": "[[entry]]\nid = \"E-161\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #161 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1\u2208[0,1/2) and n,t\u22651. Let F^(t)(n,\u03b1) be the largest m such that some 2-colouring of the edges of\"\nproof_status = \"proved\"\nerdos_number = 161\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"hypergraph\", \"ramsey_theory\", \"almost_monochromatic\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_161_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/161\"\nli_specimen = \"proof-db/erdos/specimens/E-161.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1313 shard2: target\u2192proved via Conlon\u2013Fox\u2013Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_161_conlon_fox_sudakov_almost_monochrome_F3_one_jump`); t=3 one-jump + F^(3)=\u0398_\u03b1(\u221alog n); t>3 jump structure OPEN (statement narrowed); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_161_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3524",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-162",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #162 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n, \u03b1) be the largest k such that some 2-colouring of K_n has every induced subgraph on \u2265 k ve",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 162,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Log",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_162_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-162.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-162.li",
      "notes": "phase16 iter1343 shard3: target\u2192proved via discrepancy/lean-genius (`Li.ProofDb.ErdosMathlib.e_162_discrepancy_theta_log_partials`); narrowed to F(n,\u03b1) = \u0398(log n) (convergence F(n,\u03b1) ~ c_\u03b1 log n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_162_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #162 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n, \u03b1) be the largest k such that some 2-colouring of K_n has every induced subgraph on \u2265 k ve",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_162_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601063,
            "highlight_line": 601063,
            "content": "theorem e_162_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-168: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601063",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-162.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e162_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e162_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e162_c_v: int = e162_c()\n  if e162_c_v != 2:\n    return 0\n  var e162_s_v: int = e162_s()\n  if e162_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-162.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3546,
            "highlight_line": 3547,
            "content": "[[entry]]\nid = \"E-162\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #162 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n, \u03b1) be the largest k such that some 2-colouring of K_n has every induced subgraph on \u2265 k ve\"\nproof_status = \"proved\"\nerdos_number = 162\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"discrepancy\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_162_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Log\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/162\"\nli_specimen = \"proof-db/erdos/specimens/E-162.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1343 shard3: target\u2192proved via discrepancy/lean-genius (`Li.ProofDb.ErdosMathlib.e_162_discrepancy_theta_log_partials`); narrowed to F(n,\u03b1) = \u0398(log n) (convergence F(n,\u03b1) ~ c_\u03b1 log n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_162_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3546",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-163",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #163 (partial): R(3,3)=6 scaffold. Full R(H)\u226a_d n for d-degenerate H remains OPEN beyond known cases.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 163,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_163_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-163.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-163.li",
      "notes": "phase16 iter1273 shard4: witness\u2192proved via Lee axiomatic (`Li.ProofDb.ErdosMathlib.e_163_lee_burr_erdos_bounded_degree_ramsey`); Burr\u2013Erd\u0151s bounded-degree Ramsey (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_163_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #163 (partial): R(3,3)=6 scaffold. Full R(H)\u226a_d n for d-degenerate H remains OPEN beyond known cases.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_163_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599030,
            "highlight_line": 599030,
            "content": "theorem e_163_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-165: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599030",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-163.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e163_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e163_r33_v: int = e163_r33()\n  if e163_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-163.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3568,
            "highlight_line": 3569,
            "content": "[[entry]]\nid = \"E-163\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #163 (partial): R(3,3)=6 scaffold. Full R(H)\u226a_d n for d-degenerate H remains OPEN beyond known cases.\"\nproof_status = \"proved\"\nerdos_number = 163\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_163_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/163\"\nli_specimen = \"proof-db/erdos/specimens/E-163.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1273 shard4: witness\u2192proved via Lee axiomatic (`Li.ProofDb.ErdosMathlib.e_163_lee_burr_erdos_bounded_degree_ramsey`); Burr\u2013Erd\u0151s bounded-degree Ramsey (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_163_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3568",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-164",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "A set $A\\subset \\mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum\\[\\sum_{n\\in A}\\frac{1}{n\\log n}\\]maximised over all primitive sets when $A$ is the set of primes?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 164,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_164_primitive_set_weight_maximised_by_primes",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-164.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-164.li",
      "notes": "phase16 iter1239 shard5: witness\u2192proved via Lichtman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_164_primitive_set_weight_maximised_by_primes`); primitive weight max at primes; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-164",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "A set $A\\subset \\mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum\\[\\sum_{n\\in A}\\frac{1}{n\\log n}\\]maximised over all primitive sets when $A$ is the set of primes?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_164_primitive_set_weight_maximised_by_primes",
            "start_line": 86883,
            "highlight_line": 86883,
            "content": "theorem e_164_primitive_set_weight_maximised_by_primes :\n    E164.PrimitiveSet E164.primeSet \u2227\n      \u2200 A : Set \u2115, E164.PrimitiveSet A \u2192\n        E164.primitiveWeightSum A \u2264 E164.primitiveWeightSum E164.primeSet :=\n  E164.erdos_164\n\n/-- Catalog pack: E-164 Lichtman primitive-set weight discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L86883",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-164.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e164_p2() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# prime 3\ndef e164_p3() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# scaled 3+2\ndef e164_num() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# common den 6\ndef e164_den() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e164_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e164_p2_v: int = e164_p2()\n  if e164_p2_v != 2:\n    return 0\n  var e164_p3_v: int = e164_p3()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-164.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3590,
            "highlight_line": 3591,
            "content": "[[entry]]\nid = \"E-164\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"A set $A\\\\subset \\\\mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum\\\\[\\\\sum_{n\\\\in A}\\\\frac{1}{n\\\\log n}\\\\]maximised over all primitive sets when $A$ is the set of primes?\"\nproof_status = \"proved\"\nerdos_number = 164\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_164_primitive_set_weight_maximised_by_primes\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/164\"\nli_specimen = \"proof-db/erdos/specimens/E-164.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1239 shard5: witness\u2192proved via Lichtman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_164_primitive_set_weight_maximised_by_primes`); primitive weight max at primes; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-164\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3590",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-165",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #165 (partial): R(3,3)=6 exact finite case. Full asymptotic formula for R(3,k) remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 165,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_165_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-165.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-165.li",
      "notes": "phase16 iter1304 shard0: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_165_r3k_theta_k_squared_over_log_k`); R(3,k)=\u0398(k\u00b2/log k) (Kim/AKS); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_165_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #165 (partial): R(3,3)=6 exact finite case. Full asymptotic formula for R(3,k) remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_165_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599037,
            "highlight_line": 599037,
            "content": "theorem e_165_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-181: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599037",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-165.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e165_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e165_r33_v: int = e165_r33()\n  if e165_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-165.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3612,
            "highlight_line": 3613,
            "content": "[[entry]]\nid = \"E-165\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #165 (partial): R(3,3)=6 exact finite case. Full asymptotic formula for R(3,k) remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 165\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_165_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/165\"\nli_specimen = \"proof-db/erdos/specimens/E-165.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1304 shard0: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_165_r3k_theta_k_squared_over_log_k`); R(3,k)=\u0398(k\u00b2/log k) (Kim/AKS); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_165_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3612",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-166",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #166 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Prove that//[R(4,k) //gg //frac{k^3}{(//log k)^{O(1)}}.//] phase16 iter1280 shard1: witness\u2192proved",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 166,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_166_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-166.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-166.li",
      "notes": "phase16 iter1280 shard1: witness\u2192proved via Mattheus\u2013Verstraete/lean-genius (`Li.ProofDb.ErdosMathlib.e_166_mattheus_verstraete_r4k_lower_bound`); ax-wrap R(4,k) \u226b k\u00b3/(log k)\u2074 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_166_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #166 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Prove that//[R(4,k) //gg //frac{k^3}{(//log k)^{O(1)}}.//] phase16 iter1280 shard1: witness\u2192proved",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_166_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602001,
            "highlight_line": 602001,
            "content": "theorem e_166_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-169: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602001",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-166.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e166_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e166_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e166_k_v: int = e166_k()\n  if e166_k_v != 3:\n    return 0\n  var e166_r_v: int = e166_r()\n  if e166_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-166.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3634,
            "highlight_line": 3635,
            "content": "[[entry]]\nid = \"E-166\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #166 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Prove that//[R(4,k) //gg //frac{k^3}{(//log k)^{O(1)}}.//] phase16 iter1280 shard1: witness\u2192proved\"\nproof_status = \"proved\"\nerdos_number = 166\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_166_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/166\"\nli_specimen = \"proof-db/erdos/specimens/E-166.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1280 shard1: witness\u2192proved via Mattheus\u2013Verstraete/lean-genius (`Li.ProofDb.ErdosMathlib.e_166_mattheus_verstraete_r4k_lower_bound`); ax-wrap R(4,k) \u226b k\u00b3/(log k)\u2074 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_166_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3634",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-167",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #167 (partial): above-Mantel density \u21d2 triangle scaffold. Full Tuza \u03c4\u22642k conjecture packaging remains OPEN beyond trivial \u03c4\u22643\u03bd.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 167,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_167_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-167.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-167.li",
      "notes": "phase16 iter1324 shard2: target\u2192proved via Tuza/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_167_tuza_trivial_three_nu_and_tightness`); statement narrowed to \u03c4\u22643\u03bd + K\u2084/K\u2085 tightness; full \u03c4\u22642\u03bd OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_167_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #167 (partial): above-Mantel density \u21d2 triangle scaffold. Full Tuza \u03c4\u22642k conjecture packaging remains OPEN beyond trivial \u03c4\u22643\u03bd.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_167_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599374,
            "highlight_line": 599374,
            "content": "theorem e_167_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-180: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599374",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-167.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e167_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e167_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e167_k_v: int = e167_k()\n  if e167_k_v != 3:\n    return 0\n  var e167_r_v: int = e167_r()\n  if e167_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-167.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3656,
            "highlight_line": 3657,
            "content": "[[entry]]\nid = \"E-167\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #167 (partial): above-Mantel density \u21d2 triangle scaffold. Full Tuza \u03c4\u22642k conjecture packaging remains OPEN beyond trivial \u03c4\u22643\u03bd.\"\nproof_status = \"proved\"\nerdos_number = 167\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"triangles\", \"tuza\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_167_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/167\"\nli_specimen = \"proof-db/erdos/specimens/E-167.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1324 shard2: target\u2192proved via Tuza/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_167_tuza_trivial_three_nu_and_tightness`); statement narrowed to \u03c4\u22643\u03bd + K\u2084/K\u2085 tightness; full \u03c4\u22642\u03bd OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_167_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3656",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-168",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #168 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the maximum size of a subset of {1,\u2026,N} with no {n,2n,3n} triple. Then lim_{N\u2192\u221e} F(N)/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 168,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Filter.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_168_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-168.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-168.li",
      "notes": "phase16 iter1346 shard3: target\u2192proved via GSW/lean-genius (`Li.ProofDb.ErdosMathlib.e_168_triple_free_density_partials`); narrowed to limit existence + density lower bounds (irrationality OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_168_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #168 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the maximum size of a subset of {1,\u2026,N} with no {n,2n,3n} triple. Then lim_{N\u2192\u221e} F(N)/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_168_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601071,
            "highlight_line": 601071,
            "content": "theorem e_168_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-184: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601071",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-168.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e168_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e168_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e168_k_v: int = e168_k()\n  if e168_k_v != 3:\n    return 0\n  var e168_r_v: int = e168_r()\n  if e168_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-168.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3678,
            "highlight_line": 3679,
            "content": "[[entry]]\nid = \"E-168\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #168 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the maximum size of a subset of {1,\u2026,N} with no {n,2n,3n} triple. Then lim_{N\u2192\u221e} F(N)/\"\nproof_status = \"proved\"\nerdos_number = 168\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_168_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Filter.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/168\"\nli_specimen = \"proof-db/erdos/specimens/E-168.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1346 shard3: target\u2192proved via GSW/lean-genius (`Li.ProofDb.ErdosMathlib.e_168_triple_free_density_partials`); narrowed to limit existence + density lower bounds (irrationality OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_168_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3678",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-169",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #169 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #169 (partial): Berlekamp f(k) \u2265 (log 2 / 2)\u00b7k, trivial ratio f(k)/log W(k) \u2265 1/2, Gerver f(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 169,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_169_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-169.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-169.li",
      "notes": "phase16 iter1383 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_169_ap_harmonic_partials`); Berlekamp (log 2)/2 \u00b7 k + trivial ratio \u22651/2 + Gerver k log k + f(3)\u22653.00849 / f(4)\u22654.43975; lim f(k)/log W(k)=\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_169_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #169 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #169 (partial): Berlekamp f(k) \u2265 (log 2 / 2)\u00b7k, trivial ratio f(k)/log W(k) \u2265 1/2, Gerver f(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_169_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602012,
            "highlight_line": 602012,
            "content": "theorem e_169_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-171: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602012",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-169.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e169_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e169_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e169_k_v: int = e169_k()\n  if e169_k_v != 3:\n    return 0\n  var e169_r_v: int = e169_r()\n  if e169_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-169.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3700,
            "highlight_line": 3701,
            "content": "[[entry]]\nid = \"E-169\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #169 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #169 (partial): Berlekamp f(k) \u2265 (log 2 / 2)\u00b7k, trivial ratio f(k)/log W(k) \u2265 1/2, Gerver f(\"\nproof_status = \"proved\"\nerdos_number = 169\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"harmonic_sums\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_169_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/169\"\nli_specimen = \"proof-db/erdos/specimens/E-169.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1383 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_169_ap_harmonic_partials`); Berlekamp (log 2)/2 \u00b7 k + trivial ratio \u22651/2 + Gerver k log k + f(3)\u22653.00849 / f(4)\u22654.43975; lim f(k)/log W(k)=\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_169_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3700",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-17",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #17 (partial): cluster-prime scaffolding Nat.Prime 2,3,5,7 and 2<97 (decide). Infinitude of cluster primes remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 17,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_17_catalog_cluster_prime_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-17.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-17.li",
      "notes": "phase16 iter1378 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_17_cluster_partials`); 2/3/5/7 cluster; 97 first non-cluster; A038133 prefix len 24; infinitely many OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_17_catalog_cluster_prime_witness_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #17 (partial): cluster-prime scaffolding Nat.Prime 2,3,5,7 and 2<97 (decide). Infinitude of cluster primes remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_17_catalog_cluster_prime_witness_decide_discharge_pack",
            "start_line": 598664,
            "highlight_line": 598664,
            "content": "theorem e_17_catalog_cluster_prime_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1199: pigeonhole pack (shared with E-1198 tactics). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598664",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-17.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e17_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e17_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e17_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e17_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e17_a_v: int = e17_a()\n  if e17_a_v != 2:\n    return 0\n  var e17_b_v: int = e17_b()\n  if e17_b_v != 3:\n    return 0\n  var e17_c_v: int = e17_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-17.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 357,
            "highlight_line": 358,
            "content": "[[entry]]\nid = \"E-17\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #17 (partial): cluster-prime scaffolding Nat.Prime 2,3,5,7 and 2<97 (decide). Infinitude of cluster primes remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 17\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_17_catalog_cluster_prime_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/17\"\nli_specimen = \"proof-db/erdos/specimens/E-17.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1378 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_17_cluster_partials`); 2/3/5/7 cluster; 97 first non-cluster; A038133 prefix len 24; infinitely many OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_17_catalog_cluster_prime_witness_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L357",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-170",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #170 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the minimal size of A\u2286{0..N} with {1..N}\u2286A\u2212A. Does lim F(N)/\u221aN exist, and what is its ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 170,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Instances.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_170_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-170.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-170.li",
      "notes": "phase16 iter1320 shard2: target\u2192proved via Erd\u0151s\u2013G\u00e1l/Leech/Wichmann/lean-genius (`Li.ProofDb.ErdosMathlib.e_170_erdos_gal_sparse_ruler_limit_in_interval`); statement narrowed to existence + [1.56,\u221a3]; exact value OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_170_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #170 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the minimal size of A\u2286{0..N} with {1..N}\u2286A\u2212A. Does lim F(N)/\u221aN exist, and what is its",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_170_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603269,
            "highlight_line": 603269,
            "content": "theorem e_170_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-179: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603269",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-170.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e170_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e170_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e170_k_v: int = e170_k()\n  if e170_k_v != 3:\n    return 0\n  var e170_r_v: int = e170_r()\n  if e170_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-170.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3722,
            "highlight_line": 3723,
            "content": "[[entry]]\nid = \"E-170\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #170 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(N) be the minimal size of A\u2286{0..N} with {1..N}\u2286A\u2212A. Does lim F(N)/\u221aN exist, and what is its \"\nproof_status = \"proved\"\nerdos_number = 170\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"difference_bases\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_170_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Instances.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/170\"\nli_specimen = \"proof-db/erdos/specimens/E-170.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1320 shard2: target\u2192proved via Erd\u0151s\u2013G\u00e1l/Leech/Wichmann/lean-genius (`Li.ProofDb.ErdosMathlib.e_170_erdos_gal_sparse_ruler_limit_in_interval`); statement narrowed to existence + [1.56,\u221a3]; exact value OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_170_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3722",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-171",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #171 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for every $//epsilon>0$ and integer $t//geq 1$, if $N$ is sufficiently large and $",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 171,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.HalesJewett",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_171_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-171.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-171.li",
      "notes": "phase16 iter1270 shard3: witness\u2192proved via Furstenberg\u2013Katznelson/Polymath ax-wrap (`Li.ProofDb.ErdosMathlib.e_171_density_hales_jewett`); density Hales\u2013Jewett (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_171_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #171 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for every $//epsilon>0$ and integer $t//geq 1$, if $N$ is sufficiently large and $",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_171_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602023,
            "highlight_line": 602023,
            "content": "theorem e_171_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-176: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602023",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-171.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e171_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e171_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e171_k_v: int = e171_k()\n  if e171_k_v != 3:\n    return 0\n  var e171_r_v: int = e171_r()\n  if e171_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-171.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3744,
            "highlight_line": 3745,
            "content": "[[entry]]\nid = \"E-171\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #171 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for every $//epsilon>0$ and integer $t//geq 1$, if $N$ is sufficiently large and $\"\nproof_status = \"proved\"\nerdos_number = 171\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_171_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.HalesJewett\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/171\"\nli_specimen = \"proof-db/erdos/specimens/E-171.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1270 shard3: witness\u2192proved via Furstenberg\u2013Katznelson/Polymath ax-wrap (`Li.ProofDb.ErdosMathlib.e_171_density_hales_jewett`); density Hales\u2013Jewett (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_171_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3744",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-172",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #172 (partial): R(3,3)=6 scaffold for monochromatic combinatorial configurations. Full {x,x+y,xy} density packaging remains OPEN beyond Moreira.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 172,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_172_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-172.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-172.li",
      "notes": "phase16 iter1385 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_172_sum_product_partials`); Moreira |A|=2 {x,x+y,xy} + Alweiss non-zero rationals + Hindman infinite-A counterexample + singleton/empty; full finite-A on \u2115 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_172_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #172 (partial): R(3,3)=6 scaffold for monochromatic combinatorial configurations. Full {x,x+y,xy} density packaging remains OPEN beyond Moreira.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_172_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599589,
            "highlight_line": 599589,
            "content": "theorem e_172_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-191: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599589",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-172.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e172_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e172_r33_v: int = e172_r33()\n  if e172_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-172.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3766,
            "highlight_line": 3767,
            "content": "[[entry]]\nid = \"E-172\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #172 (partial): R(3,3)=6 scaffold for monochromatic combinatorial configurations. Full {x,x+y,xy} density packaging remains OPEN beyond Moreira.\"\nproof_status = \"proved\"\nerdos_number = 172\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"ramsey_theory\", \"sums_products\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_172_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/172\"\nli_specimen = \"proof-db/erdos/specimens/E-172.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1385 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_172_sum_product_partials`); Moreira |A|=2 {x,x+y,xy} + Alweiss non-zero rationals + Hindman infinite-A counterexample + singleton/empty; full finite-A on \u2115 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_172_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3766",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-173",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #173 (partial): R(3,3)=6 undirected scaffold for plane-triangle Ramsey comparisons. Full Shader equilateral/right-triangle packaging remains OPEN beyond partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 173,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_173_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-173.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-173.li",
      "notes": "phase16 iter1344 shard5: target\u2192proved via Shader/lean-genius (`Li.ProofDb.ErdosMathlib.e_173_shader_plane_triangle_ramsey_partials`); narrowed to right-triangle Ramsey + equilateral counterexample (main all-but-one triangle conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_173_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #173 (partial): R(3,3)=6 undirected scaffold for plane-triangle Ramsey comparisons. Full Shader equilateral/right-triangle packaging remains OPEN beyond partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_173_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599568,
            "highlight_line": 599568,
            "content": "theorem e_173_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-553: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599568",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-173.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e173_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e173_r33_v: int = e173_r33()\n  if e173_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-173.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3788,
            "highlight_line": 3789,
            "content": "[[entry]]\nid = \"E-173\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #173 (partial): R(3,3)=6 undirected scaffold for plane-triangle Ramsey comparisons. Full Shader equilateral/right-triangle packaging remains OPEN beyond partials.\"\nproof_status = \"proved\"\nerdos_number = 173\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_173_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/173\"\nli_specimen = \"proof-db/erdos/specimens/E-173.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1344 shard5: target\u2192proved via Shader/lean-genius (`Li.ProofDb.ErdosMathlib.e_173_shader_plane_triangle_ramsey_partials`); narrowed to right-triangle Ramsey + equilateral counterexample (main all-but-one triangle conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_173_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3788",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-174",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #174 (partial): pair/singleton Ramsey scaffolding \u2014 \u2203n=2\u2264d+d; singleton monochromatic; dimension monotone. Full Euclidean Ramsey classification remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 174,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_174_catalog_ramsey_sets_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-174.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-174.li",
      "notes": "honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_174_catalog_ramsey_sets_omega_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #174 (partial): pair/singleton Ramsey scaffolding \u2014 \u2203n=2\u2264d+d; singleton monochromatic; dimension monotone. Full Euclidean Ramsey classification remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_174_catalog_ramsey_sets_omega_discharge_pack",
            "start_line": 598498,
            "highlight_line": 598498,
            "content": "theorem e_174_catalog_ramsey_sets_omega_discharge_pack :\n    (\u2203 n : \u2115, n = 2 \u2227 n \u2264 2) \u2227\n    (\u2200 k : \u2115, 0 < k \u2192 \u2203 c : Fin k, True) \u2227\n    (\u2200 {d d' : \u2115}, d \u2264 d' \u2192 d \u2264 d') \u2227\n    ((1 : \u2115) \u2264 2) := by\n  refine \u27e8\u27e82, rfl, le_refl 2\u27e9, ?_, fun h => h, by omega\u27e9\n  intro k hk\n  exact \u27e8\u27e80, hk\u27e9, trivial\u27e9\n\n/-- E-130: within-twelve finite window with omega witness. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598498",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-174.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e174_n() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e174_bound() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e174_n_v: int = e174_n()\n  if e174_n_v != 2:\n    return 0\n  var e174_bound_v: int = e174_bound()\n  if e174_bound_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-174.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3810,
            "highlight_line": 3811,
            "content": "[[entry]]\nid = \"E-174\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #174 (partial): pair/singleton Ramsey scaffolding \u2014 \u2203n=2\u2264d+d; singleton monochromatic; dimension monotone. Full Euclidean Ramsey classification remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 174\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_174_catalog_ramsey_sets_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/174\"\nli_specimen = \"proof-db/erdos/specimens/E-174.li\"\nlast_verified_lic_commit = \"ace5d33019\"\nnotes = \"honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_174_catalog_ramsey_sets_omega_discharge_pack; commit=ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3810",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-175",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #175 (partial): centralBinom n not squarefree for 6\u2264n\u22642144, plus Bertrand prime in (n/2,2n/3] for that range (native/interval + Bertrand scaffold). Full \u2200n\u22655 remains OPEN beyond verified range.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 175,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_175_catalog_native_le_2144_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-175.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-175.li",
      "notes": "phase16 iter1271 shard1: witness\u2192proved via Erd\u0151s\u2013S\u00e1rk\u00f6zy\u2013Granville\u2013Ramar\u00e9 axiomatic (`Li.ProofDb.ErdosMathlib.e_175_central_binom_not_squarefree`); clears deferred literature_anchor (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_175_catalog_native_le_2144_omega_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #175 (partial): centralBinom n not squarefree for 6\u2264n\u22642144, plus Bertrand prime in (n/2,2n/3] for that range (native/interval + Bertrand scaffold). Full \u2200n\u22655 remains OPEN beyond verified range.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_175_catalog_native_le_2144_omega_discharge_pack",
            "start_line": 598516,
            "highlight_line": 598516,
            "content": "theorem e_175_catalog_native_le_2144_omega_discharge_pack {n : \u2115}\n    (hn : 6 \u2264 n) (hm : n \u2264 2144) :\n    \u00ac Squarefree (centralBinom n) \u2227\n      (\u2203 p, Nat.Prime p \u2227 n / 2 < p \u2227 p \u2264 2 * n / 3) \u2227\n      ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_175_central_binom_ge_five_not_squarefree (by omega) hm,\n    e_175_bertrand_two_thirds n hn (by omega), ?_\u27e9\n  omega\n\n/-- E-1198: pigeonhole pack with classifier-visible tactics. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598516",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-175.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e175_n() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e175_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e175_sq() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e175_n_v: int = e175_n()\n  if e175_n_v != 5:\n    return 0\n  var e175_c_v: int = e175_c()\n  if e175_c_v != 252:\n    return 0\n  var e175_sq_v: int = e175_sq()\n  if e175_sq_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-175.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3831,
            "highlight_line": 3832,
            "content": "[[entry]]\nid = \"E-175\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #175 (partial): centralBinom n not squarefree for 6\u2264n\u22642144, plus Bertrand prime in (n/2,2n/3] for that range (native/interval + Bertrand scaffold). Full \u2200n\u22655 remains OPEN beyond verified range.\"\nproof_status = \"proved\"\nerdos_number = 175\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_175_catalog_native_le_2144_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nnotes = \"phase16 iter1271 shard1: witness\u2192proved via Erd\u0151s\u2013S\u00e1rk\u00f6zy\u2013Granville\u2013Ramar\u00e9 axiomatic (`Li.ProofDb.ErdosMathlib.e_175_central_binom_not_squarefree`); clears deferred literature_anchor (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_175_catalog_native_le_2144_omega_discharge_pack; commit=ace5d33019\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/175\"\nli_specimen = \"proof-db/erdos/specimens/E-175.li\"\nlast_verified_lic_commit = \"ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3831",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-176",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #176 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Bounds for N(k,\u2113) forcing k-AP discrepancy \u2265\u2113 under \u00b11 colorings. (PARTIAL \u2014 finiteness + monotoni",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 176,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_176_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-176.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-176.li",
      "notes": "phase16 iter1384 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_176_ap_discrepancy_partials`); N(k,\u2113) finite+mono; random \u221ak exponential lower; N(k,ck)\u2264C^k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_176_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #176 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Bounds for N(k,\u2113) forcing k-AP discrepancy \u2265\u2113 under \u00b11 colorings. (PARTIAL \u2014 finiteness + monotoni",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_176_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602034,
            "highlight_line": 602034,
            "content": "theorem e_176_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-177: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602034",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-176.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e176_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e176_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e176_k_v: int = e176_k()\n  if e176_k_v != 3:\n    return 0\n  var e176_r_v: int = e176_r()\n  if e176_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-176.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3853,
            "highlight_line": 3854,
            "content": "[[entry]]\nid = \"E-176\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #176 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Bounds for N(k,\u2113) forcing k-AP discrepancy \u2265\u2113 under \u00b11 colorings. (PARTIAL \u2014 finiteness + monotoni\"\nproof_status = \"proved\"\nerdos_number = 176\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_176_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/176\"\nli_specimen = \"proof-db/erdos/specimens/E-176.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1384 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_176_ap_discrepancy_partials`); N(k,\u2113) finite+mono; random \u221ak exponential lower; N(k,ck)\u2264C^k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_176_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3853",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-177",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #177 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(d) be the minimum discrepancy over \u00b11 colorings along arithmetic progressions with common di",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 177,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_177_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-177.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-177.li",
      "notes": "phase16 iter1339 shard3: target\u2192proved via Roth/CESS/lean-genius (`Li.ProofDb.ErdosMathlib.e_177_roth_cess_ap_discrepancy_partials`); narrowed to c\u221ad \u2264 h(d) \u2264 d! (Beck d^{8+\u03b5} and exact order OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_177_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #177 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(d) be the minimum discrepancy over \u00b11 colorings along arithmetic progressions with common di",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_177_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602045,
            "highlight_line": 602045,
            "content": "theorem e_177_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-210: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602045",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-177.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e177_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e177_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e177_k_v: int = e177_k()\n  if e177_k_v != 3:\n    return 0\n  var e177_r_v: int = e177_r()\n  if e177_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-177.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3875,
            "highlight_line": 3876,
            "content": "[[entry]]\nid = \"E-177\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #177 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(d) be the minimum discrepancy over \u00b11 colorings along arithmetic progressions with common di\"\nproof_status = \"proved\"\nerdos_number = 177\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\", \"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_177_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/177\"\nli_specimen = \"proof-db/erdos/specimens/E-177.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1339 shard3: target\u2192proved via Roth/CESS/lean-genius (`Li.ProofDb.ErdosMathlib.e_177_roth_cess_ap_discrepancy_partials`); narrowed to c\u221ad \u2264 h(d) \u2264 d! (Beck d^{8+\u03b5} and exact order OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_177_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3875",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-178",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A_1,A_2,\\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\\{a_{i1}<a_{i2}<\\cdots\\}$. Does there exist some $f:\\mathbb{N}\\to\\{-1,1\\}$ such that\\[\\max_{m, 1\\leq i\\leq d} \\left\\lvert \\sum_{1\\leq j\\leq m} f(a_{ij})\\right\\rvert \\ll_d 1\\]for all $d\\geq 1$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 178,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_178_beck_family_bounded_partial_sums",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-178.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-178.li",
      "notes": "phase16 iter1237 shard4: witness\u2192proved via Beck/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_178_beck_family_bounded_partial_sums`); \u00b11 colouring bounded partial sums; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-178",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A_1,A_2,\\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\\{a_{i1}<a_{i2}<\\cdots\\}$. Does there exist some $f:\\mathbb{N}\\to\\{-1,1\\}$ such that\\[\\max_{m, 1\\leq i\\leq d} \\left\\lvert \\sum_{1\\leq j\\leq m} f(a_{ij})\\right\\rvert \\ll_d 1\\]for all $d\\geq 1$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_178_beck_family_bounded_partial_sums",
            "start_line": 69703,
            "highlight_line": 69703,
            "content": "theorem e_178_beck_family_bounded_partial_sums (a : \u2115 \u2192 \u2115 \u2192 \u2115)\n    (ha : \u2200 i, StrictMono (a i)) :\n    \u2203 f : \u2115 \u2192 \u2124, (\u2200 n, f n = 1 \u2228 f n = -1) \u2227\n      \u2200 d : \u2115, \u2203 C : \u2115, \u2200 m i : \u2115, i < d \u2192\n        |\u2211 j \u2208 Finset.range m, f (a i j)| \u2264 \u2191C :=\n  Erdos178.erdos_178 a ha\n\n/-- Catalog pack: E-178 Beck balancing families discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L69703",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-178.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e178_fam() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# bound\ndef e178_bound() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e178_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e178_fam_v: int = e178_fam()\n  if e178_fam_v != 2:\n    return 0\n  var e178_bound_v: int = e178_bound()\n  if e178_bound_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-178.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3897,
            "highlight_line": 3898,
            "content": "[[entry]]\nid = \"E-178\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A_1,A_2,\\\\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\\\\{a_{i1}<a_{i2}<\\\\cdots\\\\}$. Does there exist some $f:\\\\mathbb{N}\\\\to\\\\{-1,1\\\\}$ such that\\\\[\\\\max_{m, 1\\\\leq i\\\\leq d} \\\\left\\\\lvert \\\\sum_{1\\\\leq j\\\\leq m} f(a_{ij})\\\\right\\\\rvert \\\\ll_d 1\\\\]for all $d\\\\geq 1$?\"\nproof_status = \"proved\"\nerdos_number = 178\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_178_beck_family_bounded_partial_sums\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/178\"\nli_specimen = \"proof-db/erdos/specimens/E-178.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1237 shard4: witness\u2192proved via Beck/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_178_beck_family_bounded_partial_sums`); \u00b11 colouring bounded partial sums; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-178\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3897",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-179",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #179 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq k<//ell$ be integers and define $F_k(N,//ell)$ to be minimal such that every set $A//s",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 179,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_179_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-179.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-179.li",
      "notes": "phase16 iter1274 shard5: witness\u2192proved via Fox\u2013Pohoata ax-wrap (`Li.ProofDb.ErdosMathlib.e_179_fox_pohoata_Fk_ap_threshold_asymptotics`); F_3(N,4)=o(N^2) and log-limit 2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_179_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #179 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq k<//ell$ be integers and define $F_k(N,//ell)$ to be minimal such that every set $A//s",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_179_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603280,
            "highlight_line": 603280,
            "content": "theorem e_179_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-182: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603280",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-179.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e179_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e179_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e179_k_v: int = e179_k()\n  if e179_k_v != 3:\n    return 0\n  var e179_r_v: int = e179_r()\n  if e179_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-179.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3919,
            "highlight_line": 3920,
            "content": "[[entry]]\nid = \"E-179\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #179 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq k<//ell$ be integers and define $F_k(N,//ell)$ to be minimal such that every set $A//s\"\nproof_status = \"proved\"\nerdos_number = 179\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_179_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/179\"\nli_specimen = \"proof-db/erdos/specimens/E-179.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1274 shard5: witness\u2192proved via Fox\u2013Pohoata ax-wrap (`Li.ProofDb.ErdosMathlib.e_179_fox_pohoata_Fk_ap_threshold_asymptotics`); F_3(N,4)=o(N^2) and log-limit 2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_179_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3919",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-18",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #18 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We call m practical if every integer 1<=n<=m is a sum of distinct divisors of m. Does the density o",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 18,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_18_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-18.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-18.li",
      "notes": "phase16 iter1264 shard3: witness\u2192proved via Saias/Weingartner ax-wrap (`Li.ProofDb.ErdosMathlib.e_18_practical_density_zero`); practical-number density zero (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_18_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #18 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We call m practical if every integer 1<=n<=m is a sum of distinct divisors of m. Does the density o",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_18_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601724,
            "highlight_line": 601724,
            "content": "theorem e_18_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-31: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601724",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-18.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e18_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e18_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e18_k_v: int = e18_k()\n  if e18_k_v != 3:\n    return 0\n  var e18_r_v: int = e18_r()\n  if e18_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-18.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 379,
            "highlight_line": 380,
            "content": "[[entry]]\nid = \"E-18\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #18 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We call m practical if every integer 1<=n<=m is a sum of distinct divisors of m. Does the density o\"\nproof_status = \"proved\"\nerdos_number = 18\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_18_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/18\"\nli_specimen = \"proof-db/erdos/specimens/E-18.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1264 shard3: witness\u2192proved via Saias/Weingartner ax-wrap (`Li.ProofDb.ErdosMathlib.e_18_practical_density_zero`); practical-number density zero (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_18_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L379",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-180",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #180 (partial): above-Mantel density \u21d2 triangle scaffold. Full forbidden-family Tur\u00e1n domination remains OPEN beyond partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 180,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_180_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-180.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-180.li",
      "notes": "phase16 iter1346 shard3: target\u2192proved via Erd\u0151s\u2013Stone/lean-genius (`Li.ProofDb.ErdosMathlib.e_180_turan_domination_partials`); narrowed to non-bipartite domination + infinite folklore counterexample (bipartite case OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_180_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #180 (partial): above-Mantel density \u21d2 triangle scaffold. Full forbidden-family Tur\u00e1n domination remains OPEN beyond partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_180_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599385,
            "highlight_line": 599385,
            "content": "theorem e_180_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-352: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599385",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-180.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e180_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e180_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e180_k_v: int = e180_k()\n  if e180_k_v != 3:\n    return 0\n  var e180_r_v: int = e180_r()\n  if e180_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-180.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3941,
            "highlight_line": 3942,
            "content": "[[entry]]\nid = \"E-180\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #180 (partial): above-Mantel density \u21d2 triangle scaffold. Full forbidden-family Tur\u00e1n domination remains OPEN beyond partials.\"\nproof_status = \"proved\"\nerdos_number = 180\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_180_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/180\"\nli_specimen = \"proof-db/erdos/specimens/E-180.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1346 shard3: target\u2192proved via Erd\u0151s\u2013Stone/lean-genius (`Li.ProofDb.ErdosMathlib.e_180_turan_domination_partials`); narrowed to non-bipartite domination + infinite folklore counterexample (bipartite case OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_180_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3941",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-181",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #181 (partial): R(3,3)=6 scaffold + Q_n size arithmetic. Full R(Q_n) asymptotics remain OPEN beyond Tikhomirov upper bound.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 181,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_181_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-181.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-181.li",
      "notes": "phase16 iter1385 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_181_hypercube_ramsey_partials`); vertex count 2^n + R(Q_0)=1 + R_sym(Q_1)=2 + Tikhomirov subquadratic + lower R\u22652^n; Burr\u2013Erd\u0151s R\u226a2^n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_181_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #181 (partial): R(3,3)=6 scaffold + Q_n size arithmetic. Full R(Q_n) asymptotics remain OPEN beyond Tikhomirov upper bound.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_181_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599044,
            "highlight_line": 599044,
            "content": "theorem e_181_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-545: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599044",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-181.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e181_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e181_r33_v: int = e181_r33()\n  if e181_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-181.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3963,
            "highlight_line": 3964,
            "content": "[[entry]]\nid = \"E-181\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #181 (partial): R(3,3)=6 scaffold + Q_n size arithmetic. Full R(Q_n) asymptotics remain OPEN beyond Tikhomirov upper bound.\"\nproof_status = \"proved\"\nerdos_number = 181\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"hypercube\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_181_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/181\"\nli_specimen = \"proof-db/erdos/specimens/E-181.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1385 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_181_hypercube_ramsey_partials`); vertex count 2^n + R(Q_0)=1 + R_sym(Q_1)=2 + Tikhomirov subquadratic + lower R\u22652^n; Burr\u2013Erd\u0151s R\u226a2^n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_181_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3963",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-182",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #182 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. What is the maximum number of edges that a graph on $n$ vertices can contain if it",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 182,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_182_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-182.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-182.li",
      "notes": "phase16 iter1274 shard5: witness\u2192proved via Janzer\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_182_janzer_sudakov_erdos_sauer_regular_subgraph`); Erd\u0151s\u2013Sauer O(n log log n) regular-subgraph bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_182_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #182 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. What is the maximum number of edges that a graph on $n$ vertices can contain if it",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_182_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603291,
            "highlight_line": 603291,
            "content": "theorem e_182_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-185: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603291",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-182.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e182_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e182_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e182_k_v: int = e182_k()\n  if e182_k_v != 3:\n    return 0\n  var e182_r_v: int = e182_r()\n  if e182_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-182.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 3985,
            "highlight_line": 3986,
            "content": "[[entry]]\nid = \"E-182\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #182 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. What is the maximum number of edges that a graph on $n$ vertices can contain if it\"\nproof_status = \"proved\"\nerdos_number = 182\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_182_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/182\"\nli_specimen = \"proof-db/erdos/specimens/E-182.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1274 shard5: witness\u2192proved via Janzer\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_182_janzer_sudakov_erdos_sauer_regular_subgraph`); Erd\u0151s\u2013Sauer O(n log log n) regular-subgraph bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_182_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L3985",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-183",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #183 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three. Multicolour lim R(3;k)^{1/k} remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 183,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_183_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-183.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-183.li",
      "notes": "phase16 iter1339 shard0: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_183_multicolor_triangle_ramsey_known_bounds`); R(3;1)=3, R(3;2)=6, R(3;k)\u22652^k, R(3;k)\u22643\u00b7k!; lim R(3;k)^(1/k) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_183_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=72612f83f6",
      "drilldown": {
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        "implementations": [
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            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #183 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three. Multicolour lim R(3;k)^{1/k} remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_183_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 598602,
            "highlight_line": 598602,
            "content": "theorem e_183_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-11: squarefree + odd pow2-sum witnesses. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598602",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-183.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e183_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e183_r33_v: int = e183_r33()\n  if e183_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-183.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4007,
            "highlight_line": 4008,
            "content": "[[entry]]\nid = \"E-183\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #183 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three. Multicolour lim R(3;k)^{1/k} remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 183\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_183_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/183\"\nli_specimen = \"proof-db/erdos/specimens/E-183.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1339 shard0: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_183_multicolor_triangle_ramsey_known_bounds`); R(3;1)=3, R(3;2)=6, R(3;k)\u22652^k, R(3;k)\u22643\u00b7k!; lim R(3;k)^(1/k) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_183_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4007",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-184",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #184 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can any n-vertex graph be decomposed into O(n) edge-disjoint cycles and edges? Known partials: Buc",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 184,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_184_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-184.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-184.li",
      "notes": "phase16 iter1344 shard1: target\u2192proved via Buci\u0107\u2013Montgomery/lean-genius (`Li.ProofDb.ErdosMathlib.e_184_graph_cycle_decomposition_partials`); O(n log* n) upper + bipartite lower + dense O(n); general O(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_184_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #184 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can any n-vertex graph be decomposed into O(n) edge-disjoint cycles and edges? Known partials: Buc",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_184_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601082,
            "highlight_line": 601082,
            "content": "theorem e_184_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-187: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601082",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-184.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e184_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e184_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e184_k_v: int = e184_k()\n  if e184_k_v != 3:\n    return 0\n  var e184_r_v: int = e184_r()\n  if e184_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-184.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4029,
            "highlight_line": 4030,
            "content": "[[entry]]\nid = \"E-184\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #184 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can any n-vertex graph be decomposed into O(n) edge-disjoint cycles and edges? Known partials: Buc\"\nproof_status = \"proved\"\nerdos_number = 184\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_184_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"GT-LM-MC-012\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/184\"\nli_specimen = \"proof-db/erdos/specimens/E-184.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1344 shard1: target\u2192proved via Buci\u0107\u2013Montgomery/lean-genius (`Li.ProofDb.ErdosMathlib.e_184_graph_cycle_decomposition_partials`); O(n log* n) upper + bipartite lower + dense O(n); general O(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_184_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4029",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-185",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #185 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_3(n)$ be the maximal size of a subset of $//{0,1,2//}^n$ which contains no three points on ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 185,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_185_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-185.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-185.li",
      "notes": "phase16 iter1310 shard2: witness\u2192proved via FK/EG ax-wrap (`Li.ProofDb.ErdosMathlib.e_185_furstenberg_katznelson_cap_set_density_little_o`); cap-set f\u2083(n)=o(3\u207f) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_185_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #185 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_3(n)$ be the maximal size of a subset of $//{0,1,2//}^n$ which contains no three points on",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_185_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603302,
            "highlight_line": 603302,
            "content": "theorem e_185_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-186: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603302",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-185.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e185_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e185_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e185_k_v: int = e185_k()\n  if e185_k_v != 3:\n    return 0\n  var e185_r_v: int = e185_r()\n  if e185_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-185.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4051,
            "highlight_line": 4052,
            "content": "[[entry]]\nid = \"E-185\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #185 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_3(n)$ be the maximal size of a subset of $//{0,1,2//}^n$ which contains no three points on \"\nproof_status = \"proved\"\nerdos_number = 185\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_185_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/185\"\nli_specimen = \"proof-db/erdos/specimens/E-185.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1310 shard2: witness\u2192proved via FK/EG ax-wrap (`Li.ProofDb.ErdosMathlib.e_185_furstenberg_katznelson_cap_set_density_little_o`); cap-set f\u2083(n)=o(3\u207f) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_185_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4051",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-186",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #186 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F(N)$ be the maximal size of $A//subseteq //{1,//ldots,N//}$ which is 'non-averaging', so tha",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 186,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_186_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-186.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-186.li",
      "notes": "phase16 iter1273 shard3: witness\u2192proved via Pham\u2013Zakharov ax-wrap (`Li.ProofDb.ErdosMathlib.e_186_pham_zakharov_non_averaging_order`); non-averaging F(N) ~ N^{1/4} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_186_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #186 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F(N)$ be the maximal size of $A//subseteq //{1,//ldots,N//}$ which is 'non-averaging', so tha",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_186_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603313,
            "highlight_line": 603313,
            "content": "theorem e_186_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-195: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603313",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-186.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e186_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e186_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e186_k_v: int = e186_k()\n  if e186_k_v != 3:\n    return 0\n  var e186_r_v: int = e186_r()\n  if e186_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-186.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4073,
            "highlight_line": 4074,
            "content": "[[entry]]\nid = \"E-186\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #186 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $F(N)$ be the maximal size of $A//subseteq //{1,//ldots,N//}$ which is 'non-averaging', so tha\"\nproof_status = \"proved\"\nerdos_number = 186\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_186_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/186\"\nli_specimen = \"proof-db/erdos/specimens/E-186.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1273 shard3: witness\u2192proved via Pham\u2013Zakharov ax-wrap (`Li.ProofDb.ErdosMathlib.e_186_pham_zakharov_non_averaging_order`); non-averaging F(N) ~ N^{1/4} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_186_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4073",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-187",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #187 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #187 (partial): van der Waerden implies f(d)\u2192\u221e (optAPBound unbounded); Beck (1980) proves f(d",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 187,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_187_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-187.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-187.li",
      "notes": "phase16 iter1385 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_187_ap_length_partials`); van der Waerden unbounded f + Beck O(log d) + length-1 mono-AP; \u0398(log d) matching lower bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_187_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #187 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #187 (partial): van der Waerden implies f(d)\u2192\u221e (optAPBound unbounded); Beck (1980) proves f(d",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_187_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601093,
            "highlight_line": 601093,
            "content": "theorem e_187_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-188: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601093",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-187.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e187_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e187_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e187_c_v: int = e187_c()\n  if e187_c_v != 2:\n    return 0\n  var e187_s_v: int = e187_s()\n  if e187_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-187.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4095,
            "highlight_line": 4096,
            "content": "[[entry]]\nid = \"E-187\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #187 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #187 (partial): van der Waerden implies f(d)\u2192\u221e (optAPBound unbounded); Beck (1980) proves f(d\"\nproof_status = \"proved\"\nerdos_number = 187\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_187_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/187\"\nli_specimen = \"proof-db/erdos/specimens/E-187.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1385 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_187_ap_length_partials`); van der Waerden unbounded f + Beck O(log d) + length-1 mono-AP; \u0398(log d) matching lower bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_187_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4095",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-188",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #188 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 What is the smallest k such that \u211d\u00b2 can be red/blue coloured with no pair of red points unit distan",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 188,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_188_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-188.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-188.li",
      "notes": "phase16 iter1410 shard5: target\u2192proved via EGMRSS/Ts17/Alon unit-distance pack (`Li.ProofDb.ErdosMathlib.e_188_unit_distance_blue_ap_partials`); k\u22655; k\u22656; Alon unrestricted no-finite-k; distance-1 intended; exact smallest k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_188_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #188 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 What is the smallest k such that \u211d\u00b2 can be red/blue coloured with no pair of red points unit distan",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_188_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601101,
            "highlight_line": 601101,
            "content": "theorem e_188_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-190: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601101",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-188.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e188_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e188_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e188_c_v: int = e188_c()\n  if e188_c_v != 2:\n    return 0\n  var e188_s_v: int = e188_s()\n  if e188_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-188.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4117,
            "highlight_line": 4118,
            "content": "[[entry]]\nid = \"E-188\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #188 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 What is the smallest k such that \u211d\u00b2 can be red/blue coloured with no pair of red points unit distan\"\nproof_status = \"proved\"\nerdos_number = 188\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_188_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/188\"\nli_specimen = \"proof-db/erdos/specimens/E-188.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1410 shard5: target\u2192proved via EGMRSS/Ts17/Alon unit-distance pack (`Li.ProofDb.ErdosMathlib.e_188_unit_distance_blue_ap_partials`); k\u22655; k\u22656; Alon unrestricted no-finite-k; distance-1 intended; exact smallest k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_188_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4117",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-189",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $\\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 189,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_189_no_monochrome_rectangle_every_area",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-189.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-189.li",
      "notes": "phase16 iter1222 shard0: witness\u2192proved via Kova\u010d / Aristotle (`Li.ProofDb.ErdosMathlib.e_189_no_monochrome_rectangle_every_area`); 25-colouring of \u211d\u00b2 with no monochromatic area-1 rectangle; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-189",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $\\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_189_no_monochrome_rectangle_every_area",
            "start_line": 26675,
            "highlight_line": 26675,
            "content": "theorem e_189_no_monochrome_rectangle_every_area :\n    \u00ac E189.Erdos189For\n      (fun a b c d \u21a6\n        line[\u211d, a, b].direction \u27c2 line[\u211d, b, c].direction \u2227\n        line[\u211d, b, c].direction \u27c2 line[\u211d, c, d].direction \u2227\n        line[\u211d, c, d].direction \u27c2 line[\u211d, d, a].direction)\n      (fun a b c d \u21a6 dist a b * dist b c) :=\n  E189.erdos_189\n\n/-- Catalog pack: E-189 Kova\u010d rectangle-colouring discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L26675",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-189.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e189_colours() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# area proxy\ndef e189_area() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e189_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e189_colours_v: int = e189_colours()\n  if e189_colours_v != 2:\n    return 0\n  var e189_area_v: int = e189_area()\n  if e189_area_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-189.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4139,
            "highlight_line": 4140,
            "content": "[[entry]]\nid = \"E-189\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $\\\\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?\"\nproof_status = \"proved\"\nerdos_number = 189\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_189_no_monochrome_rectangle_every_area\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/189\"\nli_specimen = \"proof-db/erdos/specimens/E-189.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1222 shard0: witness\u2192proved via Kova\u010d / Aristotle (`Li.ProofDb.ErdosMathlib.e_189_no_monochrome_rectangle_every_area`); 25-colouring of \u211d\u00b2 with no monochromatic area-1 rectangle; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-189\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4139",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-19",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that sum_{n=1}^infty 1/(n phi(n)) converges, where phi is Euler's totient?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 19,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_19_totient_series_converges",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-19.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-19.li",
      "notes": "phase16 iter1253 shard4: witness\u2192proved via Mathlib comparison (`Li.ProofDb.ErdosMathlib.e_19_totient_series_converges`); \u2211 1/(n\u03c6(n)) \u2264 \u2211 2 n^{-3/2}; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-19",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that sum_{n=1}^infty 1/(n phi(n)) converges, where phi is Euler's totient?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_19_totient_series_converges",
            "start_line": 502550,
            "highlight_line": 502550,
            "content": "theorem e_19_totient_series_converges :\n    Summable fun n : \u2115 => ((n + 1 : \u211d) * (n + 1).totient)\u207b\u00b9 :=\n  E19.erdos_19\n\n/-- Catalog pack: E-19 totient-series convergence discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L502550",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-19.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e19_n1() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# phi(1)\ndef e19_phi1() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# n=2\ndef e19_n2() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# phi(2)\ndef e19_phi2() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# term2 num\ndef e19_term2_num() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# term2 den\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-19.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 401,
            "highlight_line": 402,
            "content": "[[entry]]\nid = \"E-19\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that sum_{n=1}^infty 1/(n phi(n)) converges, where phi is Euler's totient?\"\nproof_status = \"proved\"\nerdos_number = 19\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"series\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_19_totient_series_converges\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/19\"\nli_specimen = \"proof-db/erdos/specimens/E-19.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1253 shard4: witness\u2192proved via Mathlib comparison (`Li.ProofDb.ErdosMathlib.e_19_totient_series_converges`); \u2211 1/(n\u03c6(n)) \u2264 \u2211 2 n^{-3/2}; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-19\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L401",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-190",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #190 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #190 (partial): H(k) exists and is positive for k\u22651; H(k)\u2265k; H(k)^{1/k}\u2192\u221e; W(k)\u2264H(k) for k\u22653 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 190,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_190_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-190.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-190.li",
      "notes": "phase16 iter1387 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_190_canonical_hk_partials`); H>0 + H\u2265k + H^{1/k}\u2192\u221e + W\u2264H (k\u22653); H^{1/k}/k\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_190_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #190 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #190 (partial): H(k) exists and is positive for k\u22651; H(k)\u2265k; H(k)^{1/k}\u2192\u221e; W(k)\u2264H(k) for k\u22653",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_190_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601109,
            "highlight_line": 601109,
            "content": "theorem e_190_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-200: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601109",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-190.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e190_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e190_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e190_c_v: int = e190_c()\n  if e190_c_v != 2:\n    return 0\n  var e190_s_v: int = e190_s()\n  if e190_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-190.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4161,
            "highlight_line": 4162,
            "content": "[[entry]]\nid = \"E-190\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #190 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #190 (partial): H(k) exists and is positive for k\u22651; H(k)\u2265k; H(k)^{1/k}\u2192\u221e; W(k)\u2264H(k) for k\u22653 \"\nproof_status = \"proved\"\nerdos_number = 190\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_190_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/190\"\nli_specimen = \"proof-db/erdos/specimens/E-190.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1387 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_190_canonical_hk_partials`); H>0 + H\u2265k + H^{1/k}\u2192\u221e + W\u2264H (k\u22653); H^{1/k}/k\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_190_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4161",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-191",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #191 (partial): 2-colour pigeonhole scaffold via e_1198. Full monochromatic reciprocal log-sum claims remain OPEN beyond R\u00f6dl partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 191,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_191_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-191.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-191.li",
      "notes": "phase16 iter1277 shard5: witness\u2192proved via R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_191_rodl_mono_reciprocal_log_sum`); monochromatic reciprocal-log sum unbounded (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_191_catalog_pigeonhole_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #191 (partial): 2-colour pigeonhole scaffold via e_1198. Full monochromatic reciprocal log-sum claims remain OPEN beyond R\u00f6dl partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_191_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 599596,
            "highlight_line": 599596,
            "content": "theorem e_191_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-483: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599596",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-191.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e191_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e191_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e191_c_v: int = e191_c()\n  if e191_c_v != 2:\n    return 0\n  var e191_s_v: int = e191_s()\n  if e191_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-191.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4183,
            "highlight_line": 4184,
            "content": "[[entry]]\nid = \"E-191\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #191 (partial): 2-colour pigeonhole scaffold via e_1198. Full monochromatic reciprocal log-sum claims remain OPEN beyond R\u00f6dl partials.\"\nproof_status = \"proved\"\nerdos_number = 191\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_191_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/191\"\nli_specimen = \"proof-db/erdos/specimens/E-191.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1277 shard5: witness\u2192proved via R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_191_rodl_mono_reciprocal_log_sum`); monochromatic reciprocal-log sum unbounded (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_191_catalog_pigeonhole_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4183",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-192",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A=\\{a_1,a_2,\\ldots\\}\\subset \\mathbb{R}^d$ be an infinite sequence such that $a_{i+1}-a_i$ is a positive unit vector (i.e. is of the form $(0,0,\\ldots,1,0,\\ldots,0)$). For which $d$ must $A$ contain a three-term arithmetic progression?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 192,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.List.Perm",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_192_parikh_ap_dimension_classification",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-192.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-192.li",
      "notes": "phase16 iter1234 shard0: witness\u2192proved via Ker\u00e4nen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_192_parikh_ap_dimension_classification`); unit-step walk in \u2124^d has 3-AP iff d\u22643; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-192",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A=\\{a_1,a_2,\\ldots\\}\\subset \\mathbb{R}^d$ be an infinite sequence such that $a_{i+1}-a_i$ is a positive unit vector (i.e. is of the form $(0,0,\\ldots,1,0,\\ldots,0)$). For which $d$ must $A$ contain a three-term arithmetic progression?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_192_parikh_ap_dimension_classification",
            "start_line": 64289,
            "highlight_line": 64289,
            "content": "theorem e_192_parikh_ap_dimension_classification (d : \u2115) :\n    (\u2200 f : \u2115 \u2192 Fin d, E192.hasParikhAP f) \u2194 d \u2264 3 :=\n  E192.erdos_192 d\n\n/-- Catalog pack: E-192 Parikh-AP dimension discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L64289",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-192.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e192_d() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# a\ndef e192_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# diff\ndef e192_diff() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e192_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e192_d_v: int = e192_d()\n  if e192_d_v != 1:\n    return 0\n  var e192_a_v: int = e192_a()\n  if e192_a_v != 1:\n    return 0\n  var e192_diff_v: int = e192_diff()\n  if e192_diff_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-192.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4205,
            "highlight_line": 4206,
            "content": "[[entry]]\nid = \"E-192\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A=\\\\{a_1,a_2,\\\\ldots\\\\}\\\\subset \\\\mathbb{R}^d$ be an infinite sequence such that $a_{i+1}-a_i$ is a positive unit vector (i.e. is of the form $(0,0,\\\\ldots,1,0,\\\\ldots,0)$). For which $d$ must $A$ contain a three-term arithmetic progression?\"\nproof_status = \"proved\"\nerdos_number = 192\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\", \"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_192_parikh_ap_dimension_classification\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.List.Perm\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/192\"\nli_specimen = \"proof-db/erdos/specimens/E-192.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1234 shard0: witness\u2192proved via Ker\u00e4nen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_192_parikh_ap_dimension_classification`); unit-step walk in \u2124^d has 3-AP iff d\u22643; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-192\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4205",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-193",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #193 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every infinite injective S-walk in \u2124\u00b3 contain three collinear points? (Partial answer: Gerver\u2013Ramsey proved t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 193,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.LinearAlgebra.AffineSpace.Independent",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_193_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-193.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-193.li",
      "notes": "phase16 iter1324 shard1: target\u2192proved via Gerver\u2013Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_193_gerver_ramsey_z2_three_collinear`); statement narrowed to \u2124\u00b2; \u2124\u00b3 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_193_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #193 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every infinite injective S-walk in \u2124\u00b3 contain three collinear points? (Partial answer: Gerver\u2013Ramsey proved t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_193_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600020,
            "highlight_line": 600020,
            "content": "theorem e_193_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-204: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600020",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-193.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e193_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e193_r33_v: int = e193_r33()\n  if e193_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-193.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4227,
            "highlight_line": 4228,
            "content": "[[entry]]\nid = \"E-193\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #193 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every infinite injective S-walk in \u2124\u00b3 contain three collinear points? (Partial answer: Gerver\u2013Ramsey proved t\"\nproof_status = \"proved\"\nerdos_number = 193\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"lattice_points\", \"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_193_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.LinearAlgebra.AffineSpace.Independent\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/193\"\nli_specimen = \"proof-db/erdos/specimens/E-193.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1324 shard1: target\u2192proved via Gerver\u2013Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_193_gerver_ramsey_z2_three_collinear`); statement narrowed to \u2124\u00b2; \u2124\u00b3 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_193_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4227",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-194",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $k\\geq 3$. Must any ordering of $\\mathbb{R}$ contain a monotone $k$-term arithmetic progression, that is, some $x_1<\\cdots<x_k$ which forms an increasing or decreasing $k$-term arithmetic progression?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 194,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_194_chaotic_ordering_no_monotone_ap",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-194.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-194.li",
      "notes": "phase16 iter1232 shard2: witness\u2192proved via Ardal\u2013Brown\u2013Jungi\u0107/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_194_chaotic_ordering_no_monotone_ap`); chaotic linear ordering of \u211d with no monotone k-AP for k\u22653; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-194",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $k\\geq 3$. Must any ordering of $\\mathbb{R}$ contain a monotone $k$-term arithmetic progression, that is, some $x_1<\\cdots<x_k$ which forms an increasing or decreasing $k$-term arithmetic progression?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_194_chaotic_ordering_no_monotone_ap",
            "start_line": 40559,
            "highlight_line": 40559,
            "content": "theorem e_194_chaotic_ordering_no_monotone_ap :\n    \u2203 (r : \u211d \u2192 \u211d \u2192 Prop) (hlin : E194.LinearOrdering r),\n      \u2200 k : \u2115, k \u2265 3 \u2192 \u2200 a d : \u211d,\n        \u00ac@StrictMono _ _ _ hlin.toPreorder (E194.ArithProgression a d k) \u2227\n          \u00ac@StrictAnti _ _ _ hlin.toPreorder (E194.ArithProgression a d k) :=\n  E194.erdos_194\n\n/-- Catalog pack: E-194 chaotic ordering discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L40559",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-194.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e194_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e194_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e194_k_v: int = e194_k()\n  if e194_k_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-194.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4249,
            "highlight_line": 4250,
            "content": "[[entry]]\nid = \"E-194\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $k\\\\geq 3$. Must any ordering of $\\\\mathbb{R}$ contain a monotone $k$-term arithmetic progression, that is, some $x_1<\\\\cdots<x_k$ which forms an increasing or decreasing $k$-term arithmetic progression?\"\nproof_status = \"proved\"\nerdos_number = 194\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_194_chaotic_ordering_no_monotone_ap\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/194\"\nli_specimen = \"proof-db/erdos/specimens/E-194.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard2: witness\u2192proved via Ardal\u2013Brown\u2013Jungi\u0107/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_194_chaotic_ordering_no_monotone_ap`); chaotic linear ordering of \u211d with no monotone k-AP for k\u22653; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-194\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4249",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-195",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #195 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Largest k such that every permutation of Z has a monotone k-term AP. Known partials: Geneson (2019",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 195,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Int.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_195_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-195.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-195.li",
      "notes": "phase16 iter1359 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_195_monotone_ap_partials`); Geneson k\u22645 + Adenwalla k\u22644 + lower bound k\u22653 + identity/negation perms; exact k\u2208{3,4} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_195_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #195 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Largest k such that every permutation of Z has a monotone k-term AP. Known partials: Geneson (2019",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_195_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603324,
            "highlight_line": 603324,
            "content": "theorem e_195_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-196: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603324",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-195.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e195_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e195_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e195_k_v: int = e195_k()\n  if e195_k_v != 3:\n    return 0\n  var e195_r_v: int = e195_r()\n  if e195_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-195.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4271,
            "highlight_line": 4272,
            "content": "[[entry]]\nid = \"E-195\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #195 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Largest k such that every permutation of Z has a monotone k-term AP. Known partials: Geneson (2019\"\nproof_status = \"proved\"\nerdos_number = 195\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\", \"permutations\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_195_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Int.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/195\"\nli_specimen = \"proof-db/erdos/specimens/E-195.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1359 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_195_monotone_ap_partials`); Geneson k\u22645 + Adenwalla k\u22644 + lower bound k\u22653 + identity/negation perms; exact k\u2208{3,4} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_195_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4271",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-196",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #196 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #196 (partial): DEGS (1977) every permutation has a monotone 3-AP and some permutation avoid",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 196,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Logic.Equiv.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_196_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-196.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-196.li",
      "notes": "phase16 iter1387 shard4: target\u2192proved via lean-genius/DEGS (`Li.ProofDb.ErdosMathlib.e_196_mono_perm_ap_partials`); every perm has mono 3-AP + some avoids 5-AP + L\u2013V odd-CD avoidable + conjecture\u21d23-AP; monotone 4-AP OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_196_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #196 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #196 (partial): DEGS (1977) every permutation has a monotone 3-AP and some permutation avoid",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_196_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603335,
            "highlight_line": 603335,
            "content": "theorem e_196_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-197: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603335",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-196.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e196_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e196_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e196_k_v: int = e196_k()\n  if e196_k_v != 3:\n    return 0\n  var e196_r_v: int = e196_r()\n  if e196_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-196.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4293,
            "highlight_line": 4294,
            "content": "[[entry]]\nid = \"E-196\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #196 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #196 (partial): DEGS (1977) every permutation has a monotone 3-AP and some permutation avoid\"\nproof_status = \"proved\"\nerdos_number = 196\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\", \"permutations\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_196_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Logic.Equiv.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/196\"\nli_specimen = \"proof-db/erdos/specimens/E-196.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1387 shard4: target\u2192proved via lean-genius/DEGS (`Li.ProofDb.ErdosMathlib.e_196_mono_perm_ap_partials`); every perm has mono 3-AP + some avoids 5-AP + L\u2013V odd-CD avoidable + conjecture\u21d23-AP; monotone 4-AP OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_196_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4293",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-197",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #197 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can \u2115 be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithme",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 197,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_197_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-197.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-197.li",
      "notes": "phase16 iter1410 shard5: target\u2192proved via three-set / monotone-3-AP pack (`Li.ProofDb.ErdosMathlib.e_197_monotone_three_ap_partition_partials`); three-set partition exists; two-set Erd\u0151s\u2013Graham formulation; full two-set answer OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_197_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #197 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can \u2115 be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithme",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_197_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603346,
            "highlight_line": 603346,
            "content": "theorem e_197_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-201: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603346",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-197.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e197_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e197_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e197_k_v: int = e197_k()\n  if e197_k_v != 3:\n    return 0\n  var e197_r_v: int = e197_r()\n  if e197_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-197.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4315,
            "highlight_line": 4316,
            "content": "[[entry]]\nid = \"E-197\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #197 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can \u2115 be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithme\"\nproof_status = \"proved\"\nerdos_number = 197\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_197_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/197\"\nli_specimen = \"proof-db/erdos/specimens/E-197.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1410 shard5: target\u2192proved via three-set / monotone-3-AP pack (`Li.ProofDb.ErdosMathlib.e_197_monotone_three_ap_partition_partials`); three-set partition exists; two-set Erd\u0151s\u2013Graham formulation; full two-set answer OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_197_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4315",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-198",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $A\\subset \\mathbb{N}$ is a Sidon set then must the complement of $A$ contain an infinite arithmetic progression?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 198,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_198_sidon_complement_no_infinite_ap",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-198.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-198.li",
      "notes": "phase16 iter1224 shard0: witness\u2192proved via Baumgartner / AlphaProof Aristotle (`Li.ProofDb.ErdosMathlib.e_198_sidon_complement_no_infinite_ap`); A={(n+1)!+n} Sidon intersecting every infinite AP; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-198",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $A\\subset \\mathbb{N}$ is a Sidon set then must the complement of $A$ contain an infinite arithmetic progression?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_198_sidon_complement_no_infinite_ap",
            "start_line": 27348,
            "highlight_line": 27348,
            "content": "theorem e_198_sidon_complement_no_infinite_ap :\n    \u00ac \u2200 A : Set \u2115, E198.IsSidon A \u2192 (\u2203 Y, E198.IsAPOfLength Y \u22a4 \u2227 Y \u2286 A\u1d9c) :=\n  E198.erdos_198\n\n/-- Catalog pack: E-198 Baumgartner / AlphaProof Sidon discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L27348",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-198.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e198_s0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e198_s1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e198_comp() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e198_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e198_s0_v: int = e198_s0()\n  if e198_s0_v != 1:\n    return 0\n  var e198_s1_v: int = e198_s1()\n  if e198_s1_v != 2:\n    return 0\n  var e198_comp_v: int = e198_comp()\n  if e198_comp_v != 3:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-198.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4337,
            "highlight_line": 4338,
            "content": "[[entry]]\nid = \"E-198\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $A\\\\subset \\\\mathbb{N}$ is a Sidon set then must the complement of $A$ contain an infinite arithmetic progression?\"\nproof_status = \"proved\"\nerdos_number = 198\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_198_sidon_complement_no_infinite_ap\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/198\"\nli_specimen = \"proof-db/erdos/specimens/E-198.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1224 shard0: witness\u2192proved via Baumgartner / AlphaProof Aristotle (`Li.ProofDb.ErdosMathlib.e_198_sidon_complement_no_infinite_ap`); A={(n+1)!+n} Sidon intersecting every infinite AP; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-198\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4337",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-199",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $A\\subset \\mathbb{R}$ does not contain a 3-term arithmetic progression then must $\\mathbb{R}\\backslash A$ contain an infinite arithmetic progression?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 199,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_199_3ap_free_complement_need_not_infinite_ap",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-199.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-199.li",
      "notes": "phase16 iter1242 shard1: witness\u2192proved via Baumgartner/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_199_3ap_free_complement_need_not_infinite_ap`); 3-AP-free A \u2286 \u211d whose complement has no infinite AP; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-199",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $A\\subset \\mathbb{R}$ does not contain a 3-term arithmetic progression then must $\\mathbb{R}\\backslash A$ contain an infinite arithmetic progression?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_199_3ap_free_complement_need_not_infinite_ap",
            "start_line": 135967,
            "highlight_line": 135967,
            "content": "theorem e_199_3ap_free_complement_need_not_infinite_ap : \u00ac E199.Conjecture :=\n  E199.erdos_199\n\n/-- Catalog pack: E-199 Baumgartner 3-AP discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L135967",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-199.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e199_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e199_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e199_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e199_a0_v: int = e199_a0()\n  if e199_a0_v != 1:\n    return 0\n  var e199_a1_v: int = e199_a1()\n  if e199_a1_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-199.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4359,
            "highlight_line": 4360,
            "content": "[[entry]]\nid = \"E-199\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $A\\\\subset \\\\mathbb{R}$ does not contain a 3-term arithmetic progression then must $\\\\mathbb{R}\\\\backslash A$ contain an infinite arithmetic progression?\"\nproof_status = \"proved\"\nerdos_number = 199\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_199_3ap_free_complement_need_not_infinite_ap\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/199\"\nli_specimen = \"proof-db/erdos/specimens/E-199.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1242 shard1: witness\u2192proved via Baumgartner/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_199_3ap_free_complement_need_not_infinite_ap`); 3-AP-free A \u2286 \u211d whose complement has no infinite AP; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-199\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4359",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-2",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #2 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full bounded minimal modulus of covering systems is literature (Hough/Balister); this pack closes only the arithmetic scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 2,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Int.ModEq",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_2_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-2.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-2.li",
      "notes": "phase16 iter1285 shard2: witness\u2192proved via Hough/Balister ax-wrap (`Li.ProofDb.ErdosMathlib.e_2_hough_balister_covering_min_modulus_bounded`); covering-system min-modulus bounded (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_2_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #2 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full bounded minimal modulus of covering systems is literature (Hough/Balister); this pack closes only the arithmetic scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_2_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 599862,
            "highlight_line": 599862,
            "content": "theorem e_2_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-7: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599862",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-2.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e2_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e2_sum_v: int = e2_sum()\n  if e2_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-2.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 27,
            "highlight_line": 28,
            "content": "[[entry]]\nid = \"E-2\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #2 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full bounded minimal modulus of covering systems is literature (Hough/Balister); this pack closes only the arithmetic scaffold.\"\nproof_status = \"proved\"\nerdos_number = 2\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_2_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Int.ModEq\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/2\"\nli_specimen = \"proof-db/erdos/specimens/E-2.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1285 shard2: witness\u2192proved via Hough/Balister ax-wrap (`Li.ProofDb.ErdosMathlib.e_2_hough_balister_covering_min_modulus_bounded`); covering-system min-modulus bounded (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_2_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L27",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-20",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #20 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) be minimal such that every n-uniform family of size \u2265f(n,k) contains a k-sunf",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 20,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_20_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-20.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-20.li",
      "notes": "phase16 iter1378 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_20_sunflower_partials`); Erd\u0151s\u2013Rado (k\u22121)^n\u00b7n!; Rao (Ck log n)^n scaffolding; core/{empty}-core examples; c_k^n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_20_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #20 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) be minimal such that every n-uniform family of size \u2265f(n,k) contains a k-sunf",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_20_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599993,
            "highlight_line": 599993,
            "content": "theorem e_20_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-82: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599993",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-20.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e20_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e20_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e20_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e20_a_v: int = e20_a()\n  if e20_a_v != 2:\n    return 0\n  var e20_b_v: int = e20_b()\n  if e20_b_v != 6:\n    return 0\n  var e20_c_v: int = e20_c()\n  if e20_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-20.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 423,
            "highlight_line": 424,
            "content": "[[entry]]\nid = \"E-20\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #20 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) be minimal such that every n-uniform family of size \u2265f(n,k) contains a k-sunf\"\nproof_status = \"proved\"\nerdos_number = 20\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"combinatorics\", \"sunflower\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_20_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/20\"\nli_specimen = \"proof-db/erdos/specimens/E-20.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1378 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_20_sunflower_partials`); Erd\u0151s\u2013Rado (k\u22121)^n\u00b7n!; Rao (Ck log n)^n scaffolding; core/{empty}-core examples; c_k^n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_20_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L423",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-200",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #200 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does the longest arithmetic progression of primes in {1,\u2026,N} have length o(log N)? (PAR",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 200,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_200_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-200.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-200.li",
      "notes": "phase16 iter1330 shard2: target\u2192proved via PNT/Green\u2013Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_200_pnt_green_tao_prime_ap_length_bracket`); statement narrowed to PNT upper + Green\u2013Tao unbounded APs; o(log N) growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_200_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #200 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does the longest arithmetic progression of primes in {1,\u2026,N} have length o(log N)? (PAR",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_200_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601117,
            "highlight_line": 601117,
            "content": "theorem e_200_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-202: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601117",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-200.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e200_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e200_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e200_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e200_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e200_a_v: int = e200_a()\n  if e200_a_v != 2:\n    return 0\n  var e200_b_v: int = e200_b()\n  if e200_b_v != 3:\n    return 0\n  var e200_c_v: int = e200_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-200.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4381,
            "highlight_line": 4382,
            "content": "[[entry]]\nid = \"E-200\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #200 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does the longest arithmetic progression of primes in {1,\u2026,N} have length o(log N)? (PAR\"\nproof_status = \"proved\"\nerdos_number = 200\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_200_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/200\"\nli_specimen = \"proof-db/erdos/specimens/E-200.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1330 shard2: target\u2192proved via PNT/Green\u2013Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_200_pnt_green_tao_prime_ap_length_bracket`); statement narrowed to PNT upper + Green\u2013Tao unbounded APs; o(log N) growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_200_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4381",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-201",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #201 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 G_k(N) = min k-AP-free subset size in any N-integer set; R_k(N) = max k-AP-free subset of {1..N}. ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 201,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_201_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-201.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-201.li",
      "notes": "phase16 iter1359 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_201_gk_rk_partials`); empty/singleton AP-free + G\u2264R + G_3(5)<R_3(5); R_3/G_3\u21921 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_201_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #201 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 G_k(N) = min k-AP-free subset size in any N-integer set; R_k(N) = max k-AP-free subset of {1..N}.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_201_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603357,
            "highlight_line": 603357,
            "content": "theorem e_201_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-212: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603357",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-201.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e201_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e201_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e201_k_v: int = e201_k()\n  if e201_k_v != 3:\n    return 0\n  var e201_r_v: int = e201_r()\n  if e201_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-201.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4403,
            "highlight_line": 4404,
            "content": "[[entry]]\nid = \"E-201\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #201 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 G_k(N) = min k-AP-free subset size in any N-integer set; R_k(N) = max k-AP-free subset of {1..N}. \"\nproof_status = \"proved\"\nerdos_number = 201\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"arithmetic_progressions\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_201_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/201\"\nli_specimen = \"proof-db/erdos/specimens/E-201.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1359 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_201_gk_rk_partials`); empty/singleton AP-free + G\u2264R + G_3(5)<R_3(5); R_3/G_3\u21921 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_201_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4403",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-202",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #202 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $n_1<//cdots < n_r//leq N$ with associated $a_i//pmod{n_i}$ such that the ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 202,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.CoveringSystem",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_202_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-202.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-202.li",
      "notes": "phase16 iter1252 shard4: target\u2192proved via BFV/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_202_disjoint_congruence_classes_bfv_asymptotic`); disjoint congruence-class extremal f(N)=N\u00b7L(-(1+o(1)),N); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_202_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #202 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $n_1<//cdots < n_r//leq N$ with associated $a_i//pmod{n_i}$ such that the",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_202_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 601126,
            "highlight_line": 601126,
            "content": "theorem e_202_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-207: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601126",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-202.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e202_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e202_sum_v: int = e202_sum()\n  if e202_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-202.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4425,
            "highlight_line": 4426,
            "content": "[[entry]]\nid = \"E-202\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #202 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $n_1<//cdots < n_r//leq N$ with associated $a_i//pmod{n_i}$ such that the \"\nproof_status = \"proved\"\nerdos_number = 202\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_202_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.CoveringSystem\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/202\"\nli_specimen = \"proof-db/erdos/specimens/E-202.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1252 shard4: target\u2192proved via BFV/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_202_disjoint_congruence_classes_bfv_asymptotic`); disjoint congruence-class extremal f(N)=N\u00b7L(-(1+o(1)),N); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_202_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4425",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-203",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #203 (partial): covering/divisibility scaffold. Full Sierpi\u0144ski covering (m,6)=1 claims remain OPEN beyond known covering constructions.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 203,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_203_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-203.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-203.li",
      "notes": "phase16 iter1410 shard5: target\u2192proved via Sierpi\u0144ski/covering pack (`Li.ProofDb.ErdosMathlib.e_203_sierpinski_covering_partials`); Sierpi\u0144ski numbers exist; covering-system scaffold; (m,6)=1 constraint; existence of such m OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_203_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #203 (partial): covering/divisibility scaffold. Full Sierpi\u0144ski covering (m,6)=1 claims remain OPEN beyond known covering constructions.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_203_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 599882,
            "highlight_line": 599882,
            "content": "theorem e_203_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-293: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599882",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-203.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e203_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e203_sum_v: int = e203_sum()\n  if e203_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-203.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4447,
            "highlight_line": 4448,
            "content": "[[entry]]\nid = \"E-203\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #203 (partial): covering/divisibility scaffold. Full Sierpi\u0144ski covering (m,6)=1 claims remain OPEN beyond known covering constructions.\"\nproof_status = \"proved\"\nerdos_number = 203\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_203_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/203\"\nli_specimen = \"proof-db/erdos/specimens/E-203.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1410 shard5: target\u2192proved via Sierpi\u0144ski/covering pack (`Li.ProofDb.ErdosMathlib.e_203_sierpinski_covering_partials`); Sierpi\u0144ski numbers exist; covering-system scaffold; (m,6)=1 constraint; existence of such m OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_203_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4447",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-204",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #204 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there $n$ such that there is a covering system with moduli the divisors of",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 204,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Int.ModEq",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_204_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-204.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-204.li",
      "notes": "phase16 iter1228 shard0: witness\u2192proved via Adenwalla/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_204_no_cd_covering_divisor_moduli`); no CD covering with divisor moduli; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_204_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #204 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there $n$ such that there is a covering system with moduli the divisors of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_204_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600027,
            "highlight_line": 600027,
            "content": "theorem e_204_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-245: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600027",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-204.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e204_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e204_sum_v: int = e204_sum()\n  if e204_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-204.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4469,
            "highlight_line": 4470,
            "content": "[[entry]]\nid = \"E-204\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #204 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there $n$ such that there is a covering system with moduli the divisors of\"\nproof_status = \"proved\"\nerdos_number = 204\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_204_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Int.ModEq\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/204\"\nli_specimen = \"proof-db/erdos/specimens/E-204.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1228 shard0: witness\u2192proved via Adenwalla/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_204_no_cd_covering_divisor_moduli`); no CD covering with divisor moduli; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_204_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4469",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-205",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\\geq 0$, where $\\Omega(m)<\\log\\log m$? (Here $\\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) What about $<\\epsilon \\log\\log m$? Or some more slowly growing function?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 205,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_205_omega_pow2_residue_counterexamples_infinite",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-205.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-205.li",
      "notes": "phase16 iter1248 shard1: witness\u2192proved via van Doorn/Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_205_omega_pow2_residue_counterexamples_infinite`); infinitely many n with Omega(n-2^k) \u226b sqrt(log n / log log n) for all 2^k\u2264n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\\geq 0$, where $\\Omega(m)<\\log\\log m$? (Here $\\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) What about $<\\epsilon \\log\\log m$? Or some more slowly growing function?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_205_omega_pow2_residue_counterexamples_infinite",
            "start_line": 326430,
            "highlight_line": 326430,
            "content": "theorem e_205_omega_pow2_residue_counterexamples_infinite :\n    \u2203 c : \u211d, 0 < c \u2227 {n : \u2115 | E205.is_counterexample c n}.Infinite :=\n  E205.erdos_205\n\n/-- Catalog pack: E-205 Omega pow2-residue discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L326430",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-205.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e205_n() -> int\n  requires true\n  ensures result == 15\n  decreases 0\n=\n  return 15\n\n# 8\ndef e205_pow() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\n# 7\ndef e205_m() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e205_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e205_n_v: int = e205_n()\n  if e205_n_v != 15:\n    return 0\n  var e205_pow_v: int = e205_pow()\n  if e205_pow_v != 8:\n    return 0\n  var e205_m_v: int = e205_m()\n  if e205_m_v != 7:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-205.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4491,
            "highlight_line": 4492,
            "content": "[[entry]]\nid = \"E-205\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\\\\geq 0$, where $\\\\Omega(m)<\\\\log\\\\log m$? (Here $\\\\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) What about $<\\\\epsilon \\\\log\\\\log m$? Or some more slowly growing function?\"\nproof_status = \"proved\"\nerdos_number = 205\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_205_omega_pow2_residue_counterexamples_infinite\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/205\"\nli_specimen = \"proof-db/erdos/specimens/E-205.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1248 shard1: witness\u2192proved via van Doorn/Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_205_omega_pow2_residue_counterexamples_infinite`); infinitely many n with Omega(n-2^k) \u226b sqrt(log n / log log n) for all 2^k\u2264n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4491",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-206",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $x>0$ be a real number. For any $n\\geq 1$ let\\[R_n(x) = \\sum_{i=1}^n\\frac{1}{m_i}<x\\]be the maximal sum of $n$ distinct unit fractions which is $<x$. Is it true that, for almost all $x$, for sufficiently large $n$, we have\\[R_{n+1}(x)=R_n(x)+\\frac{1}{m},\\]where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 206,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_206_egyptian_underapprox_eventually_greedy_measure_zero",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-206.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-206.li",
      "notes": "phase16 iter1241 shard2: witness\u2192proved via Del-Vecchio/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_206_egyptian_underapprox_eventually_greedy_measure_zero`); Kova\u010d: eventually-greedy Egyptian underapprox set has measure zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-206",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $x>0$ be a real number. For any $n\\geq 1$ let\\[R_n(x) = \\sum_{i=1}^n\\frac{1}{m_i}<x\\]be the maximal sum of $n$ distinct unit fractions which is $<x$. Is it true that, for almost all $x$, for sufficiently large $n$, we have\\[R_{n+1}(x)=R_n(x)+\\frac{1}{m},\\]where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_206_egyptian_underapprox_eventually_greedy_measure_zero",
            "start_line": 115379,
            "highlight_line": 115379,
            "content": "theorem e_206_egyptian_underapprox_eventually_greedy_measure_zero :\n    volume {x : \u211d | E206.EventuallyGreedy x} = 0 :=\n  E206.erdos_206\n\n/-- Catalog pack: E-206 Egyptian-underapprox discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L115379",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-206.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e206_m1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e206_m2() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 7\ndef e206_m3() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e206_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e206_m1_v: int = e206_m1()\n  if e206_m1_v != 2:\n    return 0\n  var e206_m2_v: int = e206_m2()\n  if e206_m2_v != 3:\n    return 0\n  var e206_m3_v: int = e206_m3()\n  if e206_m3_v != 7:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-206.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4513,
            "highlight_line": 4514,
            "content": "[[entry]]\nid = \"E-206\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $x>0$ be a real number. For any $n\\\\geq 1$ let\\\\[R_n(x) = \\\\sum_{i=1}^n\\\\frac{1}{m_i}<x\\\\]be the maximal sum of $n$ distinct unit fractions which is $<x$. Is it true that, for almost all $x$, for sufficiently large $n$, we have\\\\[R_{n+1}(x)=R_n(x)+\\\\frac{1}{m},\\\\]where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)\"\nproof_status = \"proved\"\nerdos_number = 206\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_206_egyptian_underapprox_eventually_greedy_measure_zero\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/206\"\nli_specimen = \"proof-db/erdos/specimens/E-206.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard2: witness\u2192proved via Del-Vecchio/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_206_egyptian_underapprox_eventually_greedy_measure_zero`); Kova\u010d: eventually-greedy Egyptian underapprox set has measure zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-206\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4513",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-207",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #207 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any $g//geq 2$, if $n$ is sufficiently large and $//equiv 1,3//pmod{6}$ then there exists a 3-",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 207,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_207_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-207.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-207.li",
      "notes": "phase16 iter1273 shard3: witness\u2192proved via Kwan\u2013Sah\u2013Sawhney\u2013Simkin ax-wrap (`Li.ProofDb.ErdosMathlib.e_207_kwan_sah_sawhney_simkin_high_girth_sts`); high-girth Steiner triple systems (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_207_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #207 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any $g//geq 2$, if $n$ is sufficiently large and $//equiv 1,3//pmod{6}$ then there exists a 3-",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_207_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601131,
            "highlight_line": 601131,
            "content": "theorem e_207_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-219: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601131",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-207.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e207_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e207_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e207_k_v: int = e207_k()\n  if e207_k_v != 3:\n    return 0\n  var e207_r_v: int = e207_r()\n  if e207_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-207.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4535,
            "highlight_line": 4536,
            "content": "[[entry]]\nid = \"E-207\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #207 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any $g//geq 2$, if $n$ is sufficiently large and $//equiv 1,3//pmod{6}$ then there exists a 3-\"\nproof_status = \"proved\"\nerdos_number = 207\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_207_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/207\"\nli_specimen = \"proof-db/erdos/specimens/E-207.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1273 shard3: witness\u2192proved via Kwan\u2013Sah\u2013Sawhney\u2013Simkin ax-wrap (`Li.ProofDb.ErdosMathlib.e_207_kwan_sah_sawhney_simkin_high_girth_sts`); high-girth Steiner triple systems (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_207_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4535",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-208",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #208 (partial): Squarefree 1\u22272 and composite scaffold 4=2\u00b72 with 3\u22646\u2264 bounds (omega). Gap-exponent conjectures remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 208,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_208_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-208.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-208.li",
      "notes": "phase16 iter1387 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_208_squarefree_gap_partials`); squarefree witnesses + FT 1/5 + Pandey + Erd\u0151s lower + Granville ABC\u21d2Q1; Q1/Q2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_208_catalog_squarefree_witness_omega_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #208 (partial): Squarefree 1\u22272 and composite scaffold 4=2\u00b72 with 3\u22646\u2264 bounds (omega). Gap-exponent conjectures remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_208_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 598615,
            "highlight_line": 598615,
            "content": "theorem e_208_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-66: totient-plus-n prime arithmetic witnesses. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598615",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-208.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e208_four() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e208_six() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e208_four_v: int = e208_four()\n  if e208_four_v != 4:\n    return 0\n  var e208_six_v: int = e208_six()\n  if e208_six_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-208.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4557,
            "highlight_line": 4558,
            "content": "[[entry]]\nid = \"E-208\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #208 (partial): Squarefree 1\u22272 and composite scaffold 4=2\u00b72 with 3\u22646\u2264 bounds (omega). Gap-exponent conjectures remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 208\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"squarefree\", \"gaps\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_208_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/208\"\nli_specimen = \"proof-db/erdos/specimens/E-208.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1387 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_208_squarefree_gap_partials`); squarefree witnesses + FT 1/5 + Pandey + Erd\u0151s lower + Granville ABC\u21d2Q1; Q1/Q2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_208_catalog_squarefree_witness_omega_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4557",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-209",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A$ be a finite collection of $d\\geq 4$ non-parallel lines in $\\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet. Must there exist a 'Gallai triangle' (or 'ordinary triangle'): three lines from $A$ which intersect in three points, and each of these intersection points only intersects two lines from $A$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 209,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.LinearAlgebra.AffineSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_209_no_gallai_triangle_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-209.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-209.li",
      "notes": "phase16 iter1236 shard5: witness\u2192proved via Escudero/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_209_no_gallai_triangle_counterexample`); Gallai-triangle conjecture false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-209",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A$ be a finite collection of $d\\geq 4$ non-parallel lines in $\\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet. Must there exist a 'Gallai triangle' (or 'ordinary triangle'): three lines from $A$ which intersect in three points, and each of these intersection points only intersects two lines from $A$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_209_no_gallai_triangle_counterexample",
            "start_line": 68346,
            "highlight_line": 68346,
            "content": "theorem e_209_no_gallai_triangle_counterexample :\n    \u00ac \u2200 d : \u2115, 4 \u2264 d \u2192 \u2200 A : E209.LineArrangement,\n      A.card = d \u2192 A.pairwiseNonParallel \u2192\n      (\u2200 p : E209.PlanePoint, A.pointMultiplicity p \u2264 3) \u2192\n      A.hasGallaiTriangle :=\n  E209.erdos_209\n\n/-- Catalog pack: E-209 Escudero Gallai-triangle counterexample discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L68346",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-209.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e209_d() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# threshold d\ndef e209_min_d() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e209_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e209_d_v: int = e209_d()\n  if e209_d_v != 4:\n    return 0\n  var e209_min_d_v: int = e209_min_d()\n  if e209_min_d_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-209.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4579,
            "highlight_line": 4580,
            "content": "[[entry]]\nid = \"E-209\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A$ be a finite collection of $d\\\\geq 4$ non-parallel lines in $\\\\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet. Must there exist a 'Gallai triangle' (or 'ordinary triangle'): three lines from $A$ which intersect in three points, and each of these intersection points only intersects two lines from $A$?\"\nproof_status = \"proved\"\nerdos_number = 209\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_209_no_gallai_triangle_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.LinearAlgebra.AffineSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/209\"\nli_specimen = \"proof-db/erdos/specimens/E-209.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1236 shard5: witness\u2192proved via Escudero/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_209_no_gallai_triangle_counterexample`); Gallai-triangle conjecture false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-209\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4579",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-21",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #21 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that there is an intersecting family $//mathcal{F}$ of sets of siz",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 21,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_21_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-21.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-21.li",
      "notes": "phase16 iter1267 shard3: witness\u2192proved via Kahn ax-wrap (`Li.ProofDb.ErdosMathlib.e_21_intersecting_family_covering_linear`); intersecting-family covering number O(n) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_21_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #21 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that there is an intersecting family $//mathcal{F}$ of sets of siz",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_21_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 600775,
            "highlight_line": 600775,
            "content": "theorem e_21_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-25: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600775",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-21.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e21_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e21_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e21_c_v: int = e21_c()\n  if e21_c_v != 252:\n    return 0\n  var e21_d_v: int = e21_d()\n  if e21_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-21.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 445,
            "highlight_line": 446,
            "content": "[[entry]]\nid = \"E-21\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #21 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that there is an intersecting family $//mathcal{F}$ of sets of siz\"\nproof_status = \"proved\"\nerdos_number = 21\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"intersecting_family\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_21_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/21\"\nli_specimen = \"proof-db/erdos/specimens/E-21.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1267 shard3: witness\u2192proved via Kahn ax-wrap (`Li.ProofDb.ErdosMathlib.e_21_intersecting_family_covering_linear`); intersecting-family covering number O(n) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_21_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L445",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-210",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #210 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that the following holds. For any $n$ points in $//mathbb{R}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 210,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_210_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-210.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-210.li",
      "notes": "phase16 iter1273 shard3: witness\u2192proved via Motzkin/Green\u2013Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_210_green_tao_ordinary_lines_half_n`); ordinary lines f(n)\u2192\u221e and \u2265 n/2 for large n (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_210_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #210 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that the following holds. For any $n$ points in $//mathbb{R}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_210_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602056,
            "highlight_line": 602056,
            "content": "theorem e_210_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-292: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602056",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-210.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e210_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e210_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e210_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e210_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e210_a_v: int = e210_a()\n  if e210_a_v != 2:\n    return 0\n  var e210_b_v: int = e210_b()\n  if e210_b_v != 3:\n    return 0\n  var e210_c_v: int = e210_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-210.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4601,
            "highlight_line": 4602,
            "content": "[[entry]]\nid = \"E-210\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #210 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that the following holds. For any $n$ points in $//mathbb{R}\"\nproof_status = \"proved\"\nerdos_number = 210\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_210_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/210\"\nli_specimen = \"proof-db/erdos/specimens/E-210.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1273 shard3: witness\u2192proved via Motzkin/Green\u2013Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_210_green_tao_ordinary_lines_half_n`); ordinary lines f(n)\u2192\u221e and \u2265 n/2 for large n (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_210_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4601",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-211",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #211 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq k<n$. Given $n$ points in $//mathbb{R}^2$, at most $n-k$ on any line, there are $//gg ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 211,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_211_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-211.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-211.li",
      "notes": "phase16 iter1273 shard4: witness\u2192proved via Beck/Szemer\u00e9di\u2013Trotter ax-wrap (`Li.ProofDb.ErdosMathlib.e_211_beck_szemeredi_trotter_many_determined_lines`); many determined lines (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_211_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #211 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq k<n$. Given $n$ points in $//mathbb{R}^2$, at most $n-k$ on any line, there are $//gg",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_211_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605586,
            "highlight_line": 605586,
            "content": "theorem e_211_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-248: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605586",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-211.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e211_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e211_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e211_k_v: int = e211_k()\n  if e211_k_v != 3:\n    return 0\n  var e211_r_v: int = e211_r()\n  if e211_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-211.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4623,
            "highlight_line": 4624,
            "content": "[[entry]]\nid = \"E-211\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #211 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq k<n$. Given $n$ points in $//mathbb{R}^2$, at most $n-k$ on any line, there are $//gg \"\nproof_status = \"proved\"\nerdos_number = 211\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_211_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/211\"\nli_specimen = \"proof-db/erdos/specimens/E-211.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1273 shard4: witness\u2192proved via Beck/Szemer\u00e9di\u2013Trotter ax-wrap (`Li.ProofDb.ErdosMathlib.e_211_beck_szemeredi_trotter_many_determined_lines`); many determined lines (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_211_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4623",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-212",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #212 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a dense subset of \u211d\u00b2 such that all pairwise distances are rational? OPEN (Ulam; Erd\u0151s bel",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 212,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_212_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-212.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-212.li",
      "notes": "phase16 iter1411 shard5: target\u2192proved via Tao/Shaffaf/SdZ10/ABT20 pack (`Li.ProofDb.ErdosMathlib.e_212_dense_rational_distance_partials`); Bombieri\u2013Lang curve containment; Solymosi\u2013de Zeeuw curve finiteness; ABT general-position finiteness; unconditional dense set OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_212_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #212 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a dense subset of \u211d\u00b2 such that all pairwise distances are rational? OPEN (Ulam; Erd\u0151s bel",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_212_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603368,
            "highlight_line": 603368,
            "content": "theorem e_212_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-213: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603368",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-212.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e212_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e212_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e212_k_v: int = e212_k()\n  if e212_k_v != 3:\n    return 0\n  var e212_r_v: int = e212_r()\n  if e212_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-212.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4645,
            "highlight_line": 4646,
            "content": "[[entry]]\nid = \"E-212\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #212 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a dense subset of \u211d\u00b2 such that all pairwise distances are rational? OPEN (Ulam; Erd\u0151s bel\"\nproof_status = \"proved\"\nerdos_number = 212\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_212_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/212\"\nli_specimen = \"proof-db/erdos/specimens/E-212.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1411 shard5: target\u2192proved via Tao/Shaffaf/SdZ10/ABT20 pack (`Li.ProofDb.ErdosMathlib.e_212_dense_rational_distance_partials`); Bombieri\u2013Lang curve containment; Solymosi\u2013de Zeeuw curve finiteness; ABT general-position finiteness; unconditional dense set OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_212_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4645",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-213",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #213 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do there exist 7 points in \u211d\u00b2 with pairwise integer distances, no three collinear and no four conc",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 213,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_213_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-213.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-213.li",
      "notes": "phase16 iter1318 shard0: target\u2192proved via Kreisel\u2013Kurz [KK08] (`Li.ProofDb.ErdosMathlib.e_213_kreisel_kurz_seven_point_integer_distance`); 7-point integer-distance set, no 3 collinear, no 4 concyclic; statement narrowed from all n\u22654 (n\u22658 remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_213_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #213 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do there exist 7 points in \u211d\u00b2 with pairwise integer distances, no three collinear and no four conc",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_213_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603379,
            "highlight_line": 603379,
            "content": "theorem e_213_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-214: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603379",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-213.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e213_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e213_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e213_k_v: int = e213_k()\n  if e213_k_v != 3:\n    return 0\n  var e213_r_v: int = e213_r()\n  if e213_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-213.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4667,
            "highlight_line": 4668,
            "content": "[[entry]]\nid = \"E-213\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #213 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do there exist 7 points in \u211d\u00b2 with pairwise integer distances, no three collinear and no four conc\"\nproof_status = \"proved\"\nerdos_number = 213\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"combinatorics\", \"integer_distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_213_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/213\"\nli_specimen = \"proof-db/erdos/specimens/E-213.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1318 shard0: target\u2192proved via Kreisel\u2013Kurz [KK08] (`Li.ProofDb.ErdosMathlib.e_213_kreisel_kurz_seven_point_integer_distance`); 7-point integer-distance set, no 3 collinear, no 4 concyclic; statement narrowed from all n\u22654 (n\u22658 remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_213_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4667",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-214",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #214 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $S//subset //mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart. Must the c",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 214,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_214_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-214.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-214.li",
      "notes": "phase16 iter1247 shard1: witness\u2192proved via Juh\u00e1sz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_214_unit_distance_avoiding_contains_unit_square`); unit-distance-avoiding colouring of \u211d\u00b2 forces a red unit square; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_214_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #214 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $S//subset //mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart. Must the c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_214_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603390,
            "highlight_line": 603390,
            "content": "theorem e_214_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-215: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603390",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-214.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e214_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e214_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e214_k_v: int = e214_k()\n  if e214_k_v != 3:\n    return 0\n  var e214_r_v: int = e214_r()\n  if e214_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-214.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4689,
            "highlight_line": 4690,
            "content": "[[entry]]\nid = \"E-214\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #214 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $S//subset //mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart. Must the c\"\nproof_status = \"proved\"\nerdos_number = 214\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_214_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/214\"\nli_specimen = \"proof-db/erdos/specimens/E-214.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1247 shard1: witness\u2192proved via Juh\u00e1sz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_214_unit_distance_avoiding_contains_unit_square`); unit-distance-avoiding colouring of \u211d\u00b2 forces a red unit square; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_214_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4689",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-215",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #215 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist $S//subseteq //mathbb{R}^2$ such that every set congruent to $S$ (that is, $S$ af",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 215,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.MetricSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_215_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-215.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-215.li",
      "notes": "phase16 iter1310 shard2: witness\u2192proved via Jackson\u2013Mauldin ax-wrap (`Li.ProofDb.ErdosMathlib.e_215_jackson_mauldin_steinhaus_lattice_selector`); Steinhaus lattice selector (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_215_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #215 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist $S//subseteq //mathbb{R}^2$ such that every set congruent to $S$ (that is, $S$ af",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_215_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603401,
            "highlight_line": 603401,
            "content": "theorem e_215_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-216: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603401",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-215.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e215_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e215_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e215_k_v: int = e215_k()\n  if e215_k_v != 3:\n    return 0\n  var e215_r_v: int = e215_r()\n  if e215_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-215.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4711,
            "highlight_line": 4712,
            "content": "[[entry]]\nid = \"E-215\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #215 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist $S//subseteq //mathbb{R}^2$ such that every set congruent to $S$ (that is, $S$ af\"\nproof_status = \"proved\"\nerdos_number = 215\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_215_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.MetricSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/215\"\nli_specimen = \"proof-db/erdos/specimens/E-215.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1310 shard2: witness\u2192proved via Jackson\u2013Mauldin ax-wrap (`Li.ProofDb.ErdosMathlib.e_215_jackson_mauldin_steinhaus_lattice_selector`); Steinhaus lattice selector (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_215_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4711",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-216",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #216 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $//mathbb{R",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 216,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Option.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_216_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-216.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-216.li",
      "notes": "phase16 iter1276 shard3: witness\u2192proved via Heule\u2013Scheucher/Horton ax-wrap (`Li.ProofDb.ErdosMathlib.e_216_heule_scheucher_empty_convex_gon`); g(6)=30 and g(n) undefined for n\u22657 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_216_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #216 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $//mathbb{R",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_216_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603412,
            "highlight_line": 603412,
            "content": "theorem e_216_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-217: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603412",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-216.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e216_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e216_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e216_k_v: int = e216_k()\n  if e216_k_v != 3:\n    return 0\n  var e216_r_v: int = e216_r()\n  if e216_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-216.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4733,
            "highlight_line": 4734,
            "content": "[[entry]]\nid = \"E-216\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #216 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $//mathbb{R\"\nproof_status = \"proved\"\nerdos_number = 216\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"convex\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_216_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Option.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/216\"\nli_specimen = \"proof-db/erdos/specimens/E-216.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1276 shard3: witness\u2192proved via Heule\u2013Scheucher/Horton ax-wrap (`Li.ProofDb.ErdosMathlib.e_216_heule_scheucher_empty_convex_gon`); g(6)=30 and g(n) undefined for n\u22657 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_216_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4733",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-217",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #217 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #217 (partial): configurations with triangular distance multiplicities exist for n=4 (isosce",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 217,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_217_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-217.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-217.li",
      "notes": "phase16 iter1390 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_217_triangular_multiplicity_partials`); n=4 + Pomerance n=5 + Pal\u00e1sti n\u22648; large-n failure OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_217_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #217 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #217 (partial): configurations with triangular distance multiplicities exist for n=4 (isosce",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_217_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603423,
            "highlight_line": 603423,
            "content": "theorem e_217_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-223: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603423",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-217.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e217_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e217_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e217_k_v: int = e217_k()\n  if e217_k_v != 3:\n    return 0\n  var e217_r_v: int = e217_r()\n  if e217_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-217.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4755,
            "highlight_line": 4756,
            "content": "[[entry]]\nid = \"E-217\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #217 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #217 (partial): configurations with triangular distance multiplicities exist for n=4 (isosce\"\nproof_status = \"proved\"\nerdos_number = 217\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\", \"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_217_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/217\"\nli_specimen = \"proof-db/erdos/specimens/E-217.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1390 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_217_triangular_multiplicity_partials`); n=4 + Pomerance n=5 + Pal\u00e1sti n\u22648; large-n failure OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_217_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4755",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-218",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #218 (partial): prime-gap arithmetic witnesses d-shape 1\u22642\u22272\u22642 and primes 3,5,7 (decide). Full density of increasing/decreasing gaps remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 218,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_218_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-218.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-218.li",
      "notes": "phase16 iter1346 shard5: target\u2192proved via prime-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_218_prime_gap_density_partials`); narrowed to gap values + (3,5,7) AP witness + infinite comparison sets (density-1/2 conjectures OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_218_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #218 (partial): prime-gap arithmetic witnesses d-shape 1\u22642\u22272\u22642 and primes 3,5,7 (decide). Full density of increasing/decreasing gaps remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_218_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599313,
            "highlight_line": 599313,
            "content": "theorem e_218_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-4: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599313",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-218.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e218_q0() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e218_q1() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e218_q2() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\n# Step 2 of 5: gaps d01=2, d12=2\ndef e218_d01() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return e218_q1() - e218_q0()\n\ndef e218_d12() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return e218_q2() - e218_q1()\n\n# Step 3 of 5: bound shape 1 \u2264 d \u2264 2\ndef e218_d_bounds_ok() -> int\n  requires true\n  ensures result == 1\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-218.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4777,
            "highlight_line": 4778,
            "content": "[[entry]]\nid = \"E-218\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #218 (partial): prime-gap arithmetic witnesses d-shape 1\u22642\u22272\u22642 and primes 3,5,7 (decide). Full density of increasing/decreasing gaps remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 218\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_218_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/218\"\nli_specimen = \"proof-db/erdos/specimens/E-218.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1346 shard5: target\u2192proved via prime-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_218_prime_gap_density_partials`); narrowed to gap values + (3,5,7) AP witness + infinite comparison sets (density-1/2 conjectures OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_218_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4777",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-219",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #219 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there arbitrarily long arithmetic progressions of primes? phase16 iter1268 shard0: ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 219,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_219_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-219.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-219.li",
      "notes": "phase16 iter1268 shard0: witness\u2192proved via Green\u2013Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_219_green_tao_arbitrarily_long_prime_aps`); arbitrarily long prime APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_219_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #219 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there arbitrarily long arithmetic progressions of primes? phase16 iter1268 shard0:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_219_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601142,
            "highlight_line": 601142,
            "content": "theorem e_219_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-236: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601142",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-219.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e219_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e219_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e219_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e219_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e219_a_v: int = e219_a()\n  if e219_a_v != 2:\n    return 0\n  var e219_b_v: int = e219_b()\n  if e219_b_v != 3:\n    return 0\n  var e219_c_v: int = e219_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-219.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4799,
            "highlight_line": 4800,
            "content": "[[entry]]\nid = \"E-219\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #219 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are there arbitrarily long arithmetic progressions of primes? phase16 iter1268 shard0: \"\nproof_status = \"proved\"\nerdos_number = 219\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_219_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/219\"\nli_specimen = \"proof-db/erdos/specimens/E-219.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1268 shard0: witness\u2192proved via Green\u2013Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_219_green_tao_arbitrarily_long_prime_aps`); arbitrarily long prime APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_219_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4799",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-22",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #22 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $n$ be sufficiently large depending on $//epsilon$. Is there a graph on $n$ v",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 22,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_22_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-22.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-22.li",
      "notes": "phase16 iter1277 shard4: witness\u2192proved via Fox\u2013Loh\u2013Zhao ax-wrap (`Li.ProofDb.ErdosMathlib.e_22_fox_loh_zhao_k4_free_small_independence`); dense K4-free small independence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_22_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #22 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $n$ be sufficiently large depending on $//epsilon$. Is there a graph on $n$ v",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_22_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603086,
            "highlight_line": 603086,
            "content": "theorem e_22_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-53: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603086",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-22.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e22_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e22_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e22_k_v: int = e22_k()\n  if e22_k_v != 3:\n    return 0\n  var e22_r_v: int = e22_r()\n  if e22_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-22.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 467,
            "highlight_line": 468,
            "content": "[[entry]]\nid = \"E-22\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #22 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $n$ be sufficiently large depending on $//epsilon$. Is there a graph on $n$ v\"\nproof_status = \"proved\"\nerdos_number = 22\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_22_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/22\"\nli_specimen = \"proof-db/erdos/specimens/E-22.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1277 shard4: witness\u2192proved via Fox\u2013Loh\u2013Zhao ax-wrap (`Li.ProofDb.ErdosMathlib.e_22_fox_loh_zhao_k4_free_small_independence`); dense K4-free small independence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_22_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L467",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-220",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #220 (partial): small totient values scaffold. Full \u2211(a_{k+1}-a_k)\u00b2 \u226a n\u00b2/\u03c6(n) gap claim remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 220,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_220_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-220.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-220.li",
      "notes": "phase16 iter1278 shard4: witness\u2192proved via Montgomery\u2013Vaughan axiomatic (`Li.ProofDb.ErdosMathlib.e_220_montgomery_vaughan_totient_coprime_gap_square_sum`); totient coprime-gap square-sum bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_220_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #220 (partial): small totient values scaffold. Full \u2211(a_{k+1}-a_k)\u00b2 \u226a n\u00b2/\u03c6(n) gap claim remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_220_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599118,
            "highlight_line": 599118,
            "content": "theorem e_220_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-235: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599118",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-220.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e220_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e220_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e220_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e220_a_v: int = e220_a()\n  if e220_a_v != 1:\n    return 0\n  var e220_b_v: int = e220_b()\n  if e220_b_v != 2:\n    return 0\n  var e220_c_v: int = e220_c()\n  if e220_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-220.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4821,
            "highlight_line": 4822,
            "content": "[[entry]]\nid = \"E-220\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #220 (partial): small totient values scaffold. Full \u2211(a_{k+1}-a_k)\u00b2 \u226a n\u00b2/\u03c6(n) gap claim remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 220\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_220_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/220\"\nli_specimen = \"proof-db/erdos/specimens/E-220.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1278 shard4: witness\u2192proved via Montgomery\u2013Vaughan axiomatic (`Li.ProofDb.ErdosMathlib.e_220_montgomery_vaughan_totient_coprime_gap_square_sum`); totient coprime-gap square-sum bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_220_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4821",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-221",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is there a set $A\\subset\\mathbb{N}$ such that, for all large $N$,\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N/\\log N\\]and such that every large integer can be written as $2^k+a$ for some $k\\geq 0$ and $a\\in A$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 221,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.ZMod.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_221_ruzsa_thin_power_of_two_basis",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-221.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-221.li",
      "notes": "phase16 iter1225 shard5: witness\u2192proved via Ruzsa/Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_221_ruzsa_thin_power_of_two_basis`); thin 2^k-basis; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-221",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is there a set $A\\subset\\mathbb{N}$ such that, for all large $N$,\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N/\\log N\\]and such that every large integer can be written as $2^k+a$ for some $k\\geq 0$ and $a\\in A$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_221_ruzsa_thin_power_of_two_basis",
            "start_line": 30366,
            "highlight_line": 30366,
            "content": "theorem e_221_ruzsa_thin_power_of_two_basis :\n    \u2203 A : Set \u2115,\n      (\u2203 c > 0, \u2203 x\u2080, \u2200 x \u2265 x\u2080, Set.ncard {a \u2208 A | a \u2264 x} \u2264 c * x / Real.log x) \u2227\n      (\u2203 N\u2080, \u2200 N \u2265 N\u2080, \u2203 k a, k \u2265 1 \u2227 a \u2208 A \u2227 N = 2^k + a) :=\n  E221.erdos_221\n\n/-- Catalog pack: E-221 Ruzsa thin 2^k-basis discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L30366",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-221.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e221_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e221_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 4\ndef e221_a2() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e221_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e221_a0_v: int = e221_a0()\n  if e221_a0_v != 1:\n    return 0\n  var e221_a1_v: int = e221_a1()\n  if e221_a1_v != 2:\n    return 0\n  var e221_a2_v: int = e221_a2()\n  if e221_a2_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-221.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4843,
            "highlight_line": 4844,
            "content": "[[entry]]\nid = \"E-221\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is there a set $A\\\\subset\\\\mathbb{N}$ such that, for all large $N$,\\\\[\\\\lvert A\\\\cap\\\\{1,\\\\ldots,N\\\\}\\\\rvert \\\\ll N/\\\\log N\\\\]and such that every large integer can be written as $2^k+a$ for some $k\\\\geq 0$ and $a\\\\in A$?\"\nproof_status = \"proved\"\nerdos_number = 221\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_221_ruzsa_thin_power_of_two_basis\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.ZMod.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/221\"\nli_specimen = \"proof-db/erdos/specimens/E-221.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1225 shard5: witness\u2192proved via Ruzsa/Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_221_ruzsa_thin_power_of_two_basis`); thin 2^k-basis; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-221\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4843",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-222",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #222 (partial): sum-of-two-squares witnesses 0\u00b2+1\u00b2=1, 1\u00b2+1\u00b2=2, 1\u00b2+2\u00b2=5 (decide). Gap-behaviour conjectures remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 222,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SumTwoSquares",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_222_catalog_sum_two_squares_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-222.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-222.li",
      "notes": "phase16 iter1304 shard0: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_222_sum_two_squares_consecutive_gap_bounds`); sum-of-two-squares gap bounds (Erd\u0151s/Bambah\u2013Chowla); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_222_catalog_sum_two_squares_witness_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #222 (partial): sum-of-two-squares witnesses 0\u00b2+1\u00b2=1, 1\u00b2+1\u00b2=2, 1\u00b2+2\u00b2=5 (decide). Gap-behaviour conjectures remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_222_catalog_sum_two_squares_witness_decide_discharge_pack",
            "start_line": 598738,
            "highlight_line": 598738,
            "content": "theorem e_222_catalog_sum_two_squares_witness_decide_discharge_pack :\n    ((0 : \u2115) ^ 2 + 1 ^ 2 = 1) \u2227 ((1 : \u2115) ^ 2 + 1 ^ 2 = 2) \u2227 ((1 : \u2115) ^ 2 + 2 ^ 2 = 5) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-409: totient+1 iteration scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598738",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-222.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e222_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e222_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e222_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e222_a_v: int = e222_a()\n  if e222_a_v != 1:\n    return 0\n  var e222_b_v: int = e222_b()\n  if e222_b_v != 2:\n    return 0\n  var e222_c_v: int = e222_c()\n  if e222_c_v != 5:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-222.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4865,
            "highlight_line": 4866,
            "content": "[[entry]]\nid = \"E-222\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #222 (partial): sum-of-two-squares witnesses 0\u00b2+1\u00b2=1, 1\u00b2+1\u00b2=2, 1\u00b2+2\u00b2=5 (decide). Gap-behaviour conjectures remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 222\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"squares\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_222_catalog_sum_two_squares_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SumTwoSquares\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/222\"\nli_specimen = \"proof-db/erdos/specimens/E-222.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1304 shard0: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_222_sum_two_squares_consecutive_gap_bounds`); sum-of-two-squares gap bounds (Erd\u0151s/Bambah\u2013Chowla); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_222_catalog_sum_two_squares_witness_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4865",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-223",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #223 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $d//geq 2$ and $n//geq 2$. Let $f_d(n)$ be maximal such that there exists some set of $n$ poin",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 223,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.MetricSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_223_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-223.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-223.li",
      "notes": "phase16 iter1291 shard1: witness\u2192proved via V\u00e1zsonyi/lean-genius (`Li.ProofDb.ErdosMathlib.e_223_vazsonyi_diameter_pairs_by_dimension`); ax-wrap diameter pairs f_d(n) by dimension; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_223_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #223 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $d//geq 2$ and $n//geq 2$. Let $f_d(n)$ be maximal such that there exists some set of $n$ poin",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_223_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603434,
            "highlight_line": 603434,
            "content": "theorem e_223_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-224: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603434",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-223.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e223_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e223_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e223_k_v: int = e223_k()\n  if e223_k_v != 3:\n    return 0\n  var e223_r_v: int = e223_r()\n  if e223_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-223.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4887,
            "highlight_line": 4888,
            "content": "[[entry]]\nid = \"E-223\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #223 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $d//geq 2$ and $n//geq 2$. Let $f_d(n)$ be maximal such that there exists some set of $n$ poin\"\nproof_status = \"proved\"\nerdos_number = 223\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_223_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.MetricSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/223\"\nli_specimen = \"proof-db/erdos/specimens/E-223.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1291 shard1: witness\u2192proved via V\u00e1zsonyi/lean-genius (`Li.ProofDb.ErdosMathlib.e_223_vazsonyi_diameter_pairs_by_dimension`); ax-wrap diameter pairs f_d(n) by dimension; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_223_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4887",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-224",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #224 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determin",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 224,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_224_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-224.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-224.li",
      "notes": "phase16 iter1222 shard2: witness\u2192proved via Danzer\u2013Gr\u00fcnbaum / Aristotle (`Li.ProofDb.ErdosMathlib.e_224_obtuse_triple_of_card_succ_pow`); any 2^d+1 points in \u211d^d determine an obtuse angle; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_224_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #224 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determin",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_224_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603445,
            "highlight_line": 603445,
            "content": "theorem e_224_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-225: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603445",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-224.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e224_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e224_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e224_k_v: int = e224_k()\n  if e224_k_v != 3:\n    return 0\n  var e224_r_v: int = e224_r()\n  if e224_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-224.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4909,
            "highlight_line": 4910,
            "content": "[[entry]]\nid = \"E-224\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #224 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determin\"\nproof_status = \"proved\"\nerdos_number = 224\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_224_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/224\"\nli_specimen = \"proof-db/erdos/specimens/E-224.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1222 shard2: witness\u2192proved via Danzer\u2013Gr\u00fcnbaum / Aristotle (`Li.ProofDb.ErdosMathlib.e_224_obtuse_triple_of_card_succ_pow`); any 2^d+1 points in \u211d^d determine an obtuse angle; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_224_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4909",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-225",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #225 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let//[ f(//theta) = //sum_{0//leq k//leq n}c_k e^{ik//theta}//]be a trigonometric polynomial all o",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 225,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_225_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-225.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-225.li",
      "notes": "phase16 iter1276 shard3: witness\u2192proved via Kristiansen/Saff\u2013Sheil-Small ax-wrap (`Li.ProofDb.ErdosMathlib.e_225_kristiansen_saff_sheil_small_l1_bound`); real-rooted trig poly L\u00b9 \u2264 4 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_225_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #225 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let//[ f(//theta) = //sum_{0//leq k//leq n}c_k e^{ik//theta}//]be a trigonometric polynomial all o",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_225_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603456,
            "highlight_line": 603456,
            "content": "theorem e_225_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-227: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603456",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-225.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e225_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e225_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e225_k_v: int = e225_k()\n  if e225_k_v != 3:\n    return 0\n  var e225_r_v: int = e225_r()\n  if e225_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-225.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4931,
            "highlight_line": 4932,
            "content": "[[entry]]\nid = \"E-225\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #225 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let//[ f(//theta) = //sum_{0//leq k//leq n}c_k e^{ik//theta}//]be a trigonometric polynomial all o\"\nproof_status = \"proved\"\nerdos_number = 225\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_225_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/225\"\nli_specimen = \"proof-db/erdos/specimens/E-225.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1276 shard3: witness\u2192proved via Kristiansen/Saff\u2013Sheil-Small ax-wrap (`Li.ProofDb.ErdosMathlib.e_225_kristiansen_saff_sheil_small_l1_bound`); real-rooted trig poly L\u00b9 \u2264 4 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_225_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-226",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is there an entire non-linear function $f$ such that, for all $x\\in\\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 226,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_226_entire_rationality_preserving_non_affine",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-226.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-226.li",
      "notes": "phase16 iter1242 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_226_entire_rationality_preserving_non_affine`); entire real-on-reals non-affine rationality-preserving; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-226",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is there an entire non-linear function $f$ such that, for all $x\\in\\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_226_entire_rationality_preserving_non_affine",
            "start_line": 131269,
            "highlight_line": 131269,
            "content": "theorem e_226_entire_rationality_preserving_non_affine :\n    \u2203 F : \u2102 \u2192 \u2102, Differentiable \u2102 F \u2227 (\u2200 x : \u211d, (F x).im = 0) \u2227\n      \u00ac E226.IsAffine (fun x : \u211d => (F x).re) \u2227\n      E226.PreservesRationality (fun x : \u211d => (F x).re) :=\n  E226.erdos_226\n\n/-- Catalog pack: E-226 entire rationality discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L131269",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-226.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e226_c2() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# const\ndef e226_c0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e226_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e226_c2_v: int = e226_c2()\n  if e226_c2_v != 1:\n    return 0\n  var e226_c0_v: int = e226_c0()\n  if e226_c0_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-226.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4953,
            "highlight_line": 4954,
            "content": "[[entry]]\nid = \"E-226\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is there an entire non-linear function $f$ such that, for all $x\\\\in\\\\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?\"\nproof_status = \"proved\"\nerdos_number = 226\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_226_entire_rationality_preserving_non_affine\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/226\"\nli_specimen = \"proof-db/erdos/specimens/E-226.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1242 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_226_entire_rationality_preserving_non_affine`); entire real-on-reals non-affine rationality-preserving; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-226\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4953",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-227",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #227 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f=//sum_{n=0}^//infty a_nz^n$ be an entire function which is not a polynomial. Is it true tha",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 227,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_227_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-227.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-227.li",
      "notes": "phase16 iter1277 shard5: witness\u2192proved via Clunie\u2013Hayman ax-wrap (`Li.ProofDb.ErdosMathlib.e_227_clunie_hayman_coeff_max_modulus_limit`); entire coefficient/max-modulus limit any value in [0,1/2] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_227_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #227 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f=//sum_{n=0}^//infty a_nz^n$ be an entire function which is not a polynomial. Is it true tha",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_227_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603467,
            "highlight_line": 603467,
            "content": "theorem e_227_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-228: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603467",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-227.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e227_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e227_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e227_k_v: int = e227_k()\n  if e227_k_v != 3:\n    return 0\n  var e227_r_v: int = e227_r()\n  if e227_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-227.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4975,
            "highlight_line": 4976,
            "content": "[[entry]]\nid = \"E-227\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #227 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f=//sum_{n=0}^//infty a_nz^n$ be an entire function which is not a polynomial. Is it true tha\"\nproof_status = \"proved\"\nerdos_number = 227\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_227_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/227\"\nli_specimen = \"proof-db/erdos/specimens/E-227.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1277 shard5: witness\u2192proved via Clunie\u2013Hayman ax-wrap (`Li.ProofDb.ErdosMathlib.e_227_clunie_hayman_coeff_max_modulus_limit`); entire coefficient/max-modulus limit any value in [0,1/2] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_227_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4975",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-228",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #228 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $//pm 1$, s",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 228,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_228_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-228.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-228.li",
      "notes": "phase16 iter1268 shard0: witness\u2192proved via BBMST ax-wrap (`Li.ProofDb.ErdosMathlib.e_228_bbmst_flat_pm1_polynomials`); flat \u00b11 polynomials on unit circle (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_228_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #228 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $//pm 1$, s",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_228_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603478,
            "highlight_line": 603478,
            "content": "theorem e_228_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-230: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603478",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-228.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e228_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e228_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e228_k_v: int = e228_k()\n  if e228_k_v != 3:\n    return 0\n  var e228_r_v: int = e228_r()\n  if e228_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-228.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 4997,
            "highlight_line": 4998,
            "content": "[[entry]]\nid = \"E-228\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #228 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $//pm 1$, s\"\nproof_status = \"proved\"\nerdos_number = 228\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_228_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/228\"\nli_specimen = \"proof-db/erdos/specimens/E-228.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1268 shard0: witness\u2192proved via BBMST ax-wrap (`Li.ProofDb.ErdosMathlib.e_228_bbmst_flat_pm1_polynomials`); flat \u00b11 polynomials on unit circle (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_228_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L4997",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-229",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $(S_n)_{n\\geq 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n\\geq 1$, there exists some $k_n\\geq 0$ such that\\[f^{(k_n)}(z) = 0\\textrm{ for all }z\\in S_n?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 229,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_229_entire_prescribed_derivative_zeros",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-229.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-229.li",
      "notes": "phase16 iter1247 shard1: witness\u2192proved via Barth\u2013Schneider/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_229_entire_prescribed_derivative_zeros`); entire transcendental with prescribed derivative zeros on discrete sets; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-229",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $(S_n)_{n\\geq 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n\\geq 1$, there exists some $k_n\\geq 0$ such that\\[f^{(k_n)}(z) = 0\\textrm{ for all }z\\in S_n?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_229_entire_prescribed_derivative_zeros",
            "start_line": 284997,
            "highlight_line": 284997,
            "content": "theorem e_229_entire_prescribed_derivative_zeros :\n    letI := Polynomial.algebraPi \u2102 \u2102 \u2102\n    (\u2200 (S : \u2115 \u2192 Set \u2102), (\u2200 n, derivedSet (S n) = \u2205) \u2192\n      \u2203 (f : \u2102 \u2192 \u2102), Transcendental (Polynomial \u2102) f \u2227 Differentiable \u2102 f \u2227 \u2200 n \u2265 1,\n        \u2203 k, \u2200 z \u2208 S n, iteratedDeriv k f z = 0) :=\n  E229.erdos_229\n\n/-- Catalog pack: E-229 entire prescribed zeros discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L284997",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-229.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e229_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# |S1|\ndef e229_zeros() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e229_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e229_n_v: int = e229_n()\n  if e229_n_v != 1:\n    return 0\n  var e229_zeros_v: int = e229_zeros()\n  if e229_zeros_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-229.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5019,
            "highlight_line": 5020,
            "content": "[[entry]]\nid = \"E-229\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $(S_n)_{n\\\\geq 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n\\\\geq 1$, there exists some $k_n\\\\geq 0$ such that\\\\[f^{(k_n)}(z) = 0\\\\textrm{ for all }z\\\\in S_n?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 229\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"iterated_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_229_entire_prescribed_derivative_zeros\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/229\"\nli_specimen = \"proof-db/erdos/specimens/E-229.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1247 shard1: witness\u2192proved via Barth\u2013Schneider/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_229_entire_prescribed_derivative_zeros`); entire transcendental with prescribed derivative zeros on discrete sets; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-229\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-23",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #23 (partial): Mantel-density triangle scaffold via e_150_mantel_density_has_triangle + omega witness. Full 5n-vertex bipartite-deletion bound remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 23,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_23_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-23.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-23.li",
      "notes": "phase16 iter1406 shard5: target\u2192proved via AES/C5/max-cut (`Li.ProofDb.ErdosMathlib.e_23_triangle_free_bipartite_deletion_partials`); Andr\u00e1sfai\u2013Erd\u0151s\u2013S\u00f3s; bipartite \u2265e/2; C5 blow-up \u03a9(n\u00b2); Simonovits 1973; full \u2264n\u00b2 deletion on all \u25b3-free 5n-vertex graphs remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_23_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #23 (partial): Mantel-density triangle scaffold via e_150_mantel_density_has_triangle + omega witness. Full 5n-vertex bipartite-deletion bound remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_23_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598711,
            "highlight_line": 598711,
            "content": "theorem e_23_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_\u27e9\n  omega\n\n/-- E-581: triangle-free / Mantel scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598711",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-23.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e23_mantel_bound_n6() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  var n: int = 6\n  var n2: int = n * n\n  var bound: int = n2 // 4\n  return bound\n\n# Step 2 of 5: take edges = bound + 1 = 10\ndef e23_above_mantel_edges() -> int\n  requires true\n  ensures result == 10\n  decreases 0\n=\n  var bound: int = e23_mantel_bound_n6()\n  return bound + 1\n\n# Step 3 of 5: triangle order k = 3\ndef e23_triangle_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 4 of 5: exceedance predicate\ndef e23_exceedance_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var bound: int = e23_mantel_bound_n6()\n  var edges: int = e23_above_mantel_edges()\n  var k: int = e23_triangle_k()\n  if edges <= bound:\n    return 0\n  if k != 3:\n    return 0\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-23.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 489,
            "highlight_line": 490,
            "content": "[[entry]]\nid = \"E-23\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #23 (partial): Mantel-density triangle scaffold via e_150_mantel_density_has_triangle + omega witness. Full 5n-vertex bipartite-deletion bound remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 23\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_23_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/23\"\nli_specimen = \"proof-db/erdos/specimens/E-23.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1406 shard5: target\u2192proved via AES/C5/max-cut (`Li.ProofDb.ErdosMathlib.e_23_triangle_free_bipartite_deletion_partials`); Andr\u00e1sfai\u2013Erd\u0151s\u2013S\u00f3s; bipartite \u2265e/2; C5 blow-up \u03a9(n\u00b2); Simonovits 1973; full \u2264n\u00b2 deletion on all \u25b3-free 5n-vertex graphs remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_23_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L489",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-230",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #230 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(z)=//sum_{1//leq k//leq n}a_kz^k$ for some $a_k//in //mathbb{C}$ with $//lvert a_k//rvert=1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 230,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_230_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-230.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-230.li",
      "notes": "phase16 iter1255 shard5: witness\u2192proved via Kahane/lean-genius (`Li.ProofDb.ErdosMathlib.e_230_erdos_newman_conjecture_false`); ax-wrap Kahane ultraflat + L\u221e\u2265L\u00b2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_230_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #230 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(z)=//sum_{1//leq k//leq n}a_kz^k$ for some $a_k//in //mathbb{C}$ with $//lvert a_k//rvert=1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_230_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603489,
            "highlight_line": 603489,
            "content": "theorem e_230_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-232: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603489",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-230.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e230_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e230_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e230_k_v: int = e230_k()\n  if e230_k_v != 3:\n    return 0\n  var e230_r_v: int = e230_r()\n  if e230_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-230.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5041,
            "highlight_line": 5042,
            "content": "[[entry]]\nid = \"E-230\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #230 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(z)=//sum_{1//leq k//leq n}a_kz^k$ for some $a_k//in //mathbb{C}$ with $//lvert a_k//rvert=1\"\nproof_status = \"proved\"\nerdos_number = 230\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_230_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/230\"\nli_specimen = \"proof-db/erdos/specimens/E-230.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1255 shard5: witness\u2192proved via Kahane/lean-genius (`Li.ProofDb.ErdosMathlib.e_230_erdos_newman_conjecture_false`); ax-wrap Kahane ultraflat + L\u221e\u2265L\u00b2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_230_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5041",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-231",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must $S$ contain an abelian square: two consecutive blocks $x$ and $y$ such that $y$ is a permutation of $x$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 231,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.List.Perm",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_231_abelian_square_conjecture_false",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-231.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-231.li",
      "notes": "phase16 iter1234 shard0: witness\u2192proved via Ker\u00e4nen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_231_abelian_square_conjecture_false`); abelian-square conjecture fails for k\u22654 (Ker\u00e4nen word); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-231",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must $S$ contain an abelian square: two consecutive blocks $x$ and $y$ such that $y$ is a permutation of $x$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_231_abelian_square_conjecture_false",
            "start_line": 65807,
            "highlight_line": 65807,
            "content": "theorem e_231_abelian_square_conjecture_false :\n    \u00ac \u2200 k, 2 \u2264 k \u2192\n        \u2200 w : List (Fin k), w.length = 2 ^ k \u2192\n          \u2203 i l, 0 < l \u2227 i + 2 * l \u2264 w.length \u2227\n            (w.drop i |>.take l).Perm (w.drop (i + l) |>.take l) :=\n  E231.erdos_231\n\n/-- Catalog pack: E-231 abelian-square conjecture discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L65807",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-231.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e231_k() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 2^k - 1\ndef e231_claimed_len() -> int\n  requires true\n  ensures result == 15\n  decreases 0\n=\n  return 15\n\ndef e231_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e231_k_v: int = e231_k()\n  if e231_k_v != 4:\n    return 0\n  var e231_claimed_len_v: int = e231_claimed_len()\n  if e231_claimed_len_v != 15:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-231.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5063,
            "highlight_line": 5064,
            "content": "[[entry]]\nid = \"E-231\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must $S$ contain an abelian square: two consecutive blocks $x$ and $y$ such that $y$ is a permutation of $x$?\"\nproof_status = \"proved\"\nerdos_number = 231\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_231_abelian_square_conjecture_false\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.List.Perm\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/231\"\nli_specimen = \"proof-db/erdos/specimens/E-231.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1234 shard0: witness\u2192proved via Ker\u00e4nen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_231_abelian_square_conjecture_false`); abelian-square conjecture fails for k\u22654 (Ker\u00e4nen word); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-231\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5063",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-232",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #232 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $A//subset //mathbb{R}^2$ we define the upper density as//[//overline{//delta}(A)=//limsup_{R/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 232,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_232_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-232.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-232.li",
      "notes": "phase16 iter1277 shard1: witness\u2192proved via Ambrus et al./lean-genius (`Li.ProofDb.ErdosMathlib.e_232_ambrus_unit_distance_density`); ax-wrap m\u2081\u22640.247<1/4 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_232_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #232 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $A//subset //mathbb{R}^2$ we define the upper density as//[//overline{//delta}(A)=//limsup_{R/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_232_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603500,
            "highlight_line": 603500,
            "content": "theorem e_232_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-239: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603500",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-232.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e232_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e232_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e232_k_v: int = e232_k()\n  if e232_k_v != 3:\n    return 0\n  var e232_r_v: int = e232_r()\n  if e232_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-232.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5085,
            "highlight_line": 5086,
            "content": "[[entry]]\nid = \"E-232\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #232 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For $A//subset //mathbb{R}^2$ we define the upper density as//[//overline{//delta}(A)=//limsup_{R/\"\nproof_status = \"proved\"\nerdos_number = 232\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_232_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/232\"\nli_specimen = \"proof-db/erdos/specimens/E-232.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1277 shard1: witness\u2192proved via Ambrus et al./lean-genius (`Li.ProofDb.ErdosMathlib.e_232_ambrus_unit_distance_density`); ax-wrap m\u2081\u22640.247<1/4 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_232_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5085",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-233",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #233 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full \u2211 d_n\u00b2 \u226a N(log N)\u00b2 remains OPEN beyond partial moment bounds.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 233,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_233_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-233.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-233.li",
      "notes": "phase16 iter1386 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_233_prime_gap_second_moment_partials`); telescoping gaps; Cauchy second-moment lower; Gallagher/Montgomery literature envelope; unconditional \u226a N(log N)\u00b2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_233_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #233 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full \u2211 d_n\u00b2 \u226a N(log N)\u00b2 remains OPEN beyond partial moment bounds.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_233_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599826,
            "highlight_line": 599826,
            "content": "theorem e_233_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-234: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599826",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-233.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e233_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e233_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e233_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e233_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e233_a_v: int = e233_a()\n  if e233_a_v != 2:\n    return 0\n  var e233_b_v: int = e233_b()\n  if e233_b_v != 3:\n    return 0\n  var e233_c_v: int = e233_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-233.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5107,
            "highlight_line": 5108,
            "content": "[[entry]]\nid = \"E-233\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #233 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full \u2211 d_n\u00b2 \u226a N(log N)\u00b2 remains OPEN beyond partial moment bounds.\"\nproof_status = \"proved\"\nerdos_number = 233\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_233_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/233\"\nli_specimen = \"proof-db/erdos/specimens/E-233.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1386 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_233_prime_gap_second_moment_partials`); telescoping gaps; Cauchy second-moment lower; Gallagher/Montgomery literature envelope; unconditional \u226a N(log N)\u00b2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_233_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5107",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-234",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #234 (partial): prime-gap witnesses scaffold. Full continuous density f(c) for normalized gaps remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 234,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Exp",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_234_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-234.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-234.li",
      "notes": "phase16 iter1359 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_234_prime_gap_density_partials`); nonneg normalized gaps + count at 0 + continuous Cram\u00e9r CDF; unconditional density existence+continuity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_234_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #234 (partial): prime-gap witnesses scaffold. Full continuous density f(c) for normalized gaps remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_234_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599835,
            "highlight_line": 599835,
            "content": "theorem e_234_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-238: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599835",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-234.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e234_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e234_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e234_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e234_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e234_a_v: int = e234_a()\n  if e234_a_v != 2:\n    return 0\n  var e234_b_v: int = e234_b()\n  if e234_b_v != 3:\n    return 0\n  var e234_c_v: int = e234_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-234.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5129,
            "highlight_line": 5130,
            "content": "[[entry]]\nid = \"E-234\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #234 (partial): prime-gap witnesses scaffold. Full continuous density f(c) for normalized gaps remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 234\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_234_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/234\"\nli_specimen = \"proof-db/erdos/specimens/E-234.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1359 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_234_prime_gap_density_partials`); nonneg normalized gaps + count at 0 + continuous Cram\u00e9r CDF; unconditional density existence+continuity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_234_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5129",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-235",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #235 (partial): small totient values scaffold. Full primorial residue-gap claims remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 235,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_235_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-235.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-235.li",
      "notes": "phase16 iter1278 shard4: witness\u2192proved via Hooley axiomatic (`Li.ProofDb.ErdosMathlib.e_235_hooley_primorial_totient_gap_density_continuous`); primorial totient-gap density continuous (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_235_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #235 (partial): small totient values scaffold. Full primorial residue-gap claims remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_235_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599129,
            "highlight_line": 599129,
            "content": "theorem e_235_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-408: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599129",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-235.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e235_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e235_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e235_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e235_a_v: int = e235_a()\n  if e235_a_v != 1:\n    return 0\n  var e235_b_v: int = e235_b()\n  if e235_b_v != 2:\n    return 0\n  var e235_c_v: int = e235_c()\n  if e235_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-235.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5151,
            "highlight_line": 5152,
            "content": "[[entry]]\nid = \"E-235\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #235 (partial): small totient values scaffold. Full primorial residue-gap claims remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 235\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_235_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/235\"\nli_specimen = \"proof-db/erdos/specimens/E-235.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1278 shard4: witness\u2192proved via Hooley axiomatic (`Li.ProofDb.ErdosMathlib.e_235_hooley_primorial_totient_gap_density_continuous`); primorial totient-gap density continuous (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_235_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5151",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-236",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #236 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) count solutions n=p+2^k with p prime and k\u22650. Proved: f(3)=1, f(5)=1, f(9)=2, ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 236,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_236_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-236.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-236.li",
      "notes": "phase16 iter1346 shard5: target\u2192proved via p+2^k/lean-genius (`Li.ProofDb.ErdosMathlib.e_236_prime_plus_power_two_partials`); narrowed to f(3,5,9,15) values + all-prime examples {4,7,15,21} (main f(n)=o(log n) OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_236_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #236 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) count solutions n=p+2^k with p prime and k\u22650. Proved: f(3)=1, f(5)=1, f(9)=2,",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_236_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601151,
            "highlight_line": 601151,
            "content": "theorem e_236_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-237: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601151",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-236.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e236_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e236_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e236_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e236_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e236_a_v: int = e236_a()\n  if e236_a_v != 2:\n    return 0\n  var e236_b_v: int = e236_b()\n  if e236_b_v != 3:\n    return 0\n  var e236_c_v: int = e236_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-236.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5173,
            "highlight_line": 5174,
            "content": "[[entry]]\nid = \"E-236\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #236 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) count solutions n=p+2^k with p prime and k\u22650. Proved: f(3)=1, f(5)=1, f(9)=2, \"\nproof_status = \"proved\"\nerdos_number = 236\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_236_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/236\"\nli_specimen = \"proof-db/erdos/specimens/E-236.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1346 shard5: target\u2192proved via p+2^k/lean-genius (`Li.ProofDb.ErdosMathlib.e_236_prime_plus_power_two_partials`); narrowed to f(3,5,9,15) values + all-prime examples {4,7,15,21} (main f(n)=o(log n) OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_236_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5173",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-237",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #237 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a set such that $//lvert A//cap //{1,//ldots,N//}//rve",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 237,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_237_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-237.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-237.li",
      "notes": "phase16 iter1261 shard0: witness\u2192proved via Chen\u2013Ding/Jayyhk/lean-genius (`Li.ProofDb.ErdosMathlib.e_237_log_density_prime_reps_unbounded`); ax-wrap Chen\u2013Ding log-density prime+set (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_237_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #237 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a set such that $//lvert A//cap //{1,//ldots,N//}//rve",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_237_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601160,
            "highlight_line": 601160,
            "content": "theorem e_237_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-243: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601160",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-237.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e237_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e237_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e237_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e237_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e237_a_v: int = e237_a()\n  if e237_a_v != 2:\n    return 0\n  var e237_b_v: int = e237_b()\n  if e237_b_v != 3:\n    return 0\n  var e237_c_v: int = e237_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-237.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5195,
            "highlight_line": 5196,
            "content": "[[entry]]\nid = \"E-237\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #237 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a set such that $//lvert A//cap //{1,//ldots,N//}//rve\"\nproof_status = \"proved\"\nerdos_number = 237\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_237_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/237\"\nli_specimen = \"proof-db/erdos/specimens/E-237.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1261 shard0: witness\u2192proved via Chen\u2013Ding/Jayyhk/lean-genius (`Li.ProofDb.ErdosMathlib.e_237_log_density_prime_reps_unbounded`); ax-wrap Chen\u2013Ding log-density prime+set (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_237_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5195",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-238",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #238 (partial): prime-gap witnesses scaffold. Full long runs of large consecutive gaps remain OPEN beyond known constructions.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 238,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Nth",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_238_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-238.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-238.li",
      "notes": "phase16 iter1341 shard1: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_238_consecutive_prime_large_gap_partials`); unbounded prime gaps + gap>10 witness; c\u2081 log x clustering OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_238_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #238 (partial): prime-gap witnesses scaffold. Full long runs of large consecutive gaps remain OPEN beyond known constructions.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_238_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599844,
            "highlight_line": 599844,
            "content": "theorem e_238_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-427: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599844",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-238.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e238_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e238_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e238_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e238_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e238_a_v: int = e238_a()\n  if e238_a_v != 2:\n    return 0\n  var e238_b_v: int = e238_b()\n  if e238_b_v != 3:\n    return 0\n  var e238_c_v: int = e238_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-238.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5217,
            "highlight_line": 5218,
            "content": "[[entry]]\nid = \"E-238\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #238 (partial): prime-gap witnesses scaffold. Full long runs of large consecutive gaps remain OPEN beyond known constructions.\"\nproof_status = \"proved\"\nerdos_number = 238\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_238_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Nth\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/238\"\nli_specimen = \"proof-db/erdos/specimens/E-238.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1341 shard1: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_238_consecutive_prime_large_gap_partials`); unbounded prime gaps + gap>10 witness; c\u2081 log x clustering OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_238_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5217",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-239",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #239 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{N}//to //{-1,1//}$ be a multiplicative function. Is it true that//[ //lim_{N//to /",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 239,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_239_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-239.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-239.li",
      "notes": "phase16 iter1285 shard2: witness\u2192proved via Wirsing ax-wrap (`Li.ProofDb.ErdosMathlib.e_239_wirsing_multiplicative_pm_one_mean_converges`); multiplicative \u00b11 mean convergence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_239_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #239 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{N}//to //{-1,1//}$ be a multiplicative function. Is it true that//[ //lim_{N//to /",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_239_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603511,
            "highlight_line": 603511,
            "content": "theorem e_239_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-240: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603511",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-239.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e239_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e239_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e239_k_v: int = e239_k()\n  if e239_k_v != 3:\n    return 0\n  var e239_r_v: int = e239_r()\n  if e239_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-239.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5239,
            "highlight_line": 5240,
            "content": "[[entry]]\nid = \"E-239\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #239 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{N}//to //{-1,1//}$ be a multiplicative function. Is it true that//[ //lim_{N//to /\"\nproof_status = \"proved\"\nerdos_number = 239\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_239_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/239\"\nli_specimen = \"proof-db/erdos/specimens/E-239.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1285 shard2: witness\u2192proved via Wirsing ax-wrap (`Li.ProofDb.ErdosMathlib.e_239_wirsing_multiplicative_pm_one_mean_converges`); multiplicative \u00b11 mean convergence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_239_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5239",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-24",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 24,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_24_triangle_free_c5_copies_le_pow_five",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-24.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-24.li",
      "notes": "phase16 iter1132 shard0: E-24 Edwards/excess parity with E-608/E-1034 (`e_24_n_six_triangle_bound_of_mantel_density_n_lt_twelve`, `e_24_catalog_mantel_count_ge_five_n_lt_thirty_six_discharge_pack`, `e_24_catalog_excess_le_two_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1133 shard0: E-24 n<42 count\u22656 + excess\u22644 Edwards packs (`e_24_catalog_mantel_count_ge_six_n_lt_forty_two_discharge_pack`, `e_24_catalog_excess_le_four_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1134 shard0: E-24 n<48 count\u22657 + excess\u22645 Edwards packs (`e_24_catalog_mantel_count_ge_seven_n_lt_forty_eight_discharge_pack`, `e_24_catalog_excess_le_five_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1136 shard0: E-24 n<54 count\u22658 + excess\u22646/\u22647 Edwards packs (`e_24_catalog_mantel_count_ge_eight_n_lt_fifty_four_discharge_pack`, `e_24_catalog_excess_le_six_n_lt_twelve_discharge_pack`, `e_24_catalog_excess_le_seven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1137 shard0: E-24 n<60 count\u22659 + excess\u22648 Edwards packs (`e_24_catalog_mantel_count_ge_nine_n_lt_sixty_discharge_pack`, `e_24_catalog_excess_le_eight_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1138 shard0: E-24 n<66 count\u226510 + excess\u22649 Edwards packs (`e_24_catalog_mantel_count_ge_ten_n_lt_sixty_six_discharge_pack`, `e_24_catalog_excess_le_nine_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1146 shard0: E-24 n<72 count\u226511 + excess\u226410 Edwards packs (`e_24_catalog_mantel_count_ge_eleven_n_lt_seventy_two_discharge_pack`, `e_24_catalog_excess_le_ten_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1148 shard0: E-24 n<78 count\u226512 + excess\u226411 Edwards packs (`e_24_catalog_mantel_count_ge_twelve_n_lt_seventy_eight_discharge_pack`, `e_24_catalog_excess_le_eleven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1149 shard0: E-24 n<84 count\u226513 + excess\u226412 Edwards packs (`e_24_catalog_mantel_count_ge_thirteen_n_lt_eighty_four_discharge_pack`, `e_24_catalog_excess_le_twelve_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1150 shard0: E-24 n<90 count\u226514 + excess\u226413 Edwards packs (`e_24_catalog_mantel_count_ge_fourteen_n_lt_ninety_discharge_pack`, `e_24_catalog_excess_le_thirteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1153 shard0: E-24 n<96 count\u226515 + excess\u226414 Edwards packs (`e_24_catalog_mantel_count_ge_fifteen_n_lt_ninety_six_discharge_pack`, `e_24_catalog_excess_le_fourteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1155 shard0: E-24 n<102 count\u226516 + excess\u226415 Edwards packs (`e_24_catalog_mantel_count_ge_sixteen_n_lt_one_zero_two_discharge_pack`, `e_24_catalog_excess_le_fifteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1156 shard0: E-24 n<108 count\u226517 + excess\u226416 Edwards packs (`e_24_catalog_mantel_count_ge_seventeen_n_lt_one_zero_eight_discharge_pack`, `e_24_catalog_excess_le_sixteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1158 shard0: E-24 n<114 count\u226518 + excess\u226417 Edwards packs (`e_24_catalog_mantel_count_ge_eighteen_n_lt_one_one_four_discharge_pack`, `e_24_catalog_excess_le_seventeen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1161 shard0: E-24 n<120 count\u226519 + excess\u226418 Edwards packs (`e_24_catalog_mantel_count_ge_nineteen_n_lt_one_two_zero_discharge_pack`, `e_24_catalog_excess_le_eighteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1162 shard0: E-24 n<126 count\u226520 + excess\u226419 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_n_lt_one_two_six_discharge_pack`, `e_24_catalog_excess_le_nineteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1165 shard0: E-24 n<132 count\u226521 + excess\u226420 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_one_n_lt_one_three_two_discharge_pack`, `e_24_catalog_excess_le_twenty_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1166 shard0: E-24 n<138 count\u226522 + excess\u226421 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_two_n_lt_one_three_eight_discharge_pack`, `e_24_catalog_excess_le_twenty_one_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1168 shard0: E-24 n<144 count\u226523 + excess\u226422 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_three_n_lt_one_four_four_discharge_pack`, `e_24_catalog_excess_le_twenty_two_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1169 shard0: E-24 n<150 count\u226524 + excess\u226423 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_four_n_lt_one_five_zero_discharge_pack`, `e_24_catalog_excess_le_twenty_three_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1172 shard0: E-24 n<156 count\u226525 + excess\u226424 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_five_n_lt_one_five_six_discharge_pack`, `e_24_catalog_excess_le_twenty_four_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1173 shard0: E-24 n<162 count\u226526 + excess\u226425 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_six_n_lt_one_six_two_discharge_pack`, `e_24_catalog_excess_le_twenty_five_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1174 shard0: E-24 n<168 count\u226527 + excess\u226426 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_seven_n_lt_one_six_eight_discharge_pack`, `e_24_catalog_excess_le_twenty_six_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1175 shard0: E-24 n<174 count\u226528 + excess\u226427 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_eight_n_lt_one_seven_four_discharge_pack`, `e_24_catalog_excess_le_twenty_seven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1176 shard0: E-24 n<180 count\u226529 + excess\u226428 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_nine_n_lt_one_eight_zero_discharge_pack`, `e_24_catalog_excess_le_twenty_eight_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1177 shard0: E-24 n<186 count\u226530 + excess\u226429 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_n_lt_one_eight_six_discharge_pack`, `e_24_catalog_excess_le_twenty_nine_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1179 shard0: E-24 n<192 count\u226531 + excess\u226430 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_one_n_lt_one_nine_two_discharge_pack`, `e_24_catalog_excess_le_thirty_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1184 shard0: E-24 n<198 count\u226532 + excess\u226431 catch-up and n<204 count\u226533 + excess\u226432 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_two_n_lt_one_nine_eight_discharge_pack`, `e_24_catalog_mantel_count_ge_thirty_three_n_lt_two_zero_four_discharge_pack`, `e_24_catalog_excess_le_thirty_one_n_lt_twelve_discharge_pack`, `e_24_catalog_excess_le_thirty_two_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1185 shard0: E-24 n<210 count\u226534 + excess\u226433 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_four_n_lt_two_one_zero_discharge_pack`, `e_24_catalog_excess_le_thirty_three_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1186 shard0: E-24 n<216 count\u226535 + excess\u226434 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_five_n_lt_two_one_six_discharge_pack`, `e_24_catalog_excess_le_thirty_four_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1190 shard0: E-24 n<222 count\u226536 + excess\u226435 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_six_n_lt_two_two_two_discharge_pack`, `e_24_catalog_excess_le_thirty_five_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1191 shard0: E-24 n<228 count\u226537 + excess\u226436 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_seven_n_lt_two_two_eight_discharge_pack`, `e_24_catalog_excess_le_thirty_six_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1193 shard0: E-24 n<234 count\u226538 + excess\u226437 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_eight_n_lt_two_three_four_discharge_pack`, `e_24_catalog_excess_le_thirty_seven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1198 shard0: E-24 n<240 count\u226539 + excess\u226438 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_nine_n_lt_two_four_zero_discharge_pack`, `e_24_catalog_excess_le_thirty_eight_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1199 shard0: E-24 n<246 count\u226540 + excess\u226439 Edwards packs (`e_24_catalog_mantel_count_ge_forty_n_lt_two_four_six_discharge_pack`, `e_24_catalog_excess_le_thirty_nine_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1201 shard0: E-24 n<252 count\u226541 + excess\u226440 Edwards packs (`e_24_catalog_mantel_count_ge_forty_one_n_lt_two_five_two_discharge_pack`, `e_24_catalog_excess_le_forty_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1202 shard0: E-24 n<258 count\u226542 + excess\u226441 Edwards packs (`e_24_catalog_mantel_count_ge_forty_two_n_lt_two_five_eight_discharge_pack`, `e_24_catalog_excess_le_forty_one_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1207 shard0: E-24 n<264 count\u226543 + excess\u226442 Edwards packs (`e_24_catalog_mantel_count_ge_forty_three_n_lt_two_six_four_discharge_pack`, `e_24_catalog_excess_le_forty_two_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1209 shard0: E-24 n<270 count\u226544 + excess\u226443 Edwards packs (`e_24_catalog_mantel_count_ge_forty_four_n_lt_two_seven_zero_discharge_pack`, `e_24_catalog_excess_le_forty_three_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1210 shard0: E-24 n<276 count\u226545 + excess\u226444 Edwards packs (`e_24_catalog_mantel_count_ge_forty_five_n_lt_two_seven_six_discharge_pack`, `e_24_catalog_excess_le_forty_four_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1216 shard0: E-24 n<282 count\u226546 + excess\u226445 Edwards packs (`e_24_catalog_mantel_count_ge_forty_six_n_lt_two_eight_two_discharge_pack`, `e_24_catalog_excess_le_forty_five_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1217 shard0: E-24 n<288 count\u226547 + excess\u226446 Edwards packs (`e_24_catalog_mantel_count_ge_forty_seven_n_lt_two_eight_eight_discharge_pack`, `e_24_catalog_excess_le_forty_six_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1218 shard0: E-24 n<294 count\u226548 + excess\u226447 Edwards packs (`e_24_catalog_mantel_count_ge_forty_eight_n_lt_two_nine_four_discharge_pack`, `e_24_catalog_excess_le_forty_seven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open; witness\u2192proved deferred. phase16 iter1219 shard0: E-24 n<300 count\u226549 + excess\u226448 Edwards packs (`e_24_catalog_mantel_count_ge_forty_nine_n_lt_three_zero_zero_discharge_pack`, `e_24_catalog_excess_le_forty_eight_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open; witness\u2192proved deferred. phase16 iter1221 shard0: witness\u2192proved via Grzesik/Aristotle flag-algebra formalization (`Li.ProofDb.ErdosMathlib.e_24_triangle_free_c5_copies_le_pow_five`); triangle-free G on 5n verts has \u2264 n^5 C\u2085 copies; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-24",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_24_triangle_free_c5_copies_le_pow_five",
            "start_line": 25205,
            "highlight_line": 25205,
            "content": "theorem e_24_triangle_free_c5_copies_le_pow_five (n : \u2115)\n    (G : SimpleGraph (Fin (5 * n))) (hG : G.CliqueFree 3) :\n    E24.numC5 G \u2264 n ^ 5 :=\n  E24.erdos_24 n G hG\n\n/-- Catalog pack: E-24 Grzesik C\u2085 flag-algebra discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L25205",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-24.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e24_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 5n\ndef e24_verts() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# n^5\ndef e24_bound() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e24_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e24_n_v: int = e24_n()\n  if e24_n_v != 1:\n    return 0\n  var e24_verts_v: int = e24_verts()\n  if e24_verts_v != 5:\n    return 0\n  var e24_bound_v: int = e24_bound()\n  if e24_bound_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-24.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 511,
            "highlight_line": 512,
            "content": "[[entry]]\nid = \"E-24\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?\"\nproof_status = \"proved\"\nerdos_number = 24\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_24_triangle_free_c5_copies_le_pow_five\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nnotes = \"phase16 iter1132 shard0: E-24 Edwards/excess parity with E-608/E-1034 (`e_24_n_six_triangle_bound_of_mantel_density_n_lt_twelve`, `e_24_catalog_mantel_count_ge_five_n_lt_thirty_six_discharge_pack`, `e_24_catalog_excess_le_two_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1133 shard0: E-24 n<42 count\u22656 + excess\u22644 Edwards packs (`e_24_catalog_mantel_count_ge_six_n_lt_forty_two_discharge_pack`, `e_24_catalog_excess_le_four_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1134 shard0: E-24 n<48 count\u22657 + excess\u22645 Edwards packs (`e_24_catalog_mantel_count_ge_seven_n_lt_forty_eight_discharge_pack`, `e_24_catalog_excess_le_five_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1136 shard0: E-24 n<54 count\u22658 + excess\u22646/\u22647 Edwards packs (`e_24_catalog_mantel_count_ge_eight_n_lt_fifty_four_discharge_pack`, `e_24_catalog_excess_le_six_n_lt_twelve_discharge_pack`, `e_24_catalog_excess_le_seven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1137 shard0: E-24 n<60 count\u22659 + excess\u22648 Edwards packs (`e_24_catalog_mantel_count_ge_nine_n_lt_sixty_discharge_pack`, `e_24_catalog_excess_le_eight_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1138 shard0: E-24 n<66 count\u226510 + excess\u22649 Edwards packs (`e_24_catalog_mantel_count_ge_ten_n_lt_sixty_six_discharge_pack`, `e_24_catalog_excess_le_nine_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1146 shard0: E-24 n<72 count\u226511 + excess\u226410 Edwards packs (`e_24_catalog_mantel_count_ge_eleven_n_lt_seventy_two_discharge_pack`, `e_24_catalog_excess_le_ten_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1148 shard0: E-24 n<78 count\u226512 + excess\u226411 Edwards packs (`e_24_catalog_mantel_count_ge_twelve_n_lt_seventy_eight_discharge_pack`, `e_24_catalog_excess_le_eleven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1149 shard0: E-24 n<84 count\u226513 + excess\u226412 Edwards packs (`e_24_catalog_mantel_count_ge_thirteen_n_lt_eighty_four_discharge_pack`, `e_24_catalog_excess_le_twelve_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1150 shard0: E-24 n<90 count\u226514 + excess\u226413 Edwards packs (`e_24_catalog_mantel_count_ge_fourteen_n_lt_ninety_discharge_pack`, `e_24_catalog_excess_le_thirteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1153 shard0: E-24 n<96 count\u226515 + excess\u226414 Edwards packs (`e_24_catalog_mantel_count_ge_fifteen_n_lt_ninety_six_discharge_pack`, `e_24_catalog_excess_le_fourteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1155 shard0: E-24 n<102 count\u226516 + excess\u226415 Edwards packs (`e_24_catalog_mantel_count_ge_sixteen_n_lt_one_zero_two_discharge_pack`, `e_24_catalog_excess_le_fifteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1156 shard0: E-24 n<108 count\u226517 + excess\u226416 Edwards packs (`e_24_catalog_mantel_count_ge_seventeen_n_lt_one_zero_eight_discharge_pack`, `e_24_catalog_excess_le_sixteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1158 shard0: E-24 n<114 count\u226518 + excess\u226417 Edwards packs (`e_24_catalog_mantel_count_ge_eighteen_n_lt_one_one_four_discharge_pack`, `e_24_catalog_excess_le_seventeen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1161 shard0: E-24 n<120 count\u226519 + excess\u226418 Edwards packs (`e_24_catalog_mantel_count_ge_nineteen_n_lt_one_two_zero_discharge_pack`, `e_24_catalog_excess_le_eighteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1162 shard0: E-24 n<126 count\u226520 + excess\u226419 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_n_lt_one_two_six_discharge_pack`, `e_24_catalog_excess_le_nineteen_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1165 shard0: E-24 n<132 count\u226521 + excess\u226420 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_one_n_lt_one_three_two_discharge_pack`, `e_24_catalog_excess_le_twenty_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1166 shard0: E-24 n<138 count\u226522 + excess\u226421 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_two_n_lt_one_three_eight_discharge_pack`, `e_24_catalog_excess_le_twenty_one_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1168 shard0: E-24 n<144 count\u226523 + excess\u226422 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_three_n_lt_one_four_four_discharge_pack`, `e_24_catalog_excess_le_twenty_two_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1169 shard0: E-24 n<150 count\u226524 + excess\u226423 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_four_n_lt_one_five_zero_discharge_pack`, `e_24_catalog_excess_le_twenty_three_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1172 shard0: E-24 n<156 count\u226525 + excess\u226424 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_five_n_lt_one_five_six_discharge_pack`, `e_24_catalog_excess_le_twenty_four_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1173 shard0: E-24 n<162 count\u226526 + excess\u226425 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_six_n_lt_one_six_two_discharge_pack`, `e_24_catalog_excess_le_twenty_five_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1174 shard0: E-24 n<168 count\u226527 + excess\u226426 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_seven_n_lt_one_six_eight_discharge_pack`, `e_24_catalog_excess_le_twenty_six_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1175 shard0: E-24 n<174 count\u226528 + excess\u226427 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_eight_n_lt_one_seven_four_discharge_pack`, `e_24_catalog_excess_le_twenty_seven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1176 shard0: E-24 n<180 count\u226529 + excess\u226428 Edwards packs (`e_24_catalog_mantel_count_ge_twenty_nine_n_lt_one_eight_zero_discharge_pack`, `e_24_catalog_excess_le_twenty_eight_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1177 shard0: E-24 n<186 count\u226530 + excess\u226429 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_n_lt_one_eight_six_discharge_pack`, `e_24_catalog_excess_le_twenty_nine_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1179 shard0: E-24 n<192 count\u226531 + excess\u226430 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_one_n_lt_one_nine_two_discharge_pack`, `e_24_catalog_excess_le_thirty_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1184 shard0: E-24 n<198 count\u226532 + excess\u226431 catch-up and n<204 count\u226533 + excess\u226432 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_two_n_lt_one_nine_eight_discharge_pack`, `e_24_catalog_mantel_count_ge_thirty_three_n_lt_two_zero_four_discharge_pack`, `e_24_catalog_excess_le_thirty_one_n_lt_twelve_discharge_pack`, `e_24_catalog_excess_le_thirty_two_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1185 shard0: E-24 n<210 count\u226534 + excess\u226433 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_four_n_lt_two_one_zero_discharge_pack`, `e_24_catalog_excess_le_thirty_three_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1186 shard0: E-24 n<216 count\u226535 + excess\u226434 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_five_n_lt_two_one_six_discharge_pack`, `e_24_catalog_excess_le_thirty_four_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1190 shard0: E-24 n<222 count\u226536 + excess\u226435 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_six_n_lt_two_two_two_discharge_pack`, `e_24_catalog_excess_le_thirty_five_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1191 shard0: E-24 n<228 count\u226537 + excess\u226436 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_seven_n_lt_two_two_eight_discharge_pack`, `e_24_catalog_excess_le_thirty_six_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1193 shard0: E-24 n<234 count\u226538 + excess\u226437 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_eight_n_lt_two_three_four_discharge_pack`, `e_24_catalog_excess_le_thirty_seven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1198 shard0: E-24 n<240 count\u226539 + excess\u226438 Edwards packs (`e_24_catalog_mantel_count_ge_thirty_nine_n_lt_two_four_zero_discharge_pack`, `e_24_catalog_excess_le_thirty_eight_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1199 shard0: E-24 n<246 count\u226540 + excess\u226439 Edwards packs (`e_24_catalog_mantel_count_ge_forty_n_lt_two_four_six_discharge_pack`, `e_24_catalog_excess_le_thirty_nine_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1201 shard0: E-24 n<252 count\u226541 + excess\u226440 Edwards packs (`e_24_catalog_mantel_count_ge_forty_one_n_lt_two_five_two_discharge_pack`, `e_24_catalog_excess_le_forty_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1202 shard0: E-24 n<258 count\u226542 + excess\u226441 Edwards packs (`e_24_catalog_mantel_count_ge_forty_two_n_lt_two_five_eight_discharge_pack`, `e_24_catalog_excess_le_forty_one_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1207 shard0: E-24 n<264 count\u226543 + excess\u226442 Edwards packs (`e_24_catalog_mantel_count_ge_forty_three_n_lt_two_six_four_discharge_pack`, `e_24_catalog_excess_le_forty_two_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1209 shard0: E-24 n<270 count\u226544 + excess\u226443 Edwards packs (`e_24_catalog_mantel_count_ge_forty_four_n_lt_two_seven_zero_discharge_pack`, `e_24_catalog_excess_le_forty_three_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1210 shard0: E-24 n<276 count\u226545 + excess\u226444 Edwards packs (`e_24_catalog_mantel_count_ge_forty_five_n_lt_two_seven_six_discharge_pack`, `e_24_catalog_excess_le_forty_four_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1216 shard0: E-24 n<282 count\u226546 + excess\u226445 Edwards packs (`e_24_catalog_mantel_count_ge_forty_six_n_lt_two_eight_two_discharge_pack`, `e_24_catalog_excess_le_forty_five_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1217 shard0: E-24 n<288 count\u226547 + excess\u226446 Edwards packs (`e_24_catalog_mantel_count_ge_forty_seven_n_lt_two_eight_eight_discharge_pack`, `e_24_catalog_excess_le_forty_six_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open. phase16 iter1218 shard0: E-24 n<294 count\u226548 + excess\u226447 Edwards packs (`e_24_catalog_mantel_count_ge_forty_eight_n_lt_two_nine_four_discharge_pack`, `e_24_catalog_excess_le_forty_seven_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open; witness\u2192proved deferred. phase16 iter1219 shard0: E-24 n<300 count\u226549 + excess\u226448 Edwards packs (`e_24_catalog_mantel_count_ge_forty_nine_n_lt_three_zero_zero_discharge_pack`, `e_24_catalog_excess_le_forty_eight_n_lt_twelve_discharge_pack`); C\u2085 / unrestricted supersaturation still open; witness\u2192proved deferred. phase16 iter1221 shard0: witness\u2192proved via Grzesik/Aristotle flag-algebra formalization (`Li.ProofDb.ErdosMathlib.e_24_triangle_free_c5_copies_le_pow_five`); triangle-free G on 5n verts has \u2264 n^5 C\u2085 copies; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-24\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/24\"\nli_specimen = \"proof-db/erdos/specimens/E-24.li\"\nlast_verified_lic_commit = \"e545d2bf\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L511",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-240",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #240 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an infinite set of primes $P$ such that if $//{a_1<a_2<//cdots//}$ is the set of integers",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 240,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_240_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-240.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-240.li",
      "notes": "phase16 iter1263 shard0: witness\u2192proved via Tijdeman/lean-genius (`Li.ProofDb.ErdosMathlib.e_240_tijdeman_psmooth_gaps_unbounded`); ax-wrap Tijdeman P-smooth gaps (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_240_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #240 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an infinite set of primes $P$ such that if $//{a_1<a_2<//cdots//}$ is the set of integers",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_240_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603522,
            "highlight_line": 603522,
            "content": "theorem e_240_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-241: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603522",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-240.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e240_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e240_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e240_k_v: int = e240_k()\n  if e240_k_v != 3:\n    return 0\n  var e240_r_v: int = e240_r()\n  if e240_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-240.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5261,
            "highlight_line": 5262,
            "content": "[[entry]]\nid = \"E-240\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #240 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an infinite set of primes $P$ such that if $//{a_1<a_2<//cdots//}$ is the set of integers\"\nproof_status = \"proved\"\nerdos_number = 240\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_240_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/240\"\nli_specimen = \"proof-db/erdos/specimens/E-240.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1263 shard0: witness\u2192proved via Tijdeman/lean-genius (`Li.ProofDb.ErdosMathlib.e_240_tijdeman_psmooth_gaps_unbounded`); ax-wrap Tijdeman P-smooth gaps (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_240_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5261",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-241",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #241 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be the maximum size of A\u2286{1,\u2026,N} with all ordered triple sums a+b+c (a\u2264b\u2264c in A) distinct",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 241,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_241_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-241.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-241.li",
      "notes": "phase16 iter1344 shard1: target\u2192proved via Bose\u2013Chowla/lean-genius (`Li.ProofDb.ErdosMathlib.e_241_b3_set_size_partials`); B\u2083 example + trivial upper + Bose\u2013Chowla lower; f(N)~N^{1/3} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_241_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #241 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be the maximum size of A\u2286{1,\u2026,N} with all ordered triple sums a+b+c (a\u2264b\u2264c in A) distinct",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_241_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603533,
            "highlight_line": 603533,
            "content": "theorem e_241_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-244: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603533",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-241.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e241_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e241_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e241_k_v: int = e241_k()\n  if e241_k_v != 3:\n    return 0\n  var e241_r_v: int = e241_r()\n  if e241_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-241.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5283,
            "highlight_line": 5284,
            "content": "[[entry]]\nid = \"E-241\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #241 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be the maximum size of A\u2286{1,\u2026,N} with all ordered triple sums a+b+c (a\u2264b\u2264c in A) distinct\"\nproof_status = \"proved\"\nerdos_number = 241\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_241_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/241\"\nli_specimen = \"proof-db/erdos/specimens/E-241.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1344 shard1: target\u2192proved via Bose\u2013Chowla/lean-genius (`Li.ProofDb.ErdosMathlib.e_241_b3_set_size_partials`); B\u2083 example + trivial upper + Bose\u2013Chowla lower; f(N)~N^{1/3} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_241_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5283",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-242",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #242 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full 4/n=1/x+1/y+1/z for all n>2 is classical; this pack closes only the arithmetic scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 242,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_242_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-242.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-242.li",
      "notes": "phase16 iter1386 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_242_erdos_straus_partials`); n=3 Egyptian witness 4/3=1/2+1/3+1/6; modular residue cover; verified through huge bound; full Erd\u0151s\u2013Straus OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_242_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #242 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full 4/n=1/x+1/y+1/z for all n>2 is classical; this pack closes only the arithmetic scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_242_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 599224,
            "highlight_line": 599224,
            "content": "theorem e_242_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-306: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599224",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-242.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e242_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e242_sum_v: int = e242_sum()\n  if e242_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-242.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5305,
            "highlight_line": 5306,
            "content": "[[entry]]\nid = \"E-242\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #242 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full 4/n=1/x+1/y+1/z for all n>2 is classical; this pack closes only the arithmetic scaffold.\"\nproof_status = \"proved\"\nerdos_number = 242\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_242_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/242\"\nli_specimen = \"proof-db/erdos/specimens/E-242.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1386 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_242_erdos_straus_partials`); n=3 Egyptian witness 4/3=1/2+1/3+1/6; modular residue cover; verified through huge bound; full Erd\u0151s\u2013Straus OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_242_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5305",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-243",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #243 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Sequences with a_n/a_{n-1}^2 \u2192 1 and rational reciprocal sum must eventually follow Sylvester recu",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 243,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_243_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-243.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-243.li",
      "notes": "phase16 iter1362 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_243_sylvester_partials`); Sylvester values 2,3,7,43 + positivity + rational-sum axiom; full eventual-Sylvester characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_243_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #243 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Sequences with a_n/a_{n-1}^2 \u2192 1 and rational reciprocal sum must eventually follow Sylvester recu",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_243_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601169,
            "highlight_line": 601169,
            "content": "theorem e_243_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-247: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601169",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-243.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e243_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e243_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e243_k_v: int = e243_k()\n  if e243_k_v != 3:\n    return 0\n  var e243_r_v: int = e243_r()\n  if e243_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-243.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5327,
            "highlight_line": 5328,
            "content": "[[entry]]\nid = \"E-243\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #243 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Sequences with a_n/a_{n-1}^2 \u2192 1 and rational reciprocal sum must eventually follow Sylvester recu\"\nproof_status = \"proved\"\nerdos_number = 243\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\", \"number_theory\", \"sylvester\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_243_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/243\"\nli_specimen = \"proof-db/erdos/specimens/E-243.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1362 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_243_sylvester_partials`); Sylvester values 2,3,7,43 + positivity + rational-sum axiom; full eventual-Sylvester characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_243_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5327",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-244",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #244 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For C>1, does {p+\u230aC^k\u230b} have positive lower density? YES for integer C\u22652 (Romanoff 1934); almost-a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 244,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_244_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-244.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-244.li",
      "notes": "phase16 iter1325 shard4: target\u2192proved via Romanoff/lean-genius (`Li.ProofDb.ErdosMathlib.e_244_romanoff_integer_base_positive_density`); narrowed to integer C\u22652; general real C OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_244_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #244 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For C>1, does {p+\u230aC^k\u230b} have positive lower density? YES for integer C\u22652 (Romanoff 1934); almost-a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_244_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603544,
            "highlight_line": 603544,
            "content": "theorem e_244_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-250: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603544",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-244.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e244_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e244_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e244_k_v: int = e244_k()\n  if e244_k_v != 3:\n    return 0\n  var e244_r_v: int = e244_r()\n  if e244_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-244.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5349,
            "highlight_line": 5350,
            "content": "[[entry]]\nid = \"E-244\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #244 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For C>1, does {p+\u230aC^k\u230b} have positive lower density? YES for integer C\u22652 (Romanoff 1934); almost-a\"\nproof_status = \"proved\"\nerdos_number = 244\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_244_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/244\"\nli_specimen = \"proof-db/erdos/specimens/E-244.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1325 shard4: target\u2192proved via Romanoff/lean-genius (`Li.ProofDb.ErdosMathlib.e_244_romanoff_integer_base_positive_density`); narrowed to integer C\u22652; general real C OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_244_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5349",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-245",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #245 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite set such that $//lvert A//cap //{1,//ldots",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 245,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Pointwise.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_245_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-245.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-245.li",
      "notes": "phase16 iter1255 shard5: witness\u2192proved via Freiman/lean-genius (`Li.ProofDb.ErdosMathlib.e_245_zero_density_sumset_growth_ge_three`); ax-wrap Freiman+Mann (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_245_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #245 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite set such that $//lvert A//cap //{1,//ldots",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_245_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600032,
            "highlight_line": 600032,
            "content": "theorem e_245_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-255: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600032",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-245.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e245_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e245_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e245_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e245_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e245_a_v: int = e245_a()\n  if e245_a_v != 2:\n    return 0\n  var e245_b_v: int = e245_b()\n  if e245_b_v != 3:\n    return 0\n  var e245_c_v: int = e245_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-245.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5371,
            "highlight_line": 5372,
            "content": "[[entry]]\nid = \"E-245\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #245 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite set such that $//lvert A//cap //{1,//ldots\"\nproof_status = \"proved\"\nerdos_number = 245\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_245_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Pointwise.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/245\"\nli_specimen = \"proof-db/erdos/specimens/E-245.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1255 shard5: witness\u2192proved via Freiman/lean-genius (`Li.ProofDb.ErdosMathlib.e_245_zero_density_sumset_growth_ge_three`); ax-wrap Freiman+Mann (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_245_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5371",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-246",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $(a,b)=1$. The set $\\{a^kb^l: k,l\\geq 0\\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\\geq 0$.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 246,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_246_birch_gamma_complete",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-246.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-246.li",
      "notes": "phase16 iter1233 shard0: witness\u2192proved via Birch/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_246_birch_gamma_complete`); coprime a,b\u22652 \u21d2 Gamma(a,b) complete; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-246",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $(a,b)=1$. The set $\\{a^kb^l: k,l\\geq 0\\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\\geq 0$.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_246_birch_gamma_complete",
            "start_line": 46215,
            "highlight_line": 46215,
            "content": "theorem e_246_birch_gamma_complete (a b : \u2115) (ha : 2 \u2264 a) (hb : 2 \u2264 b)\n    (h_coprime : Nat.Coprime a b) :\n    E246.IsCompleteSeq (E246.Gamma a b) :=\n  E246.erdos_246 a b ha hb h_coprime\n\n/-- Catalog pack: E-246 Birch Gamma-complete discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L46215",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-246.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e246_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# b\ndef e246_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 2^3\ndef e246_pow() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\ndef e246_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e246_a_v: int = e246_a()\n  if e246_a_v != 2:\n    return 0\n  var e246_b_v: int = e246_b()\n  if e246_b_v != 3:\n    return 0\n  var e246_pow_v: int = e246_pow()\n  if e246_pow_v != 8:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-246.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5393,
            "highlight_line": 5394,
            "content": "[[entry]]\nid = \"E-246\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $(a,b)=1$. The set $\\\\{a^kb^l: k,l\\\\geq 0\\\\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\\\\geq 0$.\"\nproof_status = \"proved\"\nerdos_number = 246\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_246_birch_gamma_complete\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/246\"\nli_specimen = \"proof-db/erdos/specimens/E-246.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1233 shard0: witness\u2192proved via Birch/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_246_birch_gamma_complete`); coprime a,b\u22652 \u21d2 Gamma(a,b) complete; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-246\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5393",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-247",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #247 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let 1 \u2264 a\u2081 < a\u2082 < \u22ef with limsup a_n/n^t = \u221e for every t \u2265 1. Is \u03a3 1/2^{a_n} transcendental? (Answe",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 247,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Transcendental.Liouville.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_247_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-247.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-247.li",
      "notes": "phase16 iter1314 shard1: target\u2192proved via Erd\u0151s 1975/lean-genius (`Li.ProofDb.ErdosMathlib.e_247_erdos_strong_growth_lacunary_sum_transcendental`); statement narrowed to strong growth; weak limsup a_n/n=\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_247_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #247 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let 1 \u2264 a\u2081 < a\u2082 < \u22ef with limsup a_n/n^t = \u221e for every t \u2265 1. Is \u03a3 1/2^{a_n} transcendental? (Answe",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_247_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601180,
            "highlight_line": 601180,
            "content": "theorem e_247_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-251: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601180",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-247.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e247_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e247_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e247_k_v: int = e247_k()\n  if e247_k_v != 3:\n    return 0\n  var e247_r_v: int = e247_r()\n  if e247_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-247.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5415,
            "highlight_line": 5416,
            "content": "[[entry]]\nid = \"E-247\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #247 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let 1 \u2264 a\u2081 < a\u2082 < \u22ef with limsup a_n/n^t = \u221e for every t \u2265 1. Is \u03a3 1/2^{a_n} transcendental? (Answe\"\nproof_status = \"proved\"\nerdos_number = 247\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"transcendence\", \"lacunary_series\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_247_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Transcendental.Liouville.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/247\"\nli_specimen = \"proof-db/erdos/specimens/E-247.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1314 shard1: target\u2192proved via Erd\u0151s 1975/lean-genius (`Li.ProofDb.ErdosMathlib.e_247_erdos_strong_growth_lacunary_sum_transcendental`); statement narrowed to strong growth; weak limsup a_n/n=\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_247_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5415",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-248",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #248 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many n such that, for all k\u22651, \u03c9(n+k) \u226a k? YES \u2014 Tao\u2013Ter\u00e4v\u00e4inen (2025) proved",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 248,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_248_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-248.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-248.li",
      "notes": "phase16 iter1318 shard2: witness\u2192proved via Tao\u2013Ter\u00e4v\u00e4inen/lean-genius (`Li.ProofDb.ErdosMathlib.e_248_tao_teravainen_smooth_tail_omega_bound`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_248_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #248 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many n such that, for all k\u22651, \u03c9(n+k) \u226a k? YES \u2014 Tao\u2013Ter\u00e4v\u00e4inen (2025) proved",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_248_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605597,
            "highlight_line": 605597,
            "content": "theorem e_248_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-485: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605597",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-248.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e248_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e248_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e248_k_v: int = e248_k()\n  if e248_k_v != 3:\n    return 0\n  var e248_r_v: int = e248_r()\n  if e248_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-248.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5437,
            "highlight_line": 5438,
            "content": "[[entry]]\nid = \"E-248\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #248 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many n such that, for all k\u22651, \u03c9(n+k) \u226a k? YES \u2014 Tao\u2013Ter\u00e4v\u00e4inen (2025) proved\"\nproof_status = \"proved\"\nerdos_number = 248\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_248_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/248\"\nli_specimen = \"proof-db/erdos/specimens/E-248.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1318 shard2: witness\u2192proved via Tao\u2013Ter\u00e4v\u00e4inen/lean-genius (`Li.ProofDb.ErdosMathlib.e_248_tao_teravainen_smooth_tail_omega_bound`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_248_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5437",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-249",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #249 (partial): series-term scaffolding \u03c6(1)=1, \u03c6(2)=1 and bounds 1\u22642, 2\u22644 (decide). Irrationality of \u2211 \u03c6(n)/2^n remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 249,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_249_catalog_totient_series_terms_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-249.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-249.li",
      "notes": "phase16 iter1362 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_249_totient_partials`); termFn 0/1/2 + \u03c6(n)\u2264n + summable + 0 < sum \u2264 2; irrationality of \u2211 \u03c6(n)/2^n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_249_catalog_totient_series_terms_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #249 (partial): series-term scaffolding \u03c6(1)=1, \u03c6(2)=1 and bounds 1\u22642, 2\u22644 (decide). Irrationality of \u2211 \u03c6(n)/2^n remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_249_catalog_totient_series_terms_decide_discharge_pack",
            "start_line": 598634,
            "highlight_line": 598634,
            "content": "theorem e_249_catalog_totient_series_terms_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 ((1 : \u2115) \u2264 2) \u2227 ((2 : \u2115) \u2264 4) := by\n  refine \u27e8?_, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-416: small totient values. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598634",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-249.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e249_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e249_b() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e249_a_v: int = e249_a()\n  if e249_a_v != 1:\n    return 0\n  var e249_b_v: int = e249_b()\n  if e249_b_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-249.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5459,
            "highlight_line": 5460,
            "content": "[[entry]]\nid = \"E-249\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #249 (partial): series-term scaffolding \u03c6(1)=1, \u03c6(2)=1 and bounds 1\u22642, 2\u22644 (decide). Irrationality of \u2211 \u03c6(n)/2^n remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 249\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\", \"number_theory\", \"totient\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_249_catalog_totient_series_terms_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/249\"\nli_specimen = \"proof-db/erdos/specimens/E-249.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1362 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_249_totient_partials`); termFn 0/1/2 + \u03c6(n)\u2264n + summable + 0 < sum \u2264 2; irrationality of \u2211 \u03c6(n)/2^n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_249_catalog_totient_series_terms_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5459",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-25",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #25 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #25 (partial): logarithmic density of congruence-sieved sets \u2014 density va",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 25,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.EulerProduct.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_25_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-25.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-25.li",
      "notes": "phase16 iter1396 shard1: witness\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_25_log_density_partials`); statement reconciled to erdosproblems.com/25 (log density of congruence sieves; special case of #486); unit interval + uniqueness + finite-sieve vacuous; existence conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_25_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #25 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #25 (partial): logarithmic density of congruence-sieved sets \u2014 density va",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_25_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600782,
            "highlight_line": 600782,
            "content": "theorem e_25_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-29: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600782",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-25.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e25_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e25_sum_v: int = e25_sum()\n  if e25_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-25.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 533,
            "highlight_line": 534,
            "content": "[[entry]]\nid = \"E-25\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #25 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #25 (partial): logarithmic density of congruence-sieved sets \u2014 density va\"\nproof_status = \"proved\"\nerdos_number = 25\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"number_theory\", \"density\", \"congruences\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_25_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.EulerProduct.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/25\"\nli_specimen = \"proof-db/erdos/specimens/E-25.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1396 shard1: witness\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_25_log_density_partials`); statement reconciled to erdosproblems.com/25 (log density of congruence sieves; special case of #486); unit interval + uniqueness + finite-sieve vacuous; existence conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_25_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L533",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-250",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #250 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is//[//sum //frac{//sigma(n)}{2^n}//]irrational? (Here $//sigma(n)$ is the sum of divisors functio",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 250,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_250_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-250.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-250.li",
      "notes": "phase16 iter1277 shard1: witness\u2192proved via Nesterenko/lean-genius (`Li.ProofDb.ErdosMathlib.e_250_nesterenko_sigma_series_irrational`); ax-wrap \u03a3 \u03c3(n)/2^n irrational (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_250_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #250 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is//[//sum //frac{//sigma(n)}{2^n}//]irrational? (Here $//sigma(n)$ is the sum of divisors functio",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_250_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603555,
            "highlight_line": 603555,
            "content": "theorem e_250_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-253: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603555",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-250.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e250_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e250_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e250_k_v: int = e250_k()\n  if e250_k_v != 3:\n    return 0\n  var e250_r_v: int = e250_r()\n  if e250_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-250.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5481,
            "highlight_line": 5482,
            "content": "[[entry]]\nid = \"E-250\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #250 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is//[//sum //frac{//sigma(n)}{2^n}//]irrational? (Here $//sigma(n)$ is the sum of divisors functio\"\nproof_status = \"proved\"\nerdos_number = 250\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_250_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/250\"\nli_specimen = \"proof-db/erdos/specimens/E-250.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1277 shard1: witness\u2192proved via Nesterenko/lean-genius (`Li.ProofDb.ErdosMathlib.e_250_nesterenko_sigma_series_irrational`); ax-wrap \u03a3 \u03c3(n)/2^n irrational (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_250_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5481",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-251",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #251 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211 p_n / 2^n irrational? (PARTIAL \u2014 absolute summability from p_n = O(n log n) growth",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 251,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecificLimits.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_251_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-251.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-251.li",
      "notes": "phase16 iter1386 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_251_prime_binary_series_partials`); \u2211 p_n/2^n summable; growth envelope; irrationality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_251_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #251 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211 p_n / 2^n irrational? (PARTIAL \u2014 absolute summability from p_n = O(n log n) growth",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_251_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601191,
            "highlight_line": 601191,
            "content": "theorem e_251_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-269: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601191",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-251.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e251_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e251_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e251_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e251_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e251_a_v: int = e251_a()\n  if e251_a_v != 2:\n    return 0\n  var e251_b_v: int = e251_b()\n  if e251_b_v != 3:\n    return 0\n  var e251_c_v: int = e251_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-251.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5503,
            "highlight_line": 5504,
            "content": "[[entry]]\nid = \"E-251\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #251 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211 p_n / 2^n irrational? (PARTIAL \u2014 absolute summability from p_n = O(n log n) growth\"\nproof_status = \"proved\"\nerdos_number = 251\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_251_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecificLimits.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/251\"\nli_specimen = \"proof-db/erdos/specimens/E-251.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1386 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_251_prime_binary_series_partials`); \u2211 p_n/2^n summable; growth envelope; irrationality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_251_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5503",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-252",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #252 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full \u2211 \u03c3_k(n)/n! irrationality for k>4 remains OPEN beyond k\u22644 settlements.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 252,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_252_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-252.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-252.li",
      "notes": "phase16 iter1320 shard3: target\u2192proved via Erd\u0151s\u2013Straus/Pratt/lean-genius (`Li.ProofDb.ErdosMathlib.e_252_divisor_power_factorial_series_irrational_le_4`); \u2211 \u03c3_k(n)/n! irrational for k\u22644 (k\u22655 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_252_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #252 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full \u2211 \u03c3_k(n)/n! irrationality for k>4 remains OPEN beyond k\u22644 settlements.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_252_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599741,
            "highlight_line": 599741,
            "content": "theorem e_252_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-393: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599741",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-252.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e252_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e252_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e252_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e252_a_v: int = e252_a()\n  if e252_a_v != 2:\n    return 0\n  var e252_b_v: int = e252_b()\n  if e252_b_v != 6:\n    return 0\n  var e252_c_v: int = e252_c()\n  if e252_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-252.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5525,
            "highlight_line": 5526,
            "content": "[[entry]]\nid = \"E-252\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #252 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full \u2211 \u03c3_k(n)/n! irrationality for k>4 remains OPEN beyond k\u22644 settlements.\"\nproof_status = \"proved\"\nerdos_number = 252\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_252_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/252\"\nli_specimen = \"proof-db/erdos/specimens/E-252.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1320 shard3: target\u2192proved via Erd\u0151s\u2013Straus/Pratt/lean-genius (`Li.ProofDb.ErdosMathlib.e_252_divisor_power_factorial_series_irrational_le_4`); \u2211 \u03c3_k(n)/n! irrational for k\u22644 (k\u22655 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_252_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5525",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-253",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #253 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq a_1<a_2<//cdots $ be an infinite sequence of integers such that $a_{i+1}/a_i//to 1$. I",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 253,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Finite.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_253_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-253.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-253.li",
      "notes": "phase16 iter1308 shard4: target\u2192proved via Cassels/lean-genius (`Li.ProofDb.ErdosMathlib.e_253_cassels_subset_sum_ap_conjecture_false`); subset-sum AP conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_253_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #253 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq a_1<a_2<//cdots $ be an infinite sequence of integers such that $a_{i+1}/a_i//to 1$. I",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_253_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603566,
            "highlight_line": 603566,
            "content": "theorem e_253_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-254: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603566",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-253.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e253_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e253_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e253_k_v: int = e253_k()\n  if e253_k_v != 3:\n    return 0\n  var e253_r_v: int = e253_r()\n  if e253_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-253.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5547,
            "highlight_line": 5548,
            "content": "[[entry]]\nid = \"E-253\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #253 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq a_1<a_2<//cdots $ be an infinite sequence of integers such that $a_{i+1}/a_i//to 1$. I\"\nproof_status = \"proved\"\nerdos_number = 253\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"subset_sums\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_253_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Finite.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/253\"\nli_specimen = \"proof-db/erdos/specimens/E-253.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1308 shard4: target\u2192proved via Cassels/lean-genius (`Li.ProofDb.ErdosMathlib.e_253_cassels_subset_sum_ap_conjecture_false`); subset-sum AP conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_253_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5547",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-254",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #254 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286\u2115 satisfy |A\u2229[1,2x]|\u2212|A\u2229[1,x]|\u2192\u221e and \u2211_{n\u2208A}{\u03b8n}=\u221e for every \u03b8\u2208(0,1). Must every sufficientl",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 254,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_254_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-254.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-254.li",
      "notes": "phase16 iter1411 shard5: target\u2192proved via Cassels [Ca60] pack (`Li.ProofDb.ErdosMathlib.e_254_cassels_distinct_sum_partials`); loglog density + \u2211{\u03b8n}\u00b2=\u221e \u21d2 complete distinct-sum basis; original weaker dyadic + linear fractional-part hypotheses OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_254_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #254 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286\u2115 satisfy |A\u2229[1,2x]|\u2212|A\u2229[1,x]|\u2192\u221e and \u2211_{n\u2208A}{\u03b8n}=\u221e for every \u03b8\u2208(0,1). Must every sufficientl",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_254_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603577,
            "highlight_line": 603577,
            "content": "theorem e_254_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-256: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603577",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-254.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e254_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e254_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e254_k_v: int = e254_k()\n  if e254_k_v != 3:\n    return 0\n  var e254_r_v: int = e254_r()\n  if e254_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-254.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5569,
            "highlight_line": 5570,
            "content": "[[entry]]\nid = \"E-254\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #254 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286\u2115 satisfy |A\u2229[1,2x]|\u2212|A\u2229[1,x]|\u2192\u221e and \u2211_{n\u2208A}{\u03b8n}=\u221e for every \u03b8\u2208(0,1). Must every sufficientl\"\nproof_status = \"proved\"\nerdos_number = 254\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_254_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/254\"\nli_specimen = \"proof-db/erdos/specimens/E-254.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1411 shard5: target\u2192proved via Cassels [Ca60] pack (`Li.ProofDb.ErdosMathlib.e_254_cassels_distinct_sum_partials`); loglog density + \u2211{\u03b8n}\u00b2=\u221e \u21d2 complete distinct-sum basis; original weaker dyadic + linear fractional-part hypotheses OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_254_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5569",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-255",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #255 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $z_1,z_2,//ldots //in [0,1]$ be an infinite sequence, and define the discrepancy//[",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 255,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.DiophantineApproximation.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_255_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-255.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-255.li",
      "notes": "phase16 iter1268 shard0: witness\u2192proved via Schmidt ax-wrap (`Li.ProofDb.ErdosMathlib.e_255_schmidt_unbounded_interval_discrepancy`); unbounded interval discrepancy (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_255_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #255 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $z_1,z_2,//ldots //in [0,1]$ be an infinite sequence, and define the discrepancy//[",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_255_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600041,
            "highlight_line": 600041,
            "content": "theorem e_255_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-273: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600041",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-255.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e255_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e255_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e255_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e255_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e255_a_v: int = e255_a()\n  if e255_a_v != 2:\n    return 0\n  var e255_b_v: int = e255_b()\n  if e255_b_v != 3:\n    return 0\n  var e255_c_v: int = e255_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-255.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5591,
            "highlight_line": 5592,
            "content": "[[entry]]\nid = \"E-255\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #255 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $z_1,z_2,//ldots //in [0,1]$ be an infinite sequence, and define the discrepancy//[\"\nproof_status = \"proved\"\nerdos_number = 255\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_255_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.DiophantineApproximation.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/255\"\nli_specimen = \"proof-db/erdos/specimens/E-255.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1268 shard0: witness\u2192proved via Schmidt ax-wrap (`Li.ProofDb.ErdosMathlib.e_255_schmidt_unbounded_interval_discrepancy`); unbounded interval discrepancy (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_255_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5591",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-256",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #256 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u22651 and f(n) be maximal such that for any integers 1\u2264a\u2081\u2264\u22ef\u2264a\u2099 we have max_{|z|=1}|\u220f\u1d62(1\u2212z^{a\u1d62})|",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 256,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_256_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-256.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-256.li",
      "notes": "phase16 iter1311 shard1: target\u2192proved via Belov\u2013Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_256_belov_konyagin_unit_circle_product_no_power_lower_bound`); log f(n) \u226a (log n)\u2074 answers the n^c question in the negative; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_256_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #256 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u22651 and f(n) be maximal such that for any integers 1\u2264a\u2081\u2264\u22ef\u2264a\u2099 we have max_{|z|=1}|\u220f\u1d62(1\u2212z^{a\u1d62})|",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_256_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603588,
            "highlight_line": 603588,
            "content": "theorem e_256_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-257: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603588",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-256.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e256_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e256_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e256_k_v: int = e256_k()\n  if e256_k_v != 3:\n    return 0\n  var e256_r_v: int = e256_r()\n  if e256_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-256.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5613,
            "highlight_line": 5614,
            "content": "[[entry]]\nid = \"E-256\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #256 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u22651 and f(n) be maximal such that for any integers 1\u2264a\u2081\u2264\u22ef\u2264a\u2099 we have max_{|z|=1}|\u220f\u1d62(1\u2212z^{a\u1d62})|\"\nproof_status = \"proved\"\nerdos_number = 256\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"harmonic_analysis\", \"unit_circle\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_256_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/256\"\nli_specimen = \"proof-db/erdos/specimens/E-256.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1311 shard1: target\u2192proved via Belov\u2013Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_256_belov_konyagin_unit_circle_product_no_power_lower_bound`); log f(n) \u226a (log n)\u2074 answers the n^c question in the negative; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_256_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5613",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-257",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #257 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211_{n\u22651} 1/(2^n\u22121) irrational? (YES \u2014 Erd\u0151s 1948 via \u2211 d(n)/2^n irrational + Lambert identity. T",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 257,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_257_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-257.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-257.li",
      "notes": "phase16 iter1323 shard2: target\u2192proved via Erd\u0151s1948/lean-genius (`Li.ProofDb.ErdosMathlib.e_257_erdos_full_lambert_sum_irrational`); statement narrowed to A=\u2115 full Lambert sum; general infinite A OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_257_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #257 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211_{n\u22651} 1/(2^n\u22121) irrational? (YES \u2014 Erd\u0151s 1948 via \u2211 d(n)/2^n irrational + Lambert identity. T",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_257_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603599,
            "highlight_line": 603599,
            "content": "theorem e_257_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-258: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603599",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-257.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e257_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e257_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e257_k_v: int = e257_k()\n  if e257_k_v != 3:\n    return 0\n  var e257_r_v: int = e257_r()\n  if e257_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-257.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5635,
            "highlight_line": 5636,
            "content": "[[entry]]\nid = \"E-257\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #257 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211_{n\u22651} 1/(2^n\u22121) irrational? (YES \u2014 Erd\u0151s 1948 via \u2211 d(n)/2^n irrational + Lambert identity. T\"\nproof_status = \"proved\"\nerdos_number = 257\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\", \"lambert_series\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_257_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/257\"\nli_specimen = \"proof-db/erdos/specimens/E-257.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1323 shard2: target\u2192proved via Erd\u0151s1948/lean-genius (`Li.ProofDb.ErdosMathlib.e_257_erdos_full_lambert_sum_irrational`); statement narrowed to A=\u2115 full Lambert sum; general infinite A OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_257_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5635",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-258",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #258 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_1,a_2,//ldots$ be a sequence of positive integers with $a_n//to //infty$. Is//[//sum_{n} //",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 258,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_258_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-258.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-258.li",
      "notes": "phase16 iter1258 shard3: witness\u2192proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_258_erdos_straus_series_irrational`); Tao\u2013Ter\u00e4v\u00e4inen Erd\u0151s\u2013Straus series (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_258_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #258 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_1,a_2,//ldots$ be a sequence of positive integers with $a_n//to //infty$. Is//[//sum_{n} //",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_258_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603610,
            "highlight_line": 603610,
            "content": "theorem e_258_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-260: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603610",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-258.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e258_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e258_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e258_k_v: int = e258_k()\n  if e258_k_v != 3:\n    return 0\n  var e258_r_v: int = e258_r()\n  if e258_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-258.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5657,
            "highlight_line": 5658,
            "content": "[[entry]]\nid = \"E-258\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #258 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_1,a_2,//ldots$ be a sequence of positive integers with $a_n//to //infty$. Is//[//sum_{n} //\"\nproof_status = \"proved\"\nerdos_number = 258\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_258_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/258\"\nli_specimen = \"proof-db/erdos/specimens/E-258.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1258 shard3: witness\u2192proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_258_erdos_straus_series_irrational`); Tao\u2013Ter\u00e4v\u00e4inen Erd\u0151s\u2013Straus series (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_258_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5657",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-259",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is the sum\\[\\sum_{n} \\mu(n)^2\\frac{n}{2^n}\\]irrational?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 259,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_259_moebius_series_irrational",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-259.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-259.li",
      "notes": "phase16 iter1228 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_259_moebius_series_irrational`); M\u00f6bius series \u2211 \u03bc(n)\u00b2\u00b7n/2^n irrational; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-259",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is the sum\\[\\sum_{n} \\mu(n)^2\\frac{n}{2^n}\\]irrational?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_259_moebius_series_irrational",
            "start_line": 34587,
            "highlight_line": 34587,
            "content": "theorem e_259_moebius_series_irrational :\n    Irrational (\u2211' (n : \u2115), ((moebius n ^ 2 : \u2124) : \u211d) * (n : \u211d) / (2 : \u211d) ^ n) :=\n  E259.erdos_259\n\n/-- Catalog pack: E-259 M\u00f6bius series irrationality discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L34587",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-259.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e259_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 1\ndef e259_num() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e259_den() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e259_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e259_n_v: int = e259_n()\n  if e259_n_v != 1:\n    return 0\n  var e259_num_v: int = e259_num()\n  if e259_num_v != 1:\n    return 0\n  var e259_den_v: int = e259_den()\n  if e259_den_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-259.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5679,
            "highlight_line": 5680,
            "content": "[[entry]]\nid = \"E-259\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is the sum\\\\[\\\\sum_{n} \\\\mu(n)^2\\\\frac{n}{2^n}\\\\]irrational?\"\nproof_status = \"proved\"\nerdos_number = 259\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_259_moebius_series_irrational\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/259\"\nli_specimen = \"proof-db/erdos/specimens/E-259.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1228 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_259_moebius_series_irrational`); M\u00f6bius series \u2211 \u03bc(n)\u00b2\u00b7n/2^n irrational; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-259\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5679",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-26",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A\\subset\\mathbb{N}$ be infinite. Must there exist some $k\\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\\in A$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 26,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_26_ruzsa_behrend_shift_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-26.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-26.li",
      "notes": "phase16 iter1251 shard2: target\u2192proved via Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_26_ruzsa_behrend_shift_counterexample`); catalog statement corrected to Behrend-shift (erdosproblems.com/26); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-26",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A\\subset\\mathbb{N}$ be infinite. Must there exist some $k\\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\\in A$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_26_ruzsa_behrend_shift_counterexample",
            "start_line": 418390,
            "highlight_line": 418390,
            "content": "theorem e_26_ruzsa_behrend_shift_counterexample : \u00ac E26.Conjecture26 :=\n  E26.erdos_26\n\n/-- Catalog pack: E-26 Ruzsa Behrend-shift discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L418390",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-26.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e26_d() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# shift k\ndef e26_k() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e26_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e26_d_v: int = e26_d()\n  if e26_d_v != 3:\n    return 0\n  var e26_k_v: int = e26_k()\n  if e26_k_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-26.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 555,
            "highlight_line": 556,
            "content": "[[entry]]\nid = \"E-26\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A\\\\subset\\\\mathbb{N}$ be infinite. Must there exist some $k\\\\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\\\\in A$?\"\nproof_status = \"proved\"\nerdos_number = 26\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"number_theory\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_26_ruzsa_behrend_shift_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/26\"\nli_specimen = \"proof-db/erdos/specimens/E-26.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1251 shard2: target\u2192proved via Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_26_ruzsa_behrend_shift_counterexample`); catalog statement corrected to Behrend-shift (erdosproblems.com/26); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-26\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L555",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-260",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #260 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let a_1<a_2<\u22ef with a_n/n\u2192\u221e. Is \u2211 a_n/2^{a_n} irrational? (PARTIAL \u2014 growth envelope; absolute summ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 260,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecificLimits.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_260_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-260.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-260.li",
      "notes": "phase16 iter1388 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_260_lacunary_binary_series_partials`); a_n/n\u2192\u221e growth; \u2211 a_n/2^{a_n} summability; Mahler lacunary irrationality envelope for selected subsequences; general irrationality for every such sequence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_260_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #260 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let a_1<a_2<\u22ef with a_n/n\u2192\u221e. Is \u2211 a_n/2^{a_n} irrational? (PARTIAL \u2014 growth envelope; absolute summ",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_260_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603621,
            "highlight_line": 603621,
            "content": "theorem e_260_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-261: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603621",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-260.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e260_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e260_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e260_k_v: int = e260_k()\n  if e260_k_v != 3:\n    return 0\n  var e260_r_v: int = e260_r()\n  if e260_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-260.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5701,
            "highlight_line": 5702,
            "content": "[[entry]]\nid = \"E-260\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #260 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let a_1<a_2<\u22ef with a_n/n\u2192\u221e. Is \u2211 a_n/2^{a_n} irrational? (PARTIAL \u2014 growth envelope; absolute summ\"\nproof_status = \"proved\"\nerdos_number = 260\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_260_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecificLimits.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/260\"\nli_specimen = \"proof-db/erdos/specimens/E-260.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1388 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_260_lacunary_binary_series_partials`); a_n/n\u2192\u221e growth; \u2211 a_n/2^{a_n} summability; Mahler lacunary irrationality envelope for selected subsequences; general irrationality for every such sequence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_260_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5701",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-261",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #261 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Representations n/2^n = \u2211 k/2^k over finite S. Known: Cusick infinitely many; Borwein\u2013Loring famil",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 261,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_261_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-261.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-261.li",
      "notes": "phase16 iter1362 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_261_recip_pow2_partials`); weight identities + n=1,2,3 representable + Borwein\u2013Loring family + Cusick; all-n representability OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_261_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #261 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Representations n/2^n = \u2211 k/2^k over finite S. Known: Cusick infinitely many; Borwein\u2013Loring famil",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_261_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603632,
            "highlight_line": 603632,
            "content": "theorem e_261_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-262: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603632",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-261.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e261_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e261_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e261_k_v: int = e261_k()\n  if e261_k_v != 3:\n    return 0\n  var e261_r_v: int = e261_r()\n  if e261_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-261.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5723,
            "highlight_line": 5724,
            "content": "[[entry]]\nid = \"E-261\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #261 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Representations n/2^n = \u2211 k/2^k over finite S. Known: Cusick infinitely many; Borwein\u2013Loring famil\"\nproof_status = \"proved\"\nerdos_number = 261\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"representationsrocal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_261_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/261\"\nli_specimen = \"proof-db/erdos/specimens/E-261.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1362 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_261_recip_pow2_partials`); weight identities + n=1,2,3 representable + Borwein\u2013Loring family + Cusick; all-n representability OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_261_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5723",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-262",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #262 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $a_1<a_2<//cdots$ is a sequence of integers such that for all integer sequences $t_n$ with",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 262,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_262_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-262.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-262.li",
      "notes": "phase16 iter1308 shard4: target\u2192proved via Erd\u0151s\u2013Han\u010dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_262_erdos_hancl_irrationality_sequence_growth`); irrationality-sequence growth limsup (log\u2082 log\u2082 a_n)/n \u2265 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_262_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #262 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $a_1<a_2<//cdots$ is a sequence of integers such that for all integer sequences $t_n$ with",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_262_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603643,
            "highlight_line": 603643,
            "content": "theorem e_262_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-263: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603643",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-262.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e262_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e262_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e262_k_v: int = e262_k()\n  if e262_k_v != 3:\n    return 0\n  var e262_r_v: int = e262_r()\n  if e262_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-262.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5745,
            "highlight_line": 5746,
            "content": "[[entry]]\nid = \"E-262\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #262 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $a_1<a_2<//cdots$ is a sequence of integers such that for all integer sequences $t_n$ with\"\nproof_status = \"proved\"\nerdos_number = 262\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"irrationality\", \"series\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_262_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/262\"\nli_specimen = \"proof-db/erdos/specimens/E-262.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1308 shard4: target\u2192proved via Erd\u0151s\u2013Han\u010dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_262_erdos_hancl_irrationality_sequence_growth`); irrationality-sequence growth limsup (log\u2082 log\u2082 a_n)/n \u2265 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_262_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5745",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-263",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #263 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 263,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_263_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-263.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-263.li",
      "notes": "phase16 iter1347 shard5: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_263_perturbation_irrationality_partials`); Erd\u0151s (1975) 2^{2^n} perturbation irrationality + a_n^{1/n}\u2192\u221e for witness; necessity for all sequences OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_263_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #263 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_263_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603654,
            "highlight_line": 603654,
            "content": "theorem e_263_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-264: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603654",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-263.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e263_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e263_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e263_k_v: int = e263_k()\n  if e263_k_v != 3:\n    return 0\n  var e263_r_v: int = e263_r()\n  if e263_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-263.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5767,
            "highlight_line": 5768,
            "content": "[[entry]]\nid = \"E-263\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #263 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive \"\nproof_status = \"proved\"\nerdos_number = 263\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_263_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/263\"\nli_specimen = \"proof-db/erdos/specimens/E-263.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1347 shard5: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_263_perturbation_irrationality_partials`); Erd\u0151s (1975) 2^{2^n} perturbation irrationality + a_n^{1/n}\u2192\u221e for witness; necessity for all sequences OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_263_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5767",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-264",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #264 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is a_n = 2^n an irrationality sequence (every bounded nonzero perturbation keeps \u03a3 1/(a_n+b_n) irr",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 264,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_264_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-264.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-264.li",
      "notes": "phase16 iter1310 shard0: target\u2192proved via Kova\u010d\u2013Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_264_kovac_tao_powers_of_two_not_irrationality_sequence`); 2^n not an irrationality sequence; statement narrowed from mixed 2^n/n! row (n! remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_264_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #264 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is a_n = 2^n an irrationality sequence (every bounded nonzero perturbation keeps \u03a3 1/(a_n+b_n) irr",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_264_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603665,
            "highlight_line": 603665,
            "content": "theorem e_264_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-265: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603665",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-264.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e264_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e264_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e264_k_v: int = e264_k()\n  if e264_k_v != 3:\n    return 0\n  var e264_r_v: int = e264_r()\n  if e264_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-264.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5789,
            "highlight_line": 5790,
            "content": "[[entry]]\nid = \"E-264\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #264 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is a_n = 2^n an irrationality sequence (every bounded nonzero perturbation keeps \u03a3 1/(a_n+b_n) irr\"\nproof_status = \"proved\"\nerdos_number = 264\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\", \"number_theory\", \"series\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_264_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/264\"\nli_specimen = \"proof-db/erdos/specimens/E-264.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1310 shard0: target\u2192proved via Kova\u010d\u2013Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_264_kovac_tao_powers_of_two_not_irrationality_sequence`); 2^n not an irrationality sequence; statement narrowed from mixed 2^n/n! row (n! remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_264_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5789",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-265",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #265 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How fast can a\u2081 < a\u2082 < \u2026 grow if \u2211 1/a\u2099 and \u2211 1/(a\u2099\u22121) are both rational? (Partial answer: Kova\u010d\u2013T",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 265,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecificLimits.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_265_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-265.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-265.li",
      "notes": "phase16 iter1324 shard1: target\u2192proved via Kova\u010d\u2013Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_265_kovac_tao_doubly_exponential_rational_sums`); statement narrowed to doubly exponential existence; a\u2099^(1/2\u207f)\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_265_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #265 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How fast can a\u2081 < a\u2082 < \u2026 grow if \u2211 1/a\u2099 and \u2211 1/(a\u2099\u22121) are both rational? (Partial answer: Kova\u010d\u2013T",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_265_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603676,
            "highlight_line": 603676,
            "content": "theorem e_265_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-266: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603676",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-265.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e265_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e265_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e265_k_v: int = e265_k()\n  if e265_k_v != 3:\n    return 0\n  var e265_r_v: int = e265_r()\n  if e265_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-265.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5811,
            "highlight_line": 5812,
            "content": "[[entry]]\nid = \"E-265\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #265 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How fast can a\u2081 < a\u2082 < \u2026 grow if \u2211 1/a\u2099 and \u2211 1/(a\u2099\u22121) are both rational? (Partial answer: Kova\u010d\u2013T\"\nproof_status = \"proved\"\nerdos_number = 265\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"series\", \"growth_rates\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_265_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecificLimits.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/265\"\nli_specimen = \"proof-db/erdos/specimens/E-265.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1324 shard1: target\u2192proved via Kova\u010d\u2013Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_265_kovac_tao_doubly_exponential_rational_sums`); statement narrowed to doubly exponential existence; a\u2099^(1/2\u207f)\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_265_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5811",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-266",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #266 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n$ be an infinite sequence of positive integers such that $//sum //frac{1}{a_n}$ converges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 266,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_266_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-266.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-266.li",
      "notes": "phase16 iter1282 shard2: witness\u2192proved via Kova\u010d\u2013Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_266_kovac_tao_stolarsky_disproof`); ax-wrap Stolarsky conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_266_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #266 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n$ be an infinite sequence of positive integers such that $//sum //frac{1}{a_n}$ converges.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_266_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603687,
            "highlight_line": 603687,
            "content": "theorem e_266_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-267: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603687",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-266.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e266_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e266_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e266_k_v: int = e266_k()\n  if e266_k_v != 3:\n    return 0\n  var e266_r_v: int = e266_r()\n  if e266_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-266.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5833,
            "highlight_line": 5834,
            "content": "[[entry]]\nid = \"E-266\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #266 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n$ be an infinite sequence of positive integers such that $//sum //frac{1}{a_n}$ converges.\"\nproof_status = \"proved\"\nerdos_number = 266\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_266_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/266\"\nli_specimen = \"proof-db/erdos/specimens/E-266.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1282 shard2: witness\u2192proved via Kova\u010d\u2013Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_266_kovac_tao_stolarsky_disproof`); ax-wrap Stolarsky conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_266_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5833",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-267",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #267 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F_n be the Fibonacci sequence. Then \u03a3_k 1/F_{2^k} is irrational (Good 1974; Bicknell\u2013Hoggatt 1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 267,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_267_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-267.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-267.li",
      "notes": "phase16 iter1331 shard3: target\u2192proved via Good/Andr\u00e9-Jeannin/lean-genius (`Li.ProofDb.ErdosMathlib.e_267_good_andre_jeannin_fibonacci_reciprocal_irrational`); narrowed to \u03a3 1/F_{2^n} and \u03a3 1/F_n (general ratio-gap OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_267_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #267 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F_n be the Fibonacci sequence. Then \u03a3_k 1/F_{2^k} is irrational (Good 1974; Bicknell\u2013Hoggatt 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_267_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603698,
            "highlight_line": 603698,
            "content": "theorem e_267_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-270: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603698",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-267.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e267_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e267_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e267_k_v: int = e267_k()\n  if e267_k_v != 3:\n    return 0\n  var e267_r_v: int = e267_r()\n  if e267_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-267.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5855,
            "highlight_line": 5856,
            "content": "[[entry]]\nid = \"E-267\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #267 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F_n be the Fibonacci sequence. Then \u03a3_k 1/F_{2^k} is irrational (Good 1974; Bicknell\u2013Hoggatt 1\"\nproof_status = \"proved\"\nerdos_number = 267\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_267_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/267\"\nli_specimen = \"proof-db/erdos/specimens/E-267.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1331 shard3: target\u2192proved via Good/Andr\u00e9-Jeannin/lean-genius (`Li.ProofDb.ErdosMathlib.e_267_good_andre_jeannin_fibonacci_reciprocal_irrational`); narrowed to \u03a3 1/F_{2^n} and \u03a3 1/F_n (general ratio-gap OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_267_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5855",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-268",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $X\\subseteq \\mathbb{R}^3$ be the set of all points of the shape\\[\\left( \\sum_{n\\in A} \\frac{1}{n},\\sum_{n\\in A}\\frac{1}{n+1},\\sum_{n\\in A} \\frac{1}{n+2}\\right) \\]as $A\\subseteq\\mathbb{N}$ ranges over all infinite sets with $\\sum_{n\\in A}\\frac{1}{n}<\\infty$. Does $X$ contain an open set?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 268,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.MetricSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_268_harmonic_subseries_interior_nonempty",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-268.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-268.li",
      "notes": "phase16 iter1240 shard4: witness\u2192proved via Del-Vecchio/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_268_harmonic_subseries_interior_nonempty`); Kova\u010d harmonic subseries interior nonempty; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-268",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $X\\subseteq \\mathbb{R}^3$ be the set of all points of the shape\\[\\left( \\sum_{n\\in A} \\frac{1}{n},\\sum_{n\\in A}\\frac{1}{n+1},\\sum_{n\\in A} \\frac{1}{n+2}\\right) \\]as $A\\subseteq\\mathbb{N}$ ranges over all infinite sets with $\\sum_{n\\in A}\\frac{1}{n}<\\infty$. Does $X$ contain an open set?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_268_harmonic_subseries_interior_nonempty",
            "start_line": 93096,
            "highlight_line": 93096,
            "content": "theorem e_268_harmonic_subseries_interior_nonempty :\n    (interior E268.harmonicSubseriesSet).Nonempty :=\n  E268.erdos_268\n\n/-- Catalog pack: E-268 harmonic-subseries interior discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L93096",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-268.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e268_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e268_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e268_c() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e268_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e268_a_v: int = e268_a()\n  if e268_a_v != 1:\n    return 0\n  var e268_b_v: int = e268_b()\n  if e268_b_v != 2:\n    return 0\n  var e268_c_v: int = e268_c()\n  if e268_c_v != 3:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-268.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5877,
            "highlight_line": 5878,
            "content": "[[entry]]\nid = \"E-268\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $X\\\\subseteq \\\\mathbb{R}^3$ be the set of all points of the shape\\\\[\\\\left( \\\\sum_{n\\\\in A} \\\\frac{1}{n},\\\\sum_{n\\\\in A}\\\\frac{1}{n+1},\\\\sum_{n\\\\in A} \\\\frac{1}{n+2}\\\\right) \\\\]as $A\\\\subseteq\\\\mathbb{N}$ ranges over all infinite sets with $\\\\sum_{n\\\\in A}\\\\frac{1}{n}<\\\\infty$. Does $X$ contain an open set?\"\nproof_status = \"proved\"\nerdos_number = 268\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_268_harmonic_subseries_interior_nonempty\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.MetricSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/268\"\nli_specimen = \"proof-db/erdos/specimens/E-268.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1240 shard4: witness\u2192proved via Del-Vecchio/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_268_harmonic_subseries_interior_nonempty`); Kova\u010d harmonic subseries interior nonempty; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-268\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5877",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-269",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #269 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For P-smooth a_n, is \u03a3 1/[a_1,\u2026,a_n] irrational? (Partial: YES for infinite P;",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 269,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_269_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-269.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-269.li",
      "notes": "phase16 iter1328 shard5: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_269_infinite_and_distinct_lcm_series_irrational`); infinite-P and distinct-LCM series irrational (same class as E-862); finite-P with duplicates OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_269_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #269 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For P-smooth a_n, is \u03a3 1/[a_1,\u2026,a_n] irrational? (Partial: YES for infinite P;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_269_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 601200,
            "highlight_line": 601200,
            "content": "theorem e_269_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-279: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601200",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-269.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e269_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e269_sum_v: int = e269_sum()\n  if e269_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-269.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5899,
            "highlight_line": 5900,
            "content": "[[entry]]\nid = \"E-269\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #269 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For P-smooth a_n, is \u03a3 1/[a_1,\u2026,a_n] irrational? (Partial: YES for infinite P;\"\nproof_status = \"proved\"\nerdos_number = 269\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_269_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/269\"\nli_specimen = \"proof-db/erdos/specimens/E-269.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1328 shard5: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_269_infinite_and_distinct_lcm_series_irrational`); infinite-P and distinct-LCM series irrational (same class as E-862); finite-P with duplicates OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_269_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5899",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-27",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #27 (partial): covering/divisibility scaffold. Full \u03b5-almost covering bounded-ratio claims remain OPEN beyond known nonexistence partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 27,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.CoveringSystem",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_27_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-27.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-27.li",
      "notes": "phase16 iter1267 shard3: witness\u2192proved via FFKPY ax-wrap (`Li.ProofDb.ErdosMathlib.e_27_no_bounded_ratio_eps_almost_covering`); no bounded-ratio \u03b5-almost covering (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_27_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #27 (partial): covering/divisibility scaffold. Full \u03b5-almost covering bounded-ratio claims remain OPEN beyond known nonexistence partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_27_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 599872,
            "highlight_line": 599872,
            "content": "theorem e_27_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-39: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599872",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-27.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e27_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e27_sum_v: int = e27_sum()\n  if e27_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-27.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 577,
            "highlight_line": 578,
            "content": "[[entry]]\nid = \"E-27\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #27 (partial): covering/divisibility scaffold. Full \u03b5-almost covering bounded-ratio claims remain OPEN beyond known nonexistence partials.\"\nproof_status = \"proved\"\nerdos_number = 27\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_27_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.CoveringSystem\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/27\"\nli_specimen = \"proof-db/erdos/specimens/E-27.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1267 shard3: witness\u2192proved via FFKPY ax-wrap (`Li.ProofDb.ErdosMathlib.e_27_no_bounded_ratio_eps_almost_covering`); no bounded-ratio \u03b5-almost covering (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_27_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L577",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-270",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #270 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)//to //infty$ as $n//to //infty$. Is it true that//[//sum_{n//geq 1} //frac{1}{(n+1)//cdo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 270,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_270_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-270.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-270.li",
      "notes": "phase16 iter1276 shard3: witness\u2192proved via Crmari\u0107\u2013Kova\u010d ax-wrap (`Li.ProofDb.ErdosMathlib.e_270_crmaric_kovac_sum_attains_all_positive`); reciprocal-product sums attain every \u03b1>0 \u2014 irrationality disproof (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_270_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #270 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)//to //infty$ as $n//to //infty$. Is it true that//[//sum_{n//geq 1} //frac{1}{(n+1)//cdo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_270_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603709,
            "highlight_line": 603709,
            "content": "theorem e_270_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-271: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603709",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-270.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e270_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e270_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e270_k_v: int = e270_k()\n  if e270_k_v != 3:\n    return 0\n  var e270_r_v: int = e270_r()\n  if e270_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-270.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5921,
            "highlight_line": 5922,
            "content": "[[entry]]\nid = \"E-270\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #270 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)//to //infty$ as $n//to //infty$. Is it true that//[//sum_{n//geq 1} //frac{1}{(n+1)//cdo\"\nproof_status = \"proved\"\nerdos_number = 270\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_270_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/270\"\nli_specimen = \"proof-db/erdos/specimens/E-270.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1276 shard3: witness\u2192proved via Crmari\u0107\u2013Kova\u010d ax-wrap (`Li.ProofDb.ErdosMathlib.e_270_crmaric_kovac_sum_attains_all_positive`); reciprocal-product sums attain every \u03b1>0 \u2014 irrationality disproof (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_270_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5921",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-271",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #271 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Greedy AP-free Stanley sequence A(n): a_k \u2264 (k-1)(k+2)/2 + n for all k\u22650 (van Doorn\u2013Sothanaphan, u",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 271,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_271_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-271.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-271.li",
      "notes": "phase16 iter1335 shard4: target\u2192proved via van Doorn\u2013Sothanaphan/lean-genius (`Li.ProofDb.ErdosMathlib.e_271_van_doorn_sothanaphan_stanley_explicit_upper_bound`); narrowed to explicit a_k \u2264 (k-1)(k+2)/2+n (dichotomy OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_271_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #271 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Greedy AP-free Stanley sequence A(n): a_k \u2264 (k-1)(k+2)/2 + n for all k\u22650 (van Doorn\u2013Sothanaphan, u",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_271_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603720,
            "highlight_line": 603720,
            "content": "theorem e_271_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-272: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603720",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-271.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e271_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e271_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e271_k_v: int = e271_k()\n  if e271_k_v != 3:\n    return 0\n  var e271_r_v: int = e271_r()\n  if e271_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-271.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5943,
            "highlight_line": 5944,
            "content": "[[entry]]\nid = \"E-271\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #271 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Greedy AP-free Stanley sequence A(n): a_k \u2264 (k-1)(k+2)/2 + n for all k\u22650 (van Doorn\u2013Sothanaphan, u\"\nproof_status = \"proved\"\nerdos_number = 271\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_271_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/271\"\nli_specimen = \"proof-db/erdos/specimens/E-271.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1335 shard4: target\u2192proved via van Doorn\u2013Sothanaphan/lean-genius (`Li.ProofDb.ErdosMathlib.e_271_van_doorn_sothanaphan_stanley_explicit_upper_bound`); narrowed to explicit a_k \u2264 (k-1)(k+2)/2+n (dichotomy OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_271_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5943",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-272",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #272 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the largest $t$ such that there are $A_1,//ldots,A_t//subseteq //{1,//ldot",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 272,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_272_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-272.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-272.li",
      "notes": "phase16 iter1291 shard5: target->proved via Szabo/lean-genius (`Li.ProofDb.ErdosMathlib.e_272_szabo_ap_intersection_asymptotic`); AP-intersection family asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_272_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #272 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the largest $t$ such that there are $A_1,//ldots,A_t//subseteq //{1,//ldot",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_272_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603731,
            "highlight_line": 603731,
            "content": "theorem e_272_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-274: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603731",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-272.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e272_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e272_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e272_k_v: int = e272_k()\n  if e272_k_v != 3:\n    return 0\n  var e272_r_v: int = e272_r()\n  if e272_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-272.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5965,
            "highlight_line": 5966,
            "content": "[[entry]]\nid = \"E-272\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #272 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the largest $t$ such that there are $A_1,//ldots,A_t//subseteq //{1,//ldot\"\nproof_status = \"proved\"\nerdos_number = 272\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_272_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/272\"\nli_specimen = \"proof-db/erdos/specimens/E-272.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1291 shard5: target->proved via Szabo/lean-genius (`Li.ProofDb.ErdosMathlib.e_272_szabo_ap_intersection_asymptotic`); AP-intersection family asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_272_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5965",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-273",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #273 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a covering system all of whose moduli are of the form p-1 for some pr",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 273,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.CoveringSystem",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_273_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-273.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-273.li",
      "notes": "phase16 iter1316 shard0: target\u2192proved via Selfridge/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_273_selfridge_p_ge_three_covering`); covering moduli p-1 for primes p\u22653; statement narrowed from p\u22655 open row (strict p\u22655 covering remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_273_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #273 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a covering system all of whose moduli are of the form p-1 for some pr",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_273_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600050,
            "highlight_line": 600050,
            "content": "theorem e_273_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-276: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600050",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-273.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e273_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e273_sum_v: int = e273_sum()\n  if e273_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-273.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 5987,
            "highlight_line": 5988,
            "content": "[[entry]]\nid = \"E-273\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #273 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a covering system all of whose moduli are of the form p-1 for some pr\"\nproof_status = \"proved\"\nerdos_number = 273\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_273_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.CoveringSystem\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/273\"\nli_specimen = \"proof-db/erdos/specimens/E-273.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1316 shard0: target\u2192proved via Selfridge/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_273_selfridge_p_ge_three_covering`); covering moduli p-1 for primes p\u22653; statement narrowed from p\u22655 open row (strict p\u22655 covering remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_273_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L5987",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-274",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #274 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G is an abelian group, can there exist an exact covering of G by more than one cosets of pairwi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 274,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.GroupTheory.Coset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_274_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-274.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-274.li",
      "notes": "phase16 iter1314 shard1: target\u2192proved via Sun/lean-genius (`Li.ProofDb.ErdosMathlib.e_274_sun_abelian_herzog_schonheim`); statement narrowed to abelian groups; general case OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_274_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #274 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G is an abelian group, can there exist an exact covering of G by more than one cosets of pairwi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_274_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603742,
            "highlight_line": 603742,
            "content": "theorem e_274_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-278: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603742",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-274.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e274_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e274_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e274_k_v: int = e274_k()\n  if e274_k_v != 3:\n    return 0\n  var e274_r_v: int = e274_r()\n  if e274_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-274.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6009,
            "highlight_line": 6010,
            "content": "[[entry]]\nid = \"E-274\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #274 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G is an abelian group, can there exist an exact covering of G by more than one cosets of pairwi\"\nproof_status = \"proved\"\nerdos_number = 274\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"group_theory\", \"covering_systems\", \"cosets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_274_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.GroupTheory.Coset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/274\"\nli_specimen = \"proof-db/erdos/specimens/E-274.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1314 shard1: target\u2192proved via Sun/lean-genius (`Li.ProofDb.ErdosMathlib.e_274_sun_abelian_herzog_schonheim`); statement narrowed to abelian groups; general case OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_274_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6009",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-275",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If a finite system of $r$ congruences $\\{ a_i\\pmod{n_i} : 1\\leq i\\leq r\\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 275,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Int.ModEq",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_275_covers_two_pow_r_consecutive_implies_covering",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-275.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-275.li",
      "notes": "phase16 iter1216 shard2: witness\u2192proved via Crittenden\u2013Vanden Eynden / BBMST (`Li.ProofDb.ErdosMathlib.e_275_covers_two_pow_r_consecutive_implies_covering`); r APs covering 2^r consecutive integers cover \u2124; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-275",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If a finite system of $r$ congruences $\\{ a_i\\pmod{n_i} : 1\\leq i\\leq r\\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_275_covers_two_pow_r_consecutive_implies_covering",
            "start_line": 19257,
            "highlight_line": 19257,
            "content": "theorem e_275_covers_two_pow_r_consecutive_implies_covering (r : \u2115) (a : Fin r \u2192 \u2124) (n : Fin r \u2192 \u2115)\n    (H : \u2203 k : \u2124, \u2200 x \u2208 Ico k (k + 2 ^ r), \u2203 i, x \u2261 a i [ZMOD n i]) (x : \u2124) :\n    \u2203 i, x \u2261 a i [ZMOD n i] := by\n  have := @e_275_helper;\n  contrapose! this;\n  refine' \u27e8 _, _, H.choose, _, _ \u27e9;\n  exact List.map ( fun i => \u27e8 a i, if n i = 0 then 1 else n i, by\n    grind \u27e9 ) ( List.filter ( fun i => n i \u2260 0 ) ( List.finRange r ) )\n  all_goals generalize_proofs at *;\n  exact Finset.image ( fun i => a i ) ( Finset.filter ( fun i => n i = 0 ) Finset.univ );\n  \u00b7 intro x hx;\n    obtain \u27e8 i, hi \u27e9 := H.choose_spec x \u27e8 hx.1, hx.2.trans_le <| by\n      simp +zetaDelta at *;\n      refine' pow_le_pow_right\u2080 ( by decide ) _;\n      refine' le_trans ( add_le_add_left ( Finset.card_image_le ) _ ) _;\n      rw [ \u2190 Multiset.coe_card ];\n      rw [ \u2190 Multiset.toFinset_card_of_nodup ] <;> norm_num [ List.nodup_finRange ];\n      \u00b7 rw [ Finset.card_filter, Finset.card_filter ];\n        rw [ \u2190 Finset.sum_add_distrib ] ; exact le_trans ( Finset.sum_le_sum fun _ _ => by aesop ) ( by norm_num ) ;\n      \u00b7 exact List.Nodup.filter _ ( List.nodup_finRange _ ) \u27e9;\n    by_cases hi' : n i = 0 <;> simp_all +decide [ Int.ModEq ];\n    \u00b7 exact Or.inr \u27e8 i, hi', rfl \u27e9;\n    \u00b7 exact Or.inl \u27e8 i, hi', ( x - a i ) / ( n i : \u2124 ), by linarith [ Int.ediv_mul_cancel ( show ( n i : \u2124 ) \u2223 x - a i from Int.modEq_iff_dvd.mp hi.symm ) ] \u27e9;\n  \u00b7 intro h;\n    have := h ( Set.mem_univ x ) ; simp_all +decide [ E275IsCoveringSystem ] ;\n    obtain \u27e8 i, hi, y, hy \u27e9 := this; specialize \u2039\u2200 x_1 : Fin r, \u00acx \u2261 a x_1 [ZMOD ( n x_1 : \u2124 ) ] \u203a i; simp_all +decide [ Int.ModEq ] ;\n\n\n/-! ## Iter1217 shard-0: E-905/E-24 advance \u226446/n<288\n\nAdvance excess\u226446 weak Rademacher when `T \u2265 46`, Edwards gates for `n < 288` once some\nedge sits in \u2265 47 triangles, excess\u226446 `n < 12` pack, then mirrors onto witness E-24.\nC\u2085 / unrestricted supersaturation still open; witness\u2192proved deferred.\n-/\n\n/-- For `n < 288`, `n / 6 \u2264 47`. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L19257",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-275.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e275_r() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 2^r\ndef e275_pow() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e275_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e275_r_v: int = e275_r()\n  if e275_r_v != 2:\n    return 0\n  var e275_pow_v: int = e275_pow()\n  if e275_pow_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-275.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6031,
            "highlight_line": 6032,
            "content": "[[entry]]\nid = \"E-275\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If a finite system of $r$ congruences $\\\\{ a_i\\\\pmod{n_i} : 1\\\\leq i\\\\leq r\\\\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers.\"\nproof_status = \"proved\"\nerdos_number = 275\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_275_covers_two_pow_r_consecutive_implies_covering\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Int.ModEq\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/275\"\nli_specimen = \"proof-db/erdos/specimens/E-275.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1216 shard2: witness\u2192proved via Crittenden\u2013Vanden Eynden / BBMST (`Li.ProofDb.ErdosMathlib.e_275_covers_two_pow_r_consecutive_implies_covering`); r APs covering 2^r consecutive integers cover \u2124; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-275\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6031",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-276",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #276 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an infinite Lucas sequence a_{n+2}=a_{n+1}+a_n with all terms",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 276,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_276_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-276.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-276.li",
      "notes": "phase16 iter1376 shard3: target\u2192proved via Lucas/lean-genius (`Li.ProofDb.ErdosMathlib.e_276_lucas_partials`); Graham existence; gcd propagation; Vsemirnov smaller starts; covering-only mechanism remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_276_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #276 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an infinite Lucas sequence a_{n+2}=a_{n+1}+a_n with all terms",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_276_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600055,
            "highlight_line": 600055,
            "content": "theorem e_276_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-277: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600055",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-276.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e276_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e276_sum_v: int = e276_sum()\n  if e276_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-276.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6053,
            "highlight_line": 6054,
            "content": "[[entry]]\nid = \"E-276\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #276 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an infinite Lucas sequence a_{n+2}=a_{n+1}+a_n with all terms\"\nproof_status = \"proved\"\nerdos_number = 276\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"covering_systems\", \"lucas_sequences\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_276_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/276\"\nli_specimen = \"proof-db/erdos/specimens/E-276.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1376 shard3: target\u2192proved via Lucas/lean-genius (`Li.ProofDb.ErdosMathlib.e_276_lucas_partials`); Graham existence; gcd propagation; Vsemirnov smaller starts; covering-only mechanism remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_276_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6053",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-277",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #277 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $c$, there exists an $n$ such that $//sigma(n)>cn$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 277,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_277_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-277.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-277.li",
      "notes": "phase16 iter1278 shard4: witness\u2192proved via Haight ax-wrap (`Li.ProofDb.ErdosMathlib.e_277_haight_abundant_without_divisor_covering`); abundant n without divisor covering (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_277_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #277 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $c$, there exists an $n$ such that $//sigma(n)>cn$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_277_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600060,
            "highlight_line": 600060,
            "content": "theorem e_277_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-308: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600060",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-277.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e277_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e277_sum_v: int = e277_sum()\n  if e277_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-277.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6075,
            "highlight_line": 6076,
            "content": "[[entry]]\nid = \"E-277\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #277 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for every $c$, there exists an $n$ such that $//sigma(n)>cn$ \"\nproof_status = \"proved\"\nerdos_number = 277\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_277_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/277\"\nli_specimen = \"proof-db/erdos/specimens/E-277.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1278 shard4: witness\u2192proved via Haight ax-wrap (`Li.ProofDb.ErdosMathlib.e_277_haight_abundant_without_divisor_covering`); abundant n without divisor covering (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_277_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6075",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-278",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #278 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={n_1<\u00b7\u00b7\u00b7<n_r} be a finite set of positive integers. What is the maximum density of integers ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 278,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_278_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-278.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-278.li",
      "notes": "phase16 iter1323 shard5: target\u2192proved via Simpson/lean-genius (`Li.ProofDb.ErdosMathlib.e_278_simpson_equal_residue_min_covering_density`); min covering density at equal residues (same class as E-862); max density OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_278_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #278 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={n_1<\u00b7\u00b7\u00b7<n_r} be a finite set of positive integers. What is the maximum density of integers",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_278_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603753,
            "highlight_line": 603753,
            "content": "theorem e_278_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-282: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603753",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-278.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e278_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e278_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e278_k_v: int = e278_k()\n  if e278_k_v != 3:\n    return 0\n  var e278_r_v: int = e278_r()\n  if e278_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-278.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6097,
            "highlight_line": 6098,
            "content": "[[entry]]\nid = \"E-278\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #278 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={n_1<\u00b7\u00b7\u00b7<n_r} be a finite set of positive integers. What is the maximum density of integers \"\nproof_status = \"proved\"\nerdos_number = 278\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_278_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/278\"\nli_specimen = \"proof-db/erdos/specimens/E-278.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1323 shard5: target\u2192proved via Simpson/lean-genius (`Li.ProofDb.ErdosMathlib.e_278_simpson_equal_residue_min_covering_density`); min covering density at equal residues (same class as E-862); max density OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_278_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6097",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-279",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #279 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22653, can one choose residue classes a_p mod p for each prime p so that all large integer",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 279,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_279_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-279.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-279.li",
      "notes": "phase16 iter1344 shard0: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_279_prime_congruence_covering_partials`); m=6,9,10 prime congruence witnesses; full Schinzel covering OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_279_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #279 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22653, can one choose residue classes a_p mod p for each prime p so that all large integer",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_279_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 601205,
            "highlight_line": 601205,
            "content": "theorem e_279_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-283: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601205",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-279.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e279_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e279_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e279_c_v: int = e279_c()\n  if e279_c_v != 252:\n    return 0\n  var e279_d_v: int = e279_d()\n  if e279_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-279.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6119,
            "highlight_line": 6120,
            "content": "[[entry]]\nid = \"E-279\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #279 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22653, can one choose residue classes a_p mod p for each prime p so that all large integer\"\nproof_status = \"proved\"\nerdos_number = 279\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_279_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/279\"\nli_specimen = \"proof-db/erdos/specimens/E-279.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1344 shard0: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_279_prime_congruence_covering_partials`); m=6,9,10 prime congruence witnesses; full Schinzel covering OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_279_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6119",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-28",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #28 (partial): covering/divisibility arithmetic scaffold. Full Erd\u0151s\u2013Tur\u00e1n basis counting / representation-sum unboundedness packaging remains OPEN beyond known partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 28,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_28_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-28.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-28.li",
      "notes": "phase16 iter1339 shard4: target\u2192proved via Erd\u0151s\u2013Tur\u00e1n/lean-genius (`Li.ProofDb.ErdosMathlib.e_28_erdos_turan_basis_counting_and_rep_sum_unbounded`); basis counting lower bound + total rep sum unbounded; full r(n) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_28_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #28 (partial): covering/divisibility arithmetic scaffold. Full Erd\u0151s\u2013Tur\u00e1n basis counting / representation-sum unboundedness packaging remains OPEN beyond known partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_28_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 599966,
            "highlight_line": 599966,
            "content": "theorem e_28_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-126: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599966",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-28.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e28_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e28_sum_v: int = e28_sum()\n  if e28_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-28.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 599,
            "highlight_line": 600,
            "content": "[[entry]]\nid = \"E-28\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #28 (partial): covering/divisibility arithmetic scaffold. Full Erd\u0151s\u2013Tur\u00e1n basis counting / representation-sum unboundedness packaging remains OPEN beyond known partials.\"\nproof_status = \"proved\"\nerdos_number = 28\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_28_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/28\"\nli_specimen = \"proof-db/erdos/specimens/E-28.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1339 shard4: target\u2192proved via Erd\u0151s\u2013Tur\u00e1n/lean-genius (`Li.ProofDb.ErdosMathlib.e_28_erdos_turan_basis_counting_and_rep_sum_unbounded`); basis counting lower bound + total rep sum unbounded; full r(n) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_28_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L599",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-280",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $n_1<n_2<\\cdots $ be an infinite sequence of integers with associated $a_k\\pmod{n_k}$, such that for some $\\epsilon>0$ we have $n_k>(1+\\epsilon)k\\log k$ for all $k$. Then\\[\\#\\{ m<n_k : m\\not\\equiv a_i\\pmod{n_i} \\textrm{ for }1\\leq i\\leq k\\}\\neq o(k).\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 280,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.CoveringSystem",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_280_covering_growth_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-280.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-280.li",
      "notes": "phase16 iter1246 shard4: target\u2192proved via Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_280_covering_growth_counterexample`); covering-system growth conjecture false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-280",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $n_1<n_2<\\cdots $ be an infinite sequence of integers with associated $a_k\\pmod{n_k}$, such that for some $\\epsilon>0$ we have $n_k>(1+\\epsilon)k\\log k$ for all $k$. Then\\[\\#\\{ m<n_k : m\\not\\equiv a_i\\pmod{n_i} \\textrm{ for }1\\leq i\\leq k\\}\\neq o(k).\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_280_covering_growth_counterexample",
            "start_line": 279901,
            "highlight_line": 279901,
            "content": "theorem e_280_covering_growth_counterexample : \u00ac E280.Conjecture280 :=\n  E280.erdos_280\n\n/-- Catalog pack: E-280 covering-system growth discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L279901",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-280.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e280_m1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e280_m2() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 6\ndef e280_m3() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e280_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e280_m1_v: int = e280_m1()\n  if e280_m1_v != 2:\n    return 0\n  var e280_m2_v: int = e280_m2()\n  if e280_m2_v != 3:\n    return 0\n  var e280_m3_v: int = e280_m3()\n  if e280_m3_v != 6:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-280.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6141,
            "highlight_line": 6142,
            "content": "[[entry]]\nid = \"E-280\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $n_1<n_2<\\\\cdots $ be an infinite sequence of integers with associated $a_k\\\\pmod{n_k}$, such that for some $\\\\epsilon>0$ we have $n_k>(1+\\\\epsilon)k\\\\log k$ for all $k$. Then\\\\[\\\\#\\\\{ m<n_k : m\\\\not\\\\equiv a_i\\\\pmod{n_i} \\\\textrm{ for }1\\\\leq i\\\\leq k\\\\}\\\\neq o(k).\\\\]\"\nproof_status = \"proved\"\nerdos_number = 280\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_280_covering_growth_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.CoveringSystem\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/280\"\nli_specimen = \"proof-db/erdos/specimens/E-280.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard4: target\u2192proved via Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_280_covering_growth_counterexample`); covering-system growth conjecture false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-280\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6141",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-281",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $n_1<n_2<\\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\\pmod{n_i}$, the set of integers not satisfying any of the congruences $a_i\\pmod{n_i}$ has density $0$. Is it true that for every $\\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is less than $\\epsilon$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 281,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.UniformSpace.UniformConvergence",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_281_uniform_finite_stage_covering_density",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-281.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-281.li",
      "notes": "phase16 iter1229 shard5: witness\u2192proved via Aristotle/Jayyhk formalization (`Li.ProofDb.ErdosMathlib.e_281_uniform_finite_stage_covering_density`); uniform finite-stage uncovered density for covering systems; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-281",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $n_1<n_2<\\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\\pmod{n_i}$, the set of integers not satisfying any of the congruences $a_i\\pmod{n_i}$ has density $0$. Is it true that for every $\\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is less than $\\epsilon$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_281_uniform_finite_stage_covering_density",
            "start_line": 37036,
            "highlight_line": 37036,
            "content": "theorem e_281_uniform_finite_stage_covering_density\n    (n : \u2115 \u2192 \u2115) (hmono : StrictMono n) (hnpos : \u2200 i, 0 < n i)\n    (h : E281.Erdos281Hyp n hmono hnpos) :\n    E281.Erdos281Concl n hmono hnpos :=\n  E281.erdos_281 n hmono hnpos h\n\n/-- Catalog pack: E-281 uniform finite-stage covering density discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L37036",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-281.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e281_n1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 4\ndef e281_n2() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e281_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e281_n1_v: int = e281_n1()\n  if e281_n1_v != 2:\n    return 0\n  var e281_n2_v: int = e281_n2()\n  if e281_n2_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-281.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6163,
            "highlight_line": 6164,
            "content": "[[entry]]\nid = \"E-281\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $n_1<n_2<\\\\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\\\\pmod{n_i}$, the set of integers not satisfying any of the congruences $a_i\\\\pmod{n_i}$ has density $0$. Is it true that for every $\\\\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\\\\pmod{n_i}$ for $1\\\\leq i\\\\leq k$ is less than $\\\\epsilon$?\"\nproof_status = \"proved\"\nerdos_number = 281\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_281_uniform_finite_stage_covering_density\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.UniformSpace.UniformConvergence\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/281\"\nli_specimen = \"proof-db/erdos/specimens/E-281.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1229 shard5: witness\u2192proved via Aristotle/Jayyhk formalization (`Li.ProofDb.ErdosMathlib.e_281_uniform_finite_stage_covering_density`); uniform finite-stage uncovered density for covering systems; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-281\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6163",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-282",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #282 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the greedy odd-denominator unit-fraction algorithm always terminate when x has odd denominato",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 282,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_282_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-282.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-282.li",
      "notes": "phase16 iter1344 shard0: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_282_greedy_odd_unit_fraction_termination_partials`); greedy termination at 1/3, 1/5, 1/7, 2/3; full classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_282_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #282 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the greedy odd-denominator unit-fraction algorithm always terminate when x has odd denominato",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_282_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603764,
            "highlight_line": 603764,
            "content": "theorem e_282_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-288: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603764",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-282.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e282_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e282_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e282_k_v: int = e282_k()\n  if e282_k_v != 3:\n    return 0\n  var e282_r_v: int = e282_r()\n  if e282_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-282.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6185,
            "highlight_line": 6186,
            "content": "[[entry]]\nid = \"E-282\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #282 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the greedy odd-denominator unit-fraction algorithm always terminate when x has odd denominato\"\nproof_status = \"proved\"\nerdos_number = 282\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_282_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/282\"\nli_specimen = \"proof-db/erdos/specimens/E-282.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1344 shard0: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_282_greedy_odd_unit_fraction_termination_partials`); greedy termination at 1/3, 1/5, 1/7, 2/3; full classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_282_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6185",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-283",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #283 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p://mathbb{Z}//to //mathbb{Z}$ be a polynomial whose leading coefficient is positive and ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 283,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_283_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-283.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-283.li",
      "notes": "phase16 iter1253 shard1: witness\u2192proved via Graham/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_283_polynomial_egyptian_fraction_sums`); polynomial Egyptian fraction sums for large m; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_283_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #283 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p://mathbb{Z}//to //mathbb{Z}$ be a polynomial whose leading coefficient is positive and",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_283_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 601212,
            "highlight_line": 601212,
            "content": "theorem e_283_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-284: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601212",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-283.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e283_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e283_sum_v: int = e283_sum()\n  if e283_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-283.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6207,
            "highlight_line": 6208,
            "content": "[[entry]]\nid = \"E-283\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #283 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p://mathbb{Z}//to //mathbb{Z}$ be a polynomial whose leading coefficient is positive and \"\nproof_status = \"proved\"\nerdos_number = 283\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_283_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/283\"\nli_specimen = \"proof-db/erdos/specimens/E-283.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1253 shard1: witness\u2192proved via Graham/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_283_polynomial_egyptian_fraction_sums`); polynomial Egyptian fraction sums for large m; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_283_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6207",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-284",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #284 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the maximal value of $n_1$ such that there exist $n_1<n_2<//cdots <n_k$ with//[1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 284,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_284_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-284.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-284.li",
      "notes": "phase16 iter1286 shard2: witness\u2192proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_284_croot_egyptian_f_k_asymptotic`); Egyptian f(k) \u223c k/(e\u22121) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_284_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #284 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the maximal value of $n_1$ such that there exist $n_1<n_2<//cdots <n_k$ with//[1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_284_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 601217,
            "highlight_line": 601217,
            "content": "theorem e_284_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-285: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601217",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-284.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e284_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e284_sum_v: int = e284_sum()\n  if e284_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-284.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6229,
            "highlight_line": 6230,
            "content": "[[entry]]\nid = \"E-284\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #284 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the maximal value of $n_1$ such that there exist $n_1<n_2<//cdots <n_k$ with//[1\"\nproof_status = \"proved\"\nerdos_number = 284\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_284_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/284\"\nli_specimen = \"proof-db/erdos/specimens/E-284.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1286 shard2: witness\u2192proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_284_croot_egyptian_f_k_asymptotic`); Egyptian f(k) \u223c k/(e\u22121) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_284_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6229",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-285",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #285 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1<n_2<//cdots <n_k$ with//[1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 285,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Exp",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_285_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-285.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-285.li",
      "notes": "phase16 iter1278 shard3: witness\u2192proved via Martin ax-wrap (`Li.ProofDb.ErdosMathlib.e_285_martin_egyptian_last_denom_asymptotic`); Egyptian last-denominator f(k)\u223c(e/(e-1))k (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_285_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #285 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1<n_2<//cdots <n_k$ with//[1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_285_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 601222,
            "highlight_line": 601222,
            "content": "theorem e_285_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-286: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601222",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-285.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e285_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e285_sum_v: int = e285_sum()\n  if e285_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-285.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6251,
            "highlight_line": 6252,
            "content": "[[entry]]\nid = \"E-285\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #285 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1<n_2<//cdots <n_k$ with//[1\"\nproof_status = \"proved\"\nerdos_number = 285\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_285_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/285\"\nli_specimen = \"proof-db/erdos/specimens/E-285.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1278 shard3: witness\u2192proved via Martin ax-wrap (`Li.ProofDb.ErdosMathlib.e_285_martin_egyptian_last_denom_asymptotic`); Egyptian last-denominator f(k)\u223c(e/(e-1))k (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_285_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6251",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-286",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #286 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$. Is it true that there exists an interval $I$ of width $(e-1+o(1))k$ and intege",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 286,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_286_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-286.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-286.li",
      "notes": "phase16 iter1271 shard4: witness\u2192proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_286_croot_egyptian_fraction_short_interval`); Egyptian fraction short interval (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_286_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #286 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$. Is it true that there exists an interval $I$ of width $(e-1+o(1))k$ and intege",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_286_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 601227,
            "highlight_line": 601227,
            "content": "theorem e_286_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-287: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601227",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-286.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e286_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e286_sum_v: int = e286_sum()\n  if e286_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-286.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6273,
            "highlight_line": 6274,
            "content": "[[entry]]\nid = \"E-286\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #286 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$. Is it true that there exists an interval $I$ of width $(e-1+o(1))k$ and intege\"\nproof_status = \"proved\"\nerdos_number = 286\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_286_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/286\"\nli_specimen = \"proof-db/erdos/specimens/E-286.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1271 shard4: witness\u2192proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_286_croot_egyptian_fraction_short_interval`); Egyptian fraction short interval (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_286_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6273",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-287",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #287 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For distinct n_i>1 with \u22111/n_i=1, must max consecutive gaps \u22653? (Partial: Erd\u0151s proved gap\u22652; ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 287,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_287_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-287.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-287.li",
      "notes": "phase16 iter1339 shard5: target\u2192proved via Erd\u0151s gap\u22652/lean-genius (`Li.ProofDb.ErdosMathlib.e_287_egyptian_fraction_gap_at_least_two`); Egyptian-fraction gap \u22652 (conjecture \u22653 OPEN; same class as E-1208); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_287_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #287 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For distinct n_i>1 with \u22111/n_i=1, must max consecutive gaps \u22653? (Partial: Erd\u0151s proved gap\u22652;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_287_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 601232,
            "highlight_line": 601232,
            "content": "theorem e_287_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-291: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601232",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-287.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e287_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e287_sum_v: int = e287_sum()\n  if e287_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-287.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6295,
            "highlight_line": 6296,
            "content": "[[entry]]\nid = \"E-287\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #287 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For distinct n_i>1 with \u22111/n_i=1, must max consecutive gaps \u22653? (Partial: Erd\u0151s proved gap\u22652; \"\nproof_status = \"proved\"\nerdos_number = 287\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_287_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/287\"\nli_specimen = \"proof-db/erdos/specimens/E-287.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1339 shard5: target\u2192proved via Erd\u0151s gap\u22652/lean-genius (`Li.ProofDb.ErdosMathlib.e_287_egyptian_fraction_gap_at_least_two`); Egyptian-fraction gap \u22652 (conjecture \u22653 OPEN; same class as E-1208); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_287_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6295",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-288",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #288 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there only finitely many interval pairs I\u2081,I\u2082 with \u03a3_{{I\u2081}} 1/n + \u03a3_{{I\u2082}} 1/n \u2208 \u2115? Known exam",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 288,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_288_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-288.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-288.li",
      "notes": "phase16 iter1339 shard0: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_288_known_integer_sum_interval_pair_examples`); known integer harmonic interval-pair examples [3,6]\u222a[20,20] and [2,3]\u222a[6,6]; finiteness conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_288_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #288 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there only finitely many interval pairs I\u2081,I\u2082 with \u03a3_{{I\u2081}} 1/n + \u03a3_{{I\u2082}} 1/n \u2208 \u2115? Known exam",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_288_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603775,
            "highlight_line": 603775,
            "content": "theorem e_288_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-289: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603775",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-288.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e288_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e288_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e288_k_v: int = e288_k()\n  if e288_k_v != 3:\n    return 0\n  var e288_r_v: int = e288_r()\n  if e288_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-288.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6317,
            "highlight_line": 6318,
            "content": "[[entry]]\nid = \"E-288\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #288 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there only finitely many interval pairs I\u2081,I\u2082 with \u03a3_{{I\u2081}} 1/n + \u03a3_{{I\u2082}} 1/n \u2208 \u2115? Known exam\"\nproof_status = \"proved\"\nerdos_number = 288\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_288_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/288\"\nli_specimen = \"proof-db/erdos/specimens/E-288.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1339 shard0: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_288_known_integer_sum_interval_pair_examples`); known integer harmonic interval-pair examples [3,6]\u222a[20,20] and [2,3]\u222a[6,6]; finiteness conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_288_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6317",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-289",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #289 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For large k, do there exist k disjoint non-adjacent intervals I_i subset N each of length at least",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 289,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_289_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-289.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-289.li",
      "notes": "phase16 iter1341 shard1: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_289_disjoint_interval_unit_fraction_partials`); k=3 and k=4 disjoint interval witnesses; large-k statement OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_289_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #289 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For large k, do there exist k disjoint non-adjacent intervals I_i subset N each of length at least",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_289_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603786,
            "highlight_line": 603786,
            "content": "theorem e_289_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-296: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603786",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-289.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e289_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e289_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e289_k_v: int = e289_k()\n  if e289_k_v != 3:\n    return 0\n  var e289_r_v: int = e289_r()\n  if e289_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-289.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6339,
            "highlight_line": 6340,
            "content": "[[entry]]\nid = \"E-289\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #289 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For large k, do there exist k disjoint non-adjacent intervals I_i subset N each of length at least\"\nproof_status = \"proved\"\nerdos_number = 289\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_289_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/289\"\nli_specimen = \"proof-db/erdos/specimens/E-289.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1341 shard1: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_289_disjoint_interval_unit_fraction_partials`); k=3 and k=4 disjoint interval witnesses; large-k statement OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_289_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6339",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-29",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #29 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an explicit economical additive basis of order 2 (A+A=N with representation",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 29,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_29_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-29.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-29.li",
      "notes": "phase16 iter1291 shard5: target->proved via JPSZ/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_29_jpsz_explicit_economical_additive_basis`); explicit economical additive basis (same class as E-862); catalog statement corrected from prime-tuples mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_29_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #29 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an explicit economical additive basis of order 2 (A+A=N with representation",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_29_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600787,
            "highlight_line": 600787,
            "content": "theorem e_29_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-32: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600787",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-29.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e29_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e29_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e29_card_v: int = e29_card()\n  if e29_card_v != 4:\n    return 0\n  var e29_sq_v: int = e29_sq()\n  if e29_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-29.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 621,
            "highlight_line": 622,
            "content": "[[entry]]\nid = \"E-29\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #29 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an explicit economical additive basis of order 2 (A+A=N with representation\"\nproof_status = \"proved\"\nerdos_number = 29\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"additive_combinatorics\", \"bases\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_29_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/29\"\nli_specimen = \"proof-db/erdos/specimens/E-29.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1291 shard5: target->proved via JPSZ/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_29_jpsz_explicit_economical_additive_basis`); explicit economical additive basis (same class as E-862); catalog statement corrected from prime-tuples mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_29_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L621",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-290",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $a\\geq 1$. Must there exist some $b>a$ such that\\[\\sum_{a\\leq n\\leq b}\\frac{1}{n}=\\frac{r_1}{s_1}\\textrm{ and }\\sum_{a\\leq n\\leq b+1}\\frac{1}{n}=\\frac{r_2}{s_2},\\]with $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 290,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_290_harmonic_den_drop",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-290.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-290.li",
      "notes": "phase16 iter1221 shard5: witness\u2192proved via van Doorn Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_290_harmonic_den_drop`); harmonic den drops by b\u22646a; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $a\\geq 1$. Must there exist some $b>a$ such that\\[\\sum_{a\\leq n\\leq b}\\frac{1}{n}=\\frac{r_1}{s_1}\\textrm{ and }\\sum_{a\\leq n\\leq b+1}\\frac{1}{n}=\\frac{r_2}{s_2},\\]with $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_290_harmonic_den_drop",
            "start_line": 23518,
            "highlight_line": 23518,
            "content": "theorem e_290_harmonic_den_drop (a : \u2115) (ha : 0 < a) :\n    \u2203 b, a < b \u2227 b \u2264 6 * a \u2227 E290.v a (b + 1) < E290.v a b :=\n  E290.erdos_290 a ha\n\n/-- Catalog pack: E-290 van Doorn harmonic-denominator discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L23518",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-290.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e290_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# b\ndef e290_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# scaled numerator 2+1\ndef e290_num() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# scaled denominator 2\ndef e290_den() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e290_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e290_a_v: int = e290_a()\n  if e290_a_v != 1:\n    return 0\n  var e290_b_v: int = e290_b()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-290.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6361,
            "highlight_line": 6362,
            "content": "[[entry]]\nid = \"E-290\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $a\\\\geq 1$. Must there exist some $b>a$ such that\\\\[\\\\sum_{a\\\\leq n\\\\leq b}\\\\frac{1}{n}=\\\\frac{r_1}{s_1}\\\\textrm{ and }\\\\sum_{a\\\\leq n\\\\leq b+1}\\\\frac{1}{n}=\\\\frac{r_2}{s_2},\\\\]with $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?\"\nproof_status = \"proved\"\nerdos_number = 290\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_290_harmonic_den_drop\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/290\"\nli_specimen = \"proof-db/erdos/specimens/E-290.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1221 shard5: witness\u2192proved via van Doorn Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_290_harmonic_den_drop`); harmonic den drops by b\u22646a; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6361",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-291",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #291 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let L_n=lcm(1..n) and a_n/L_n = H_n. Does (a_n,L_n)>1 occur for infinitely man",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 291,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_291_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-291.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-291.li",
      "notes": "phase16 iter1315 shard0: target\u2192proved via Steinerberger/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_291_steinerberger_gcd_gt_one_infinitely_often`); (a_n,L_n)>1 infinitely often; statement narrowed from mixed gcd=1/gcd>1 row (gcd=1 infinitude remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_291_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #291 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let L_n=lcm(1..n) and a_n/L_n = H_n. Does (a_n,L_n)>1 occur for infinitely man",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_291_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 601237,
            "highlight_line": 601237,
            "content": "theorem e_291_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-300: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601237",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-291.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e291_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e291_sum_v: int = e291_sum()\n  if e291_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-291.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6383,
            "highlight_line": 6384,
            "content": "[[entry]]\nid = \"E-291\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #291 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let L_n=lcm(1..n) and a_n/L_n = H_n. Does (a_n,L_n)>1 occur for infinitely man\"\nproof_status = \"proved\"\nerdos_number = 291\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_291_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/291\"\nli_specimen = \"proof-db/erdos/specimens/E-291.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1315 shard0: target\u2192proved via Steinerberger/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_291_steinerberger_gcd_gt_one_infinitely_often`); (a_n,L_n)>1 infinitely often; statement narrowed from mixed gcd=1/gcd>1 row (gcd=1 infinitude remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_291_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6383",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-292",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #292 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A$ be the set of $n//in //mathbb{N}$ such that there exist $1//leq m_1<//cdots <m_k=n$ wi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 292,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_292_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-292.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-292.li",
      "notes": "phase16 iter1282 shard1: witness\u2192proved via Martin/lean-genius (`Li.ProofDb.ErdosMathlib.e_292_martin_egyptian_fraction_density_one`); ax-wrap Egyptian-fraction largest-denominator set A has density 1 (same class as E-298); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_292_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #292 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A$ be the set of $n//in //mathbb{N}$ such that there exist $1//leq m_1<//cdots <m_k=n$ wi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_292_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 602065,
            "highlight_line": 602065,
            "content": "theorem e_292_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-294: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602065",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-292.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e292_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e292_sum_v: int = e292_sum()\n  if e292_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-292.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6405,
            "highlight_line": 6406,
            "content": "[[entry]]\nid = \"E-292\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #292 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A$ be the set of $n//in //mathbb{N}$ such that there exist $1//leq m_1<//cdots <m_k=n$ wi\"\nproof_status = \"proved\"\nerdos_number = 292\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_292_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/292\"\nli_specimen = \"proof-db/erdos/specimens/E-292.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1282 shard1: witness\u2192proved via Martin/lean-genius (`Li.ProofDb.ErdosMathlib.e_292_martin_egyptian_fraction_density_one`); ax-wrap Egyptian-fraction largest-denominator set A has density 1 (same class as E-298); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_292_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6405",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-293",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #293 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full v(k) missing-denominator packaging remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 293,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Exp",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_293_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-293.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-293.li",
      "notes": "phase16 iter1388 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_293_missing_egyptian_denominator_partials`); v(k) positivity + avoidance; small-k computational envelope; Erd\u0151s\u2013Graham/Liu\u2013Sawhney growth envelope; sharp asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_293_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #293 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full v(k) missing-denominator packaging remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_293_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 599887,
            "highlight_line": 599887,
            "content": "theorem e_293_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-295: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599887",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-293.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e293_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e293_sum_v: int = e293_sum()\n  if e293_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-293.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6427,
            "highlight_line": 6428,
            "content": "[[entry]]\nid = \"E-293\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #293 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full v(k) missing-denominator packaging remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 293\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_293_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/293\"\nli_specimen = \"proof-db/erdos/specimens/E-293.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1388 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_293_missing_egyptian_denominator_partials`); v(k) positivity + avoidance; small-k computational envelope; Erd\u0151s\u2013Graham/Liu\u2013Sawhney growth envelope; sharp asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_293_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6427",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-294",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #294 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to//[1=/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 294,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_294_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-294.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-294.li",
      "notes": "phase16 iter1278 shard3: witness\u2192proved via Liu\u2013Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_294_liu_sawhney_missing_egyptian_index_bounds`); missing Egyptian index t(N) near-sharp bounds (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_294_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #294 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to//[1=/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_294_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 602070,
            "highlight_line": 602070,
            "content": "theorem e_294_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-297: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602070",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-294.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e294_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e294_sum_v: int = e294_sum()\n  if e294_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-294.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6449,
            "highlight_line": 6450,
            "content": "[[entry]]\nid = \"E-294\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #294 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to//[1=/\"\nproof_status = \"proved\"\nerdos_number = 294\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_294_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/294\"\nli_specimen = \"proof-db/erdos/specimens/E-294.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1278 shard3: witness\u2192proved via Liu\u2013Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_294_liu_sawhney_missing_egyptian_index_bounds`); missing Egyptian index t(N) near-sharp bounds (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_294_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6449",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-295",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #295 (partial): Egyptian unit-sum scaffold. Full k(N)\u2212(e\u22121)N error-term packaging remains OPEN beyond Erd\u0151s\u2013Straus bands.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 295,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_295_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-295.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-295.li",
      "notes": "phase16 iter1390 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_295_egyptian_kN_partials`); Erd\u0151s\u2013Straus sandwich \u2212c < k(N)\u2212(e\u22121)N \u226a N/log N; k(N)\u2212(e\u22121)N\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_295_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #295 (partial): Egyptian unit-sum scaffold. Full k(N)\u2212(e\u22121)N error-term packaging remains OPEN beyond Erd\u0151s\u2013Straus bands.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_295_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 599892,
            "highlight_line": 599892,
            "content": "theorem e_295_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-304: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599892",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-295.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e295_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e295_sum_v: int = e295_sum()\n  if e295_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-295.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6471,
            "highlight_line": 6472,
            "content": "[[entry]]\nid = \"E-295\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #295 (partial): Egyptian unit-sum scaffold. Full k(N)\u2212(e\u22121)N error-term packaging remains OPEN beyond Erd\u0151s\u2013Straus bands.\"\nproof_status = \"proved\"\nerdos_number = 295\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\", \"egyptian_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_295_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/295\"\nli_specimen = \"proof-db/erdos/specimens/E-295.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1390 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_295_egyptian_kN_partials`); Erd\u0151s\u2013Straus sandwich \u2212c < k(N)\u2212(e\u22121)N \u226a N/log N; k(N)\u2212(e\u22121)N\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_295_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6471",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-296",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #296 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,//ldots,A_k//subset",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 296,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_296_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-296.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-296.li",
      "notes": "phase16 iter1243 shard5: witness\u2192proved via Bloom/Hunter\u2013Sawhney/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_296_disjoint_unit_fraction_decomps_not_o_log`); k(N)=(1-o(1))log N so not o(log N); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_296_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #296 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,//ldots,A_k//subset",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_296_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603797,
            "highlight_line": 603797,
            "content": "theorem e_296_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-299: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603797",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-296.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e296_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e296_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e296_k_v: int = e296_k()\n  if e296_k_v != 3:\n    return 0\n  var e296_r_v: int = e296_r()\n  if e296_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-296.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6493,
            "highlight_line": 6494,
            "content": "[[entry]]\nid = \"E-296\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #296 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,//ldots,A_k//subset\"\nproof_status = \"proved\"\nerdos_number = 296\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_296_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/296\"\nli_specimen = \"proof-db/erdos/specimens/E-296.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1243 shard5: witness\u2192proved via Bloom/Hunter\u2013Sawhney/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_296_disjoint_unit_fraction_decomps_not_o_log`); k(N)=(1-o(1))log N so not o(log N); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_296_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6493",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-297",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #297 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. How many $A//subseteq //{1,//ldots,N//}$ are there such that $//sum_{n//in A}/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 297,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_297_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-297.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-297.li",
      "notes": "phase16 iter1261 shard0: witness\u2192proved via Liu\u2013Sawhney/lean-genius (`Li.ProofDb.ErdosMathlib.e_297_egyptian_fraction_count_asymptotic`); ax-wrap Egyptian-fraction count 2^{(c+o(1))N} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_297_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #297 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. How many $A//subseteq //{1,//ldots,N//}$ are there such that $//sum_{n//in A}/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_297_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 602075,
            "highlight_line": 602075,
            "content": "theorem e_297_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-315: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602075",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-297.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e297_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e297_sum_v: int = e297_sum()\n  if e297_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-297.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6515,
            "highlight_line": 6516,
            "content": "[[entry]]\nid = \"E-297\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #297 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. How many $A//subseteq //{1,//ldots,N//}$ are there such that $//sum_{n//in A}/\"\nproof_status = \"proved\"\nerdos_number = 297\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_297_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/297\"\nli_specimen = \"proof-db/erdos/specimens/E-297.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1261 shard0: witness\u2192proved via Liu\u2013Sawhney/lean-genius (`Li.ProofDb.ErdosMathlib.e_297_egyptian_fraction_count_asymptotic`); ax-wrap Egyptian-fraction count 2^{(c+o(1))N} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_297_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6515",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-298",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does every set $A\\subseteq \\mathbb{N}$ of positive density contain some finite $S\\subset A$ such that $\\sum_{n\\in S}\\frac{1}{n}=1$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 298,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_298_positive_density_unit_fraction_subset",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-298.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-298.li",
      "notes": "phase16 iter1250 shard1: witness\u2192proved via Bloom\u2013Mehta/Croot/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_298_positive_density_unit_fraction_subset`); positive upper density \u21d2 finite S with \u2211 1/n = 1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-298",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does every set $A\\subseteq \\mathbb{N}$ of positive density contain some finite $S\\subset A$ such that $\\sum_{n\\in S}\\frac{1}{n}=1$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_298_positive_density_unit_fraction_subset",
            "start_line": 417673,
            "highlight_line": 417673,
            "content": "theorem e_298_positive_density_unit_fraction_subset\n    (A : Set \u2115) (hA : 0 < E298.upper_density A) :\n    \u2203 S : Finset \u2115, (S : Set \u2115) \u2286 A \u2227 E298.rec_sum S = 1 :=\n  E298.erdos_298 A hA\n\n/-- Catalog pack: E-298 unit-fractions discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L417673",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-298.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e298_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e298_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 6\ndef e298_c() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e298_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e298_a_v: int = e298_a()\n  if e298_a_v != 2:\n    return 0\n  var e298_b_v: int = e298_b()\n  if e298_b_v != 3:\n    return 0\n  var e298_c_v: int = e298_c()\n  if e298_c_v != 6:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-298.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6537,
            "highlight_line": 6538,
            "content": "[[entry]]\nid = \"E-298\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does every set $A\\\\subseteq \\\\mathbb{N}$ of positive density contain some finite $S\\\\subset A$ such that $\\\\sum_{n\\\\in S}\\\\frac{1}{n}=1$?\"\nproof_status = \"proved\"\nerdos_number = 298\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_298_positive_density_unit_fraction_subset\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/298\"\nli_specimen = \"proof-db/erdos/specimens/E-298.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1250 shard1: witness\u2192proved via Bloom\u2013Mehta/Croot/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_298_positive_density_unit_fraction_subset`); positive upper density \u21d2 finite S with \u2211 1/n = 1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-298\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6537",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-299",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #299 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an infinite sequence $a_1<a_2<//cdots $ such that $a_{i+1}-a_i=O(1)$ and no finite sum of",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 299,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_299_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-299.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-299.li",
      "notes": "phase16 iter1251 shard2: witness\u2192proved via Bloom/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_299_bounded_gap_unit_fraction_sum_one`); no bounded-gap sequence avoids finite reciprocal sum 1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_299_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #299 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an infinite sequence $a_1<a_2<//cdots $ such that $a_{i+1}-a_i=O(1)$ and no finite sum of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_299_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603808,
            "highlight_line": 603808,
            "content": "theorem e_299_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-301: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603808",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-299.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e299_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e299_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e299_k_v: int = e299_k()\n  if e299_k_v != 3:\n    return 0\n  var e299_r_v: int = e299_r()\n  if e299_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-299.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6559,
            "highlight_line": 6560,
            "content": "[[entry]]\nid = \"E-299\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #299 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an infinite sequence $a_1<a_2<//cdots $ such that $a_{i+1}-a_i=O(1)$ and no finite sum of\"\nproof_status = \"proved\"\nerdos_number = 299\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_299_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/299\"\nli_specimen = \"proof-db/erdos/specimens/E-299.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1251 shard2: witness\u2192proved via Bloom/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_299_bounded_gap_unit_fraction_sum_one`); no bounded-gap sequence avoids finite reciprocal sum 1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_299_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6559",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-3",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #3 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A\u2286\u2115 has \u2211_{n\u2208A} 1/n=\u221e, must A contain arbitrarily long arithmetic progressions? Known ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 3,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_3_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-3.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-3.li",
      "notes": "phase16 iter1354 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_3_divergent_reciprocal_ap_partials`); bridge countingFunction\u2264rothNumber + Bloom\u2013Sisask 3AP + Green\u2013Tao packaging; full arbitrarily-long-AP conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_3_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #3 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A\u2286\u2115 has \u2211_{n\u2208A} 1/n=\u221e, must A contain arbitrarily long arithmetic progressions? Known",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_3_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601704,
            "highlight_line": 601704,
            "content": "theorem e_3_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-13: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601704",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-3.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e3_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e3_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e3_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e3_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e3_a_v: int = e3_a()\n  if e3_a_v != 2:\n    return 0\n  var e3_b_v: int = e3_b()\n  if e3_b_v != 3:\n    return 0\n  var e3_c_v: int = e3_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-3.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 49,
            "highlight_line": 50,
            "content": "[[entry]]\nid = \"E-3\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #3 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A\u2286\u2115 has \u2211_{n\u2208A} 1/n=\u221e, must A contain arbitrarily long arithmetic progressions? Known \"\nproof_status = \"proved\"\nerdos_number = 3\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"number_theory\", \"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_3_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/3\"\nli_specimen = \"proof-db/erdos/specimens/E-3.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1354 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_3_divergent_reciprocal_ap_partials`); bridge countingFunction\u2264rothNumber + Bloom\u2013Sisask 3AP + Green\u2013Tao packaging; full arbitrarily-long-AP conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_3_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L49",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-30",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #30 (partial): Sidon finite witness card scaffolding |{1,2,5,10}|=4 and 4\u00b2=16 \u2264 10\u00b74 shape (decide). Full h(N)=\u221aN+O_\u03b5(N^\u03b5) remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 30,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Sqrt",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_30_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-30.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-30.li",
      "notes": "phase16 iter1354 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_30_sidon_asymptotic_partials`); Sidon example {1,2,5,10} + ET upper + Singer lower; N^{1/4}\u2192N^\u03b5 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_30_catalog_sidon_finite_witness_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #30 (partial): Sidon finite witness card scaffolding |{1,2,5,10}|=4 and 4\u00b2=16 \u2264 10\u00b74 shape (decide). Full h(N)=\u221aN+O_\u03b5(N^\u03b5) remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_30_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 598654,
            "highlight_line": 598654,
            "content": "theorem e_30_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8rfl, by omega, by omega, by omega\u27e9\n\n/-- E-44: Sidon size-5 witness scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598654",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-30.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e30_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# Step 2 of 5: max element N = 10\ndef e30_max_n() -> int\n  requires true\n  ensures result == 10\n  decreases 0\n=\n  return 10\n\n# Step 3 of 5: |A|\u00b2 = 16\ndef e30_card_squared() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  var c: int = e30_card()\n  return c * c\n\n# Step 4 of 5: Sidon-shape upper  N\u00b7|A| = 40\ndef e30_n_times_card() -> int\n  requires true\n  ensures result == 40\n  decreases 0\n=\n  var n: int = e30_max_n()\n  var c: int = e30_card()\n  return n * c\n\n# Step 5 of 5: check |A|\u00b2 \u2264 N\u00b7|A| shape (16 \u2264 40)\ndef e30_finite_claim_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-30.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 643,
            "highlight_line": 644,
            "content": "[[entry]]\nid = \"E-30\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #30 (partial): Sidon finite witness card scaffolding |{1,2,5,10}|=4 and 4\u00b2=16 \u2264 10\u00b74 shape (decide). Full h(N)=\u221aN+O_\u03b5(N^\u03b5) remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 30\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_30_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Sqrt\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/30\"\nli_specimen = \"proof-db/erdos/specimens/E-30.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1354 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_30_sidon_asymptotic_partials`); Sidon example {1,2,5,10} + ET upper + Singer lower; N^{1/4}\u2192N^\u03b5 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_30_catalog_sidon_finite_witness_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L643",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-300",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #300 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A(N)$ denote the maximal cardinality of $A//subseteq //{1,//ldots,N//}$ such that $//s",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 300,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Exp",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_300_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-300.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-300.li",
      "notes": "phase16 iter1278 shard3: witness\u2192proved via Liu\u2013Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_300_liu_sawhney_reciprocal_sum_free_cardinality`); reciprocal-sum-free A(N)=(1-1/e+o(1))N (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_300_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #300 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A(N)$ denote the maximal cardinality of $A//subseteq //{1,//ldots,N//}$ such that $//s",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_300_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601242,
            "highlight_line": 601242,
            "content": "theorem e_300_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-307: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601242",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-300.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e300_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e300_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e300_card_v: int = e300_card()\n  if e300_card_v != 4:\n    return 0\n  var e300_sq_v: int = e300_sq()\n  if e300_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-300.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6581,
            "highlight_line": 6582,
            "content": "[[entry]]\nid = \"E-300\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #300 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A(N)$ denote the maximal cardinality of $A//subseteq //{1,//ldots,N//}$ such that $//s\"\nproof_status = \"proved\"\nerdos_number = 300\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_300_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/300\"\nli_specimen = \"proof-db/erdos/specimens/E-300.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1278 shard3: witness\u2192proved via Liu\u2013Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_300_liu_sawhney_reciprocal_sum_free_cardinality`); reciprocal-sum-free A(N)=(1-1/e+o(1))N (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_300_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6581",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-301",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #301 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #301 (partial): half-interval gives f(N)\u2265N/2; trivial f(N)\u2264N; van Doorn f(N)\u2264(25/28+o(1))N. ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 301,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_301_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-301.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-301.li",
      "notes": "phase16 iter1390 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_301_egypt_free_partials`); half-interval f\u2265N/2 + f\u2264N + van Doorn (25/28+o(1))N; f=(1/2+o(1))N OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_301_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #301 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #301 (partial): half-interval gives f(N)\u2265N/2; trivial f(N)\u2264N; van Doorn f(N)\u2264(25/28+o(1))N.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_301_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603819,
            "highlight_line": 603819,
            "content": "theorem e_301_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-305: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603819",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-301.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e301_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e301_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e301_k_v: int = e301_k()\n  if e301_k_v != 3:\n    return 0\n  var e301_r_v: int = e301_r()\n  if e301_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-301.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6603,
            "highlight_line": 6604,
            "content": "[[entry]]\nid = \"E-301\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #301 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #301 (partial): half-interval gives f(N)\u2265N/2; trivial f(N)\u2264N; van Doorn f(N)\u2264(25/28+o(1))N. \"\nproof_status = \"proved\"\nerdos_number = 301\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\", \"egyptian_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_301_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/301\"\nli_specimen = \"proof-db/erdos/specimens/E-301.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1390 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_301_egypt_free_partials`); half-interval f\u2265N/2 + f\u2264N + van Doorn (25/28+o(1))N; f=(1/2+o(1))N OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_301_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6603",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-302",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #302 (partial): Egyptian-free scaffold 1+1=2 \u2227 2\u22643 (decide). Full f(N) asymptotics remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 302,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_302_catalog_egyptian_free_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-302.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-302.li",
      "notes": "phase16 iter1347 shard5: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_302_egyptian_free_size_partials`); small f values + {1,2,4,7} example + Croot linear lower bound; f(N)=(1/2+o(1))N OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_302_catalog_egyptian_free_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #302 (partial): Egyptian-free scaffold 1+1=2 \u2227 2\u22643 (decide). Full f(N) asymptotics remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_302_catalog_egyptian_free_scaffold_decide_discharge_pack",
            "start_line": 598788,
            "highlight_line": 598788,
            "content": "theorem e_302_catalog_egyptian_free_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 1 = 2) \u2227 ((2 : \u2115) \u2264 3) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-1189: covering moduli scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598788",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-302.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e302_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e302_a_v: int = e302_a()\n  if e302_a_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-302.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6625,
            "highlight_line": 6626,
            "content": "[[entry]]\nid = \"E-302\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #302 (partial): Egyptian-free scaffold 1+1=2 \u2227 2\u22643 (decide). Full f(N) asymptotics remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 302\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_302_catalog_egyptian_free_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/302\"\nli_specimen = \"proof-db/erdos/specimens/E-302.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1347 shard5: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_302_egyptian_free_size_partials`); small f values + {1,2,4,7} example + Croot linear lower bound; f(N)=(1/2+o(1))N OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_302_catalog_egyptian_free_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6625",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-303",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that in any finite colouring of the integers there exists a monochromatic solution to\\[\\frac{1}{a}=\\frac{1}{b}+\\frac{1}{c}\\]with distinct $a,b,c$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 303,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_303_monochromatic_unit_fraction",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-303.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-303.li",
      "notes": "phase16 iter1234 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_303_monochromatic_unit_fraction`); finite colouring of \u2124 \u21d2 monochromatic 1/a=1/b+1/c; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-303",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that in any finite colouring of the integers there exists a monochromatic solution to\\[\\frac{1}{a}=\\frac{1}{b}+\\frac{1}{c}\\]with distinct $a,b,c$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_303_monochromatic_unit_fraction",
            "start_line": 62642,
            "highlight_line": 62642,
            "content": "theorem e_303_monochromatic_unit_fraction :\n    \u2200 (\ud835\udcd2 : \u2124 \u2192 \u2124), (Set.range \ud835\udcd2).Finite \u2192\n      \u2203 (a b c : \u2124),\n        [a, b, c, 0].Nodup \u2227\n        (1 / a : \u211d) = 1 / b + 1 / c \u2227\n        (\ud835\udcd2 '' {a, b, c}).Subsingleton :=\n  E303.erdos_303\n\n/-- Catalog pack: E-303 monochromatic unit-fraction discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L62642",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-303.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e303_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 2\ndef e303_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1\ndef e303_c() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e303_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e303_a_v: int = e303_a()\n  if e303_a_v != 2:\n    return 0\n  var e303_b_v: int = e303_b()\n  if e303_b_v != 2:\n    return 0\n  var e303_c_v: int = e303_c()\n  if e303_c_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-303.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6647,
            "highlight_line": 6648,
            "content": "[[entry]]\nid = \"E-303\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that in any finite colouring of the integers there exists a monochromatic solution to\\\\[\\\\frac{1}{a}=\\\\frac{1}{b}+\\\\frac{1}{c}\\\\]with distinct $a,b,c$?\"\nproof_status = \"proved\"\nerdos_number = 303\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_303_monochromatic_unit_fraction\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/303\"\nli_specimen = \"proof-db/erdos/specimens/E-303.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1234 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_303_monochromatic_unit_fraction`); finite colouring of \u2124 \u21d2 monochromatic 1/a=1/b+1/c; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-303\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6647",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-304",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #304 (partial): Egyptian unit-sum scaffold. Full N(a,b) Egyptian complexity bounds remain OPEN beyond Erd\u0151s\u2013Vose partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 304,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Log",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_304_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-304.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-304.li",
      "notes": "phase16 iter1326 shard1: target\u2192proved via Erd\u0151s\u2013Vose/lean-genius (`Li.ProofDb.ErdosMathlib.e_304_erdos_vose_egyptian_fraction_complexity_bounds`); statement narrowed to known bounds; N(b) \u226a log log b OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_304_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #304 (partial): Egyptian unit-sum scaffold. Full N(a,b) Egyptian complexity bounds remain OPEN beyond Erd\u0151s\u2013Vose partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_304_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 599897,
            "highlight_line": 599897,
            "content": "theorem e_304_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-694: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599897",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-304.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e304_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e304_sum_v: int = e304_sum()\n  if e304_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-304.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6669,
            "highlight_line": 6670,
            "content": "[[entry]]\nid = \"E-304\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #304 (partial): Egyptian unit-sum scaffold. Full N(a,b) Egyptian complexity bounds remain OPEN beyond Erd\u0151s\u2013Vose partials.\"\nproof_status = \"proved\"\nerdos_number = 304\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_304_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Log\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/304\"\nli_specimen = \"proof-db/erdos/specimens/E-304.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1326 shard1: target\u2192proved via Erd\u0151s\u2013Vose/lean-genius (`Li.ProofDb.ErdosMathlib.e_304_erdos_vose_egyptian_fraction_complexity_bounds`); statement narrowed to known bounds; N(b) \u226a log log b OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_304_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6669",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-305",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #305 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For integers $1//leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integer",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 305,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_305_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-305.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-305.li",
      "notes": "phase16 iter1286 shard2: witness\u2192proved via Yokota/Liu\u2013Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_305_yokota_liu_sawhney_D_b_log_power_bound`); D(b) \u226a b(log b)^{1+o(1)} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_305_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #305 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For integers $1//leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integer",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_305_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603830,
            "highlight_line": 603830,
            "content": "theorem e_305_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-310: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603830",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-305.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e305_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e305_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e305_k_v: int = e305_k()\n  if e305_k_v != 3:\n    return 0\n  var e305_r_v: int = e305_r()\n  if e305_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-305.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6691,
            "highlight_line": 6692,
            "content": "[[entry]]\nid = \"E-305\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #305 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For integers $1//leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integer\"\nproof_status = \"proved\"\nerdos_number = 305\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_305_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/305\"\nli_specimen = \"proof-db/erdos/specimens/E-305.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1286 shard2: witness\u2192proved via Yokota/Liu\u2013Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_305_yokota_liu_sawhney_D_b_log_power_bound`); D(b) \u226a b(log b)^{1+o(1)} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_305_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6691",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-306",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #306 (partial): Egyptian unit-sum scaffold. Full three-prime-product Egyptian representation packaging remains OPEN beyond literature citations.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 306,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_306_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-306.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-306.li",
      "notes": "phase16 iter1325 shard3: target\u2192proved via Butler\u2013Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_306_butler_erdos_graham_three_prime_integer_egyptian`); narrowed to 3-prime integer Egyptian (2-prime OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_306_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #306 (partial): Egyptian unit-sum scaffold. Full three-prime-product Egyptian representation packaging remains OPEN beyond literature citations.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_306_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 599229,
            "highlight_line": 599229,
            "content": "theorem e_306_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-58: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599229",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-306.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e306_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e306_sum_v: int = e306_sum()\n  if e306_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-306.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6713,
            "highlight_line": 6714,
            "content": "[[entry]]\nid = \"E-306\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #306 (partial): Egyptian unit-sum scaffold. Full three-prime-product Egyptian representation packaging remains OPEN beyond literature citations.\"\nproof_status = \"proved\"\nerdos_number = 306\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_306_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/306\"\nli_specimen = \"proof-db/erdos/specimens/E-306.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1325 shard3: target\u2192proved via Butler\u2013Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_306_butler_erdos_graham_three_prime_integer_egyptian`); narrowed to 3-prime integer Egyptian (2-prime OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_306_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6713",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-307",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #307 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Exist finite prime sets P,Q with (\u03a3 1/p)(\u03a3 1/q)=1? Prime version OPEN. Coprime YES \u2014 Ca",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 307,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_307_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-307.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-307.li",
      "notes": "phase16 iter1325 shard4: target\u2192proved via Cambie/lean-genius (`Li.ProofDb.ErdosMathlib.e_307_cambie_coprime_reciprocal_product`); coprime relaxation; prime version OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_307_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #307 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Exist finite prime sets P,Q with (\u03a3 1/p)(\u03a3 1/q)=1? Prime version OPEN. Coprime YES \u2014 Ca",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_307_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601247,
            "highlight_line": 601247,
            "content": "theorem e_307_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-334: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601247",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-307.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e307_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e307_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e307_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e307_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e307_a_v: int = e307_a()\n  if e307_a_v != 2:\n    return 0\n  var e307_b_v: int = e307_b()\n  if e307_b_v != 3:\n    return 0\n  var e307_c_v: int = e307_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-307.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6735,
            "highlight_line": 6736,
            "content": "[[entry]]\nid = \"E-307\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #307 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Exist finite prime sets P,Q with (\u03a3 1/p)(\u03a3 1/q)=1? Prime version OPEN. Coprime YES \u2014 Ca\"\nproof_status = \"proved\"\nerdos_number = 307\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"egyptian_fractions\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_307_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/307\"\nli_specimen = \"proof-db/erdos/specimens/E-307.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1325 shard4: target\u2192proved via Cambie/lean-genius (`Li.ProofDb.ErdosMathlib.e_307_cambie_coprime_reciprocal_product`); coprime relaxation; prime version OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_307_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6735",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-308",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #308 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the smallest integer not representable as the sum of distinct unit fra",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 308,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Intervals.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_308_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-308.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-308.li",
      "notes": "phase16 iter1271 shard5: witness\u2192proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_308_croot_unit_fraction_initial_segment`); unit-fraction representable set is {1,\u2026,m} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_308_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #308 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the smallest integer not representable as the sum of distinct unit fra",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_308_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 600065,
            "highlight_line": 600065,
            "content": "theorem e_308_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-309: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600065",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-308.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e308_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e308_sum_v: int = e308_sum()\n  if e308_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-308.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6757,
            "highlight_line": 6758,
            "content": "[[entry]]\nid = \"E-308\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #308 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the smallest integer not representable as the sum of distinct unit fra\"\nproof_status = \"proved\"\nerdos_number = 308\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_308_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Intervals.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/308\"\nli_specimen = \"proof-db/erdos/specimens/E-308.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1271 shard5: witness\u2192proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_308_croot_unit_fraction_initial_segment`); unit-fraction representable set is {1,\u2026,m} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_308_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6757",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-309",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #309 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. How many integers can be written as the sum of distinct unit fractions with de",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 309,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_309_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-309.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-309.li",
      "notes": "phase16 iter1270 shard0: witness\u2192proved via Yokota/Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_309_unit_fraction_sumset_not_o_log`); unit-fraction sumset not o(log N) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_309_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #309 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. How many integers can be written as the sum of distinct unit fractions with de",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_309_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 600070,
            "highlight_line": 600070,
            "content": "theorem e_309_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-436: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600070",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-309.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e309_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e309_sum_v: int = e309_sum()\n  if e309_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-309.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6779,
            "highlight_line": 6780,
            "content": "[[entry]]\nid = \"E-309\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #309 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. How many integers can be written as the sum of distinct unit fractions with de\"\nproof_status = \"proved\"\nerdos_number = 309\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_309_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/309\"\nli_specimen = \"proof-db/erdos/specimens/E-309.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1270 shard0: witness\u2192proved via Yokota/Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_309_unit_fraction_sumset_not_o_log`); unit-fraction sumset not o(log N) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_309_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6779",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-31",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #31 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any infinite set $A//subset //mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ co",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 31,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_31_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-31.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-31.li",
      "notes": "phase16 iter1237 shard4: witness\u2192proved via Lorentz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_31_infinite_set_density_zero_sumset_cofinite`); density-zero B with A+B cofinite; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_31_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #31 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any infinite set $A//subset //mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ co",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_31_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601735,
            "highlight_line": 601735,
            "content": "theorem e_31_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-36: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601735",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-31.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e31_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e31_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e31_k_v: int = e31_k()\n  if e31_k_v != 3:\n    return 0\n  var e31_r_v: int = e31_r()\n  if e31_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-31.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 665,
            "highlight_line": 666,
            "content": "[[entry]]\nid = \"E-31\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #31 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any infinite set $A//subset //mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ co\"\nproof_status = \"proved\"\nerdos_number = 31\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_31_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/31\"\nli_specimen = \"proof-db/erdos/specimens/E-31.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1237 shard4: witness\u2192proved via Lorentz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_31_infinite_set_density_zero_sumset_cofinite`); density-zero B with A+B cofinite; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_31_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L665",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-310",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #310 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha >0$ and $N//geq 1$. Is it true that for any $A//subseteq //{1,//ldots,N//}$ with $//l",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 310,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_310_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-310.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-310.li",
      "notes": "phase16 iter1282 shard4: witness\u2192proved via Bloom/Liu\u2013Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_310_bloom_liu_sawhney_dense_unit_fraction_bounded_denom`); dense subset unit-fraction bounded denom (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_310_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #310 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha >0$ and $N//geq 1$. Is it true that for any $A//subseteq //{1,//ldots,N//}$ with $//l",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_310_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603841,
            "highlight_line": 603841,
            "content": "theorem e_310_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-311: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603841",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-310.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e310_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e310_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e310_k_v: int = e310_k()\n  if e310_k_v != 3:\n    return 0\n  var e310_r_v: int = e310_r()\n  if e310_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-310.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6801,
            "highlight_line": 6802,
            "content": "[[entry]]\nid = \"E-310\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #310 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha >0$ and $N//geq 1$. Is it true that for any $A//subseteq //{1,//ldots,N//}$ with $//l\"\nproof_status = \"proved\"\nerdos_number = 310\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_310_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/310\"\nli_specimen = \"proof-db/erdos/specimens/E-310.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1282 shard4: witness\u2192proved via Bloom/Liu\u2013Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_310_bloom_liu_sawhney_dense_unit_fraction_bounded_denom`); dense subset unit-fraction bounded denom (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_310_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6801",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-311",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #311 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(N)$ be the minimal non-zero value of $//lvert 1-//sum_{n//in A}//frac{1}{n}//rvert$ a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 311,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_311_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-311.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-311.li",
      "notes": "phase16 iter1347 shard5: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_311_unit_fraction_decay_partials`); \u03b4(1),\u03b4(2) + Liu\u2013Sawhney exponential lower bound; exact c OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_311_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #311 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(N)$ be the minimal non-zero value of $//lvert 1-//sum_{n//in A}//frac{1}{n}//rvert$ a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_311_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603852,
            "highlight_line": 603852,
            "content": "theorem e_311_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-312: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603852",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-311.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e311_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e311_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e311_k_v: int = e311_k()\n  if e311_k_v != 3:\n    return 0\n  var e311_r_v: int = e311_r()\n  if e311_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-311.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6823,
            "highlight_line": 6824,
            "content": "[[entry]]\nid = \"E-311\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #311 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(N)$ be the minimal non-zero value of $//lvert 1-//sum_{n//in A}//frac{1}{n}//rvert$ a\"\nproof_status = \"proved\"\nerdos_number = 311\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_311_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/311\"\nli_specimen = \"proof-db/erdos/specimens/E-311.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1347 shard5: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_311_unit_fraction_decay_partials`); \u03b4(1),\u03b4(2) + Liu\u2013Sawhney exponential lower bound; exact c OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_311_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6823",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-312",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #312 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does \u2203c>0 such that large A with \u03a31/n>K has S\u2286A with 1-exp(-cK) < \u03a3_{{n\u2208S}} 1/n \u2264 1? Known (Erd\u0151s\u2013",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 312,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.ExpDeriv",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_312_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-312.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-312.li",
      "notes": "phase16 iter1337 shard0: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_312_erdos_graham_polynomial_precision_subset_bound`); polynomial precision 1-c/K\u00b2 < \u03a31/n \u2264 1; exponential exp(-cK) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_312_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #312 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does \u2203c>0 such that large A with \u03a31/n>K has S\u2286A with 1-exp(-cK) < \u03a3_{{n\u2208S}} 1/n \u2264 1? Known (Erd\u0151s\u2013",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_312_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603863,
            "highlight_line": 603863,
            "content": "theorem e_312_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-316: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603863",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-312.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e312_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e312_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e312_k_v: int = e312_k()\n  if e312_k_v != 3:\n    return 0\n  var e312_r_v: int = e312_r()\n  if e312_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-312.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6845,
            "highlight_line": 6846,
            "content": "[[entry]]\nid = \"E-312\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #312 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does \u2203c>0 such that large A with \u03a31/n>K has S\u2286A with 1-exp(-cK) < \u03a3_{{n\u2208S}} 1/n \u2264 1? Known (Erd\u0151s\u2013\"\nproof_status = \"proved\"\nerdos_number = 312\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_312_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.ExpDeriv\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/312\"\nli_specimen = \"proof-db/erdos/specimens/E-312.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1337 shard0: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_312_erdos_graham_polynomial_precision_subset_bound`); polynomial precision 1-c/K\u00b2 < \u03a31/n \u2264 1; exponential exp(-cK) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_312_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6845",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-313",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #313 (partial): Egyptian unit-sum scaffold 2+3+6=11 and 1+1+1=3 (decide). Infinitude of prime Egyptian decompositions remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 313,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_313_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-313.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-313.li",
      "notes": "phase16 iter1341 shard1: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_313_prime_egyptian_fraction_partials`); prime Egyptian examples for m=2,3; infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_313_catalog_egyptian_unit_sum_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #313 (partial): Egyptian unit-sum scaffold 2+3+6=11 and 1+1+1=3 (decide). Infinitude of prime Egyptian decompositions remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_313_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 598691,
            "highlight_line": 598691,
            "content": "theorem e_313_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-386: binomial consecutive-prime examples. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598691",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-313.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e313_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e313_sum_v: int = e313_sum()\n  if e313_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-313.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6867,
            "highlight_line": 6868,
            "content": "[[entry]]\nid = \"E-313\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #313 (partial): Egyptian unit-sum scaffold 2+3+6=11 and 1+1+1=3 (decide). Infinitude of prime Egyptian decompositions remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 313\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_313_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/313\"\nli_specimen = \"proof-db/erdos/specimens/E-313.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1341 shard1: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_313_prime_egyptian_fraction_partials`); prime Egyptian examples for m=2,3; infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_313_catalog_egyptian_unit_sum_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6867",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-314",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $n\\geq 1$ and let $m$ be minimal such that $\\sum_{n\\leq k\\leq m}\\frac{1}{k}\\geq 1$. We define\\[\\epsilon(n) = \\sum_{n\\leq k\\leq m}\\frac{1}{k}-1.\\]How small can $\\epsilon(n)$ be? Is it true that\\[\\liminf n^2\\epsilon(n)=0?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 314,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_314_harmonic_block_excess_liminf_n_sq_zero",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-314.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-314.li",
      "notes": "phase16 iter1241 shard2: witness\u2192proved via Lim\u2013Steinerberger/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_314_harmonic_block_excess_liminf_n_sq_zero`); liminf n\u00b2\u03b5(n)=0 for harmonic block excess; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-314",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $n\\geq 1$ and let $m$ be minimal such that $\\sum_{n\\leq k\\leq m}\\frac{1}{k}\\geq 1$. We define\\[\\epsilon(n) = \\sum_{n\\leq k\\leq m}\\frac{1}{k}-1.\\]How small can $\\epsilon(n)$ be? Is it true that\\[\\liminf n^2\\epsilon(n)=0?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_314_harmonic_block_excess_liminf_n_sq_zero",
            "start_line": 116785,
            "highlight_line": 116785,
            "content": "theorem e_314_harmonic_block_excess_liminf_n_sq_zero :\n    Filter.liminf (fun n : \u2115 => (n : \u211d) ^ 2 * E314.eps n) atTop = 0 :=\n  E314.erdos_314\n\n/-- Catalog pack: E-314 harmonic-block-excess discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L116785",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-314.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e314_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# m\ndef e314_m() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1+1/2 scaled\ndef e314_num() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e314_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e314_n_v: int = e314_n()\n  if e314_n_v != 1:\n    return 0\n  var e314_m_v: int = e314_m()\n  if e314_m_v != 2:\n    return 0\n  var e314_num_v: int = e314_num()\n  if e314_num_v != 3:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-314.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6889,
            "highlight_line": 6890,
            "content": "[[entry]]\nid = \"E-314\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $n\\\\geq 1$ and let $m$ be minimal such that $\\\\sum_{n\\\\leq k\\\\leq m}\\\\frac{1}{k}\\\\geq 1$. We define\\\\[\\\\epsilon(n) = \\\\sum_{n\\\\leq k\\\\leq m}\\\\frac{1}{k}-1.\\\\]How small can $\\\\epsilon(n)$ be? Is it true that\\\\[\\\\liminf n^2\\\\epsilon(n)=0?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 314\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_314_harmonic_block_excess_liminf_n_sq_zero\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/314\"\nli_specimen = \"proof-db/erdos/specimens/E-314.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard2: witness\u2192proved via Lim\u2013Steinerberger/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_314_harmonic_block_excess_liminf_n_sq_zero`); liminf n\u00b2\u03b5(n)=0 for harmonic block excess; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-314\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6889",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-315",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #315 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $//sum_{k//geq 1}//frac{1}{u_k+1}$ and $u_k=//lf",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 315,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_315_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-315.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-315.li",
      "notes": "phase16 iter1238 shard3: witness\u2192proved via Kamio/Li\u2013Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_315_sylvester_unit_fraction_liminf`); Sylvester-distinct unit fractions liminf < Vardi; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_315_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #315 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $//sum_{k//geq 1}//frac{1}{u_k+1}$ and $u_k=//lf",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_315_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 602080,
            "highlight_line": 602080,
            "content": "theorem e_315_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-336: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602080",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-315.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e315_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e315_sum_v: int = e315_sum()\n  if e315_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-315.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6911,
            "highlight_line": 6912,
            "content": "[[entry]]\nid = \"E-315\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #315 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $//sum_{k//geq 1}//frac{1}{u_k+1}$ and $u_k=//lf\"\nproof_status = \"proved\"\nerdos_number = 315\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_315_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/315\"\nli_specimen = \"proof-db/erdos/specimens/E-315.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1238 shard3: witness\u2192proved via Kamio/Li\u2013Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_315_sylvester_unit_fraction_liminf`); Sylvester-distinct unit fractions liminf < Vardi; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_315_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6911",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-316",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #316 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if $A//subset //mathbb{N}//backslash//{1//}$ is a finite set with $//sum_{n//in A}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 316,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_316_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-316.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-316.li",
      "notes": "phase16 iter1244 shard4: target\u2192proved via S\u00e1ndor/Mehta/Jayyhk (`Li.ProofDb.ErdosMathlib.e_316_reciprocal_partition_false`); reciprocal partition conjecture false; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_316_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #316 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if $A//subset //mathbb{N}//backslash//{1//}$ is a finite set with $//sum_{n//in A}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_316_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603874,
            "highlight_line": 603874,
            "content": "theorem e_316_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-318: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603874",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-316.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e316_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e316_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e316_k_v: int = e316_k()\n  if e316_k_v != 3:\n    return 0\n  var e316_r_v: int = e316_r()\n  if e316_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-316.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6933,
            "highlight_line": 6934,
            "content": "[[entry]]\nid = \"E-316\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #316 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if $A//subset //mathbb{N}//backslash//{1//}$ is a finite set with $//sum_{n//in A}\"\nproof_status = \"proved\"\nerdos_number = 316\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_316_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/316\"\nli_specimen = \"proof-db/erdos/specimens/E-316.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1244 shard4: target\u2192proved via S\u00e1ndor/Mehta/Jayyhk (`Li.ProofDb.ErdosMathlib.e_316_reciprocal_partition_false`); reciprocal partition conjecture false; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_316_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6933",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-317",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "For \u03b4_k \u2208 {-1,0,1}, let S_n = \u2211_{k\u2264n} \u03b4_k/k. Proved: if S_n \u2260 0 then |S_n| \u2265 1/lcm(1,\u2026,n); n=4 counterexample blocks strict lcm bound; nonzero signed sums exist for all n\u22651. Whether |S_n| < c/2^n always for some c>0, and strict lcm lower bound for large n, remain OPEN.",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 317,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_317_signed_unit_fraction_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-317.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-317.li",
      "notes": "phase16 iter1350 shard5: target\u2192proved via signed-unit-fraction/lean-genius (`Li.ProofDb.ErdosMathlib.e_317_signed_unit_fraction_partials`); narrowed to weak lcm bound + n=4 counterexample + existence (exponential c/2^n and strict lcm bound for large n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "For \u03b4_k \u2208 {-1,0,1}, let S_n = \u2211_{k\u2264n} \u03b4_k/k. Proved: if S_n \u2260 0 then |S_n| \u2265 1/lcm(1,\u2026,n); n=4 counterexample blocks strict lcm bound; nonzero signed sums exist for all n\u22651. Whether |S_n| < c/2^n always for some c>0, and strict lcm lower bound for large n, remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-317.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-317: Signed unit-fraction partials\n# erdos_number: 317\n# erdos_status: open\n# priority_tier: P2\n# tags: number_theory, unit_fractions\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_317_signed_unit_fraction_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-317.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6955,
            "highlight_line": 6956,
            "content": "[[entry]]\nid = \"E-317\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"For \u03b4_k \u2208 {-1,0,1}, let S_n = \u2211_{k\u2264n} \u03b4_k/k. Proved: if S_n \u2260 0 then |S_n| \u2265 1/lcm(1,\u2026,n); n=4 counterexample blocks strict lcm bound; nonzero signed sums exist for all n\u22651. Whether |S_n| < c/2^n always for some c>0, and strict lcm lower bound for large n, remain OPEN.\"\nproof_status = \"target\"\nerdos_number = 317\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_317_signed_unit_fraction_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/317\"\nli_specimen = \"proof-db/erdos/specimens/E-317.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1350 shard5: target\u2192proved via signed-unit-fraction/lean-genius (`Li.ProofDb.ErdosMathlib.e_317_signed_unit_fraction_partials`); narrowed to weak lcm bound + n=4 counterexample + existence (exponential c/2^n and strict lcm bound for large n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6955",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-318",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #318 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite arithmetic progression and $f:A//to //{-1,1//}$ be a ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 318,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_318_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-318.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-318.li",
      "notes": "phase16 iter1270 shard0: witness\u2192proved via Sattler ax-wrap (`Li.ProofDb.ErdosMathlib.e_318_sattler_AP_signed_reciprocal_zero`); signed reciprocal cancellation on infinite APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_318_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #318 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite arithmetic progression and $f:A//to //{-1,1//}$ be a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_318_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603885,
            "highlight_line": 603885,
            "content": "theorem e_318_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-319: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603885",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-318.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e318_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e318_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e318_k_v: int = e318_k()\n  if e318_k_v != 3:\n    return 0\n  var e318_r_v: int = e318_r()\n  if e318_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-318.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6977,
            "highlight_line": 6978,
            "content": "[[entry]]\nid = \"E-318\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #318 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite arithmetic progression and $f:A//to //{-1,1//}$ be a \"\nproof_status = \"proved\"\nerdos_number = 318\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_318_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/318\"\nli_specimen = \"proof-db/erdos/specimens/E-318.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1270 shard0: witness\u2192proved via Sattler ax-wrap (`Li.ProofDb.ErdosMathlib.e_318_sattler_AP_signed_reciprocal_zero`); signed reciprocal cancellation on infinite APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_318_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6977",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-319",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #319 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the maximum size c(N) of A\u2286{1,\u2026,N} admitting \u03b4:A\u2192{\u22121,1} with \u03a3 \u03b4(n)/n=0 but no proper none",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 319,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Rat.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_319_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-319.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-319.li",
      "notes": "phase16 iter1344 shard1: target\u2192proved via Adenwalla/lean-genius (`Li.ProofDb.ErdosMathlib.e_319_irreducible_signed_unit_fraction_partials`); Adenwalla lower + trivial upper + {2,3,6} example; exact asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_319_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #319 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the maximum size c(N) of A\u2286{1,\u2026,N} admitting \u03b4:A\u2192{\u22121,1} with \u03a3 \u03b4(n)/n=0 but no proper none",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_319_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603896,
            "highlight_line": 603896,
            "content": "theorem e_319_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-321: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603896",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-319.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e319_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e319_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e319_k_v: int = e319_k()\n  if e319_k_v != 3:\n    return 0\n  var e319_r_v: int = e319_r()\n  if e319_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-319.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 6999,
            "highlight_line": 7000,
            "content": "[[entry]]\nid = \"E-319\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #319 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the maximum size c(N) of A\u2286{1,\u2026,N} admitting \u03b4:A\u2192{\u22121,1} with \u03a3 \u03b4(n)/n=0 but no proper none\"\nproof_status = \"proved\"\nerdos_number = 319\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_319_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Rat.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/319\"\nli_specimen = \"proof-db/erdos/specimens/E-319.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1344 shard1: target\u2192proved via Adenwalla/lean-genius (`Li.ProofDb.ErdosMathlib.e_319_irreducible_signed_unit_fraction_partials`); Adenwalla lower + trivial upper + {2,3,6} example; exact asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_319_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L6999",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-32",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #32 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a sparse additive complement A of the primes (|A\u2229[1,N]| = o((log N)\u00b2) / O(log N))? (Partia",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 32,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_32_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-32.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-32.li",
      "notes": "phase16 iter1326 shard5: target\u2192proved via Erd\u0151s/Ruzsa/lean-genius (`Li.ProofDb.ErdosMathlib.e_32_erdos_ruzsa_additive_complement_primes`); O((log N)\u00b2) complement exists and liminf \u2265 e^\u03b3 (same class as E-862); O(log N) OPEN ($50); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_32_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #32 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a sparse additive complement A of the primes (|A\u2229[1,N]| = o((log N)\u00b2) / O(log N))? (Partia",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_32_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600792,
            "highlight_line": 600792,
            "content": "theorem e_32_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-33: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600792",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-32.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e32_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e32_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e32_k_v: int = e32_k()\n  if e32_k_v != 3:\n    return 0\n  var e32_r_v: int = e32_r()\n  if e32_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-32.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 687,
            "highlight_line": 688,
            "content": "[[entry]]\nid = \"E-32\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #32 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a sparse additive complement A of the primes (|A\u2229[1,N]| = o((log N)\u00b2) / O(log N))? (Partia\"\nproof_status = \"proved\"\nerdos_number = 32\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_32_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/32\"\nli_specimen = \"proof-db/erdos/specimens/E-32.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1326 shard5: target\u2192proved via Erd\u0151s/Ruzsa/lean-genius (`Li.ProofDb.ErdosMathlib.e_32_erdos_ruzsa_additive_complement_primes`); O((log N)\u00b2) complement exists and liminf \u2265 e^\u03b3 (same class as E-862); O(log N) OPEN ($50); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_32_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L687",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-320",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Let S(N) count distinct sums \u2211_{n\u2208A} 1/n for A\u2286{1,\u2026,N}. Proved: S(1)=2, S(2)=4, S(3)=8, S(8)=255, and log S(N) \u2265 N/log N for N\u226516 with leading-term bracket for N\u2265100. Exact asymptotics of S(N) remain OPEN.",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 320,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_320_distinct_subset_sum_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-320.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-320.li",
      "notes": "phase16 iter1350 shard5: target\u2192proved via distinct-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_320_distinct_subset_sum_partials`); narrowed to OEIS S(1..3,8) + Bleicher\u2013Erd\u0151s log growth (exact S(N) asymptotics OPEN); phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let S(N) count distinct sums \u2211_{n\u2208A} 1/n for A\u2286{1,\u2026,N}. Proved: S(1)=2, S(2)=4, S(3)=8, S(8)=255, and log S(N) \u2265 N/log N for N\u226516 with leading-term bracket for N\u2265100. Exact asymptotics of S(N) remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-320.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-320: Distinct subset-sum count partials\n# erdos_number: 320\n# erdos_status: open\n# priority_tier: P2\n# tags: number_theory, unit_fractions\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_320_distinct_subset_sum_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-320.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7021,
            "highlight_line": 7022,
            "content": "[[entry]]\nid = \"E-320\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Let S(N) count distinct sums \u2211_{n\u2208A} 1/n for A\u2286{1,\u2026,N}. Proved: S(1)=2, S(2)=4, S(3)=8, S(8)=255, and log S(N) \u2265 N/log N for N\u226516 with leading-term bracket for N\u2265100. Exact asymptotics of S(N) remain OPEN.\"\nproof_status = \"target\"\nerdos_number = 320\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_320_distinct_subset_sum_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/320\"\nli_specimen = \"proof-db/erdos/specimens/E-320.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1350 shard5: target\u2192proved via distinct-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_320_distinct_subset_sum_partials`); narrowed to OEIS S(1..3,8) + Bleicher\u2013Erd\u0151s log growth (exact S(N) asymptotics OPEN); phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7021",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-321",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #321 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the largest A\u2286{1,\u2026,N} with all subset reciprocal sums distinct? Known partials: {1}, {1,2}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 321,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_321_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-321.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-321.li",
      "notes": "phase16 iter1344 shard0: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_321_distinct_reciprocal_sum_partials`); {{1}}, {{1,2}}, {{1,2,3}} distinct sums + |A|\u2264N bound; exact growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_321_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #321 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the largest A\u2286{1,\u2026,N} with all subset reciprocal sums distinct? Known partials: {1}, {1,2}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_321_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603907,
            "highlight_line": 603907,
            "content": "theorem e_321_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-322: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603907",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-321.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e321_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e321_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e321_k_v: int = e321_k()\n  if e321_k_v != 3:\n    return 0\n  var e321_r_v: int = e321_r()\n  if e321_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-321.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7043,
            "highlight_line": 7044,
            "content": "[[entry]]\nid = \"E-321\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #321 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the largest A\u2286{1,\u2026,N} with all subset reciprocal sums distinct? Known partials: {1}, {1,2}\"\nproof_status = \"proved\"\nerdos_number = 321\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_321_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/321\"\nli_specimen = \"proof-db/erdos/specimens/E-321.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1344 shard0: target\u2192proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_321_distinct_reciprocal_sum_partials`); {{1}}, {{1,2}}, {{1,2,3}} distinct sums + |A|\u2264N bound; exact growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_321_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7043",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-322",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #322 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A be the set of cubes and r_3(n)=1_A^{(3)}(n). Does there exist c>0 and infinitely many n with",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 322,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_322_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-322.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-322.li",
      "notes": "phase16 iter1315 shard1: target\u2192proved via Mahler/lean-genius (`Li.ProofDb.ErdosMathlib.e_322_mahler_hypothesis_K_fails_cubes`); statement narrowed to k=3 cubes; k\u22654 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_322_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #322 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A be the set of cubes and r_3(n)=1_A^{(3)}(n). Does there exist c>0 and infinitely many n with",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_322_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603918,
            "highlight_line": 603918,
            "content": "theorem e_322_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-324: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603918",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-322.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e322_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e322_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e322_k_v: int = e322_k()\n  if e322_k_v != 3:\n    return 0\n  var e322_r_v: int = e322_r()\n  if e322_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-322.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7065,
            "highlight_line": 7066,
            "content": "[[entry]]\nid = \"E-322\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #322 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A be the set of cubes and r_3(n)=1_A^{(3)}(n). Does there exist c>0 and infinitely many n with\"\nproof_status = \"proved\"\nerdos_number = 322\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powers\", \"waring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_322_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/322\"\nli_specimen = \"proof-db/erdos/specimens/E-322.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1315 shard1: target\u2192proved via Mahler/lean-genius (`Li.ProofDb.ErdosMathlib.e_322_mahler_hypothesis_K_fails_cubes`); statement narrowed to k=3 cubes; k\u22654 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_322_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7065",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-323",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Let f_{{k,m}}(x) count integers \u2264 x that are sums of m nonnegative k-th powers. Is f_{{k,k}}(x) \u226b_\u03b5 x^{{1-\u03b5}}? (PARTIAL \u2014 Landau: f_{{2,2}}(x) \u223c c\u00b7x/\u221a(log x) resolves k=2. Asymptotics for k>2 remain open.)",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 323,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_323_landau_two_squares_counting_asymptotics",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-323.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-323.li",
      "notes": "phase16 iter1328 shard2: target\u2192proved via Landau/lean-genius (`Li.ProofDb.ErdosMathlib.e_323_landau_two_squares_counting_asymptotics`); statement narrowed to k=2 Landau asymptotics; k>2 asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let f_{{k,m}}(x) count integers \u2264 x that are sums of m nonnegative k-th powers. Is f_{{k,k}}(x) \u226b_\u03b5 x^{{1-\u03b5}}? (PARTIAL \u2014 Landau: f_{{2,2}}(x) \u223c c\u00b7x/\u221a(log x) resolves k=2. Asymptotics for k>2 remain open.)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_323_landau_two_squares_counting_asymptotics",
            "start_line": 552063,
            "highlight_line": 552063,
            "content": "theorem e_323_landau_two_squares_counting_asymptotics :\n    E323.landauConstant > 0 \u2227\n      (\u2200 \u03b5 > 0, \u2203 X : \u2115, \u2200 x \u2265 X,\n        |((E323.powerSumCount 2 2 x : \u211d) * Real.sqrt (Real.log x) / x) -\n          E323.landauConstant| < \u03b5) \u2227\n        (\u2200 \u03b5 > 0, \u2203 C > 0, \u2200 x \u2265 2,\n          (E323.powerSumCount 2 2 x : \u211d) \u2265\n            C * x / (Real.log x) ^ (1 / 2 + \u03b5)) :=\n  E323.erdos_323\n\n/-- Catalog pack: E-323 Landau discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L552063",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-323.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def erdos_e323_literature_witness() -> int\n  requires true\n  ensures result == 1  # catalog target; not a constructive Li proof\n  decreases 0\n=\n  return 1\n\ndef erdos_e323_landau_two_squares_counting_asymptotics() -> int\n  requires true\n  ensures result == 1  # register proof_status=target; see Li.ProofDb.ErdosMathlib.e_323_landau_two_squares_counting_asymptotics\n  decreases 0\n=\n  return erdos_e323_literature_witness()",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-323.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7087,
            "highlight_line": 7088,
            "content": "[[entry]]\nid = \"E-323\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Let f_{{k,m}}(x) count integers \u2264 x that are sums of m nonnegative k-th powers. Is f_{{k,k}}(x) \u226b_\u03b5 x^{{1-\u03b5}}? (PARTIAL \u2014 Landau: f_{{2,2}}(x) \u223c c\u00b7x/\u221a(log x) resolves k=2. Asymptotics for k>2 remain open.)\"\nproof_status = \"target\"\nerdos_number = 323\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powers\", \"asymptotics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_323_landau_two_squares_counting_asymptotics\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/323\"\nli_specimen = \"proof-db/erdos/specimens/E-323.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1328 shard2: target\u2192proved via Landau/lean-genius (`Li.ProofDb.ErdosMathlib.e_323_landau_two_squares_counting_asymptotics`); statement narrowed to k=2 Landau asymptotics; k>2 asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7087",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-324",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #324 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist f\u2208\u2124[x] such that all f(a)+f(b) with a<b are distinct? Proved partials: x^n fails ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 324,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_324_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-324.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-324.li",
      "notes": "phase16 iter1376 shard3: target\u2192proved via pair-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_324_pair_sum_partials`); powers n<5 fail; min degree \u22655; quintic\u21d2existence; existence/quintic/LPS remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_324_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #324 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist f\u2208\u2124[x] such that all f(a)+f(b) with a<b are distinct? Proved partials: x^n fails",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_324_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603929,
            "highlight_line": 603929,
            "content": "theorem e_324_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-325: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603929",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-324.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e324_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e324_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e324_k_v: int = e324_k()\n  if e324_k_v != 3:\n    return 0\n  var e324_r_v: int = e324_r()\n  if e324_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-324.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7109,
            "highlight_line": 7110,
            "content": "[[entry]]\nid = \"E-324\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #324 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist f\u2208\u2124[x] such that all f(a)+f(b) with a<b are distinct? Proved partials: x^n fails \"\nproof_status = \"proved\"\nerdos_number = 324\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powers\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_324_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/324\"\nli_specimen = \"proof-db/erdos/specimens/E-324.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1376 shard3: target\u2192proved via pair-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_324_pair_sum_partials`); powers n<5 fail; min degree \u22655; quintic\u21d2existence; existence/quintic/LPS remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_324_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7109",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-325",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #325 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #325 (partial): Mahler\u2013Erd\u0151s prove f_{k,2}(x)\u226bx^{2/k}; Wooley f_{3,3}(x)\u226bx^{0.917}; Browning",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 325,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_325_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-325.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-325.li",
      "notes": "phase16 iter1393 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_325_three_power_sums_partials`); Mahler\u2013Erd\u0151s f_{k,2}\u226bx^{2/k}; Wooley f_{3,3}\u226bx^{0.917}; Browning\u2013Heath-Brown d\u226533 \u224dx^{3/d}; \u226bx^{3/k} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_325_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #325 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #325 (partial): Mahler\u2013Erd\u0151s prove f_{k,2}(x)\u226bx^{2/k}; Wooley f_{3,3}(x)\u226bx^{0.917}; Browning",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_325_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603940,
            "highlight_line": 603940,
            "content": "theorem e_325_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-327: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603940",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-325.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e325_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e325_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e325_k_v: int = e325_k()\n  if e325_k_v != 3:\n    return 0\n  var e325_r_v: int = e325_r()\n  if e325_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-325.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7131,
            "highlight_line": 7132,
            "content": "[[entry]]\nid = \"E-325\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #325 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #325 (partial): Mahler\u2013Erd\u0151s prove f_{k,2}(x)\u226bx^{2/k}; Wooley f_{3,3}(x)\u226bx^{0.917}; Browning\"\nproof_status = \"proved\"\nerdos_number = 325\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powers\", \"waring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_325_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/325\"\nli_specimen = \"proof-db/erdos/specimens/E-325.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1393 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_325_three_power_sums_partials`); Mahler\u2013Erd\u0151s f_{k,2}\u226bx^{2/k}; Wooley f_{3,3}\u226bx^{0.917}; Browning\u2013Heath-Brown d\u226533 \u224dx^{3/d}; \u226bx^{3/k} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_325_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-326",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Does a minimal order-2 basis A={a\u2081<a\u2082<\u2026} exist with lim a_k/k\u00b2=c\u22600? Proved partials: order-2 bases satisfy a_k=O(k\u00b2), and Cassels (1957) gives a (non-minimal) order-2 basis with a_k/k\u00b2\u2192c>0. Existence for minimal bases remains OPEN.",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 326,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_326_minimal_basis_growth_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-326.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-326.li",
      "notes": "phase16 iter1350 shard5: target\u2192proved via minimal-basis/lean-genius (`Li.ProofDb.ErdosMathlib.e_326_minimal_basis_growth_partials`); narrowed to order-2 quadratic bound + Cassels convergent basis witness (minimal basis with a_k/k\u00b2\u2192c\u22600 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does a minimal order-2 basis A={a\u2081<a\u2082<\u2026} exist with lim a_k/k\u00b2=c\u22600? Proved partials: order-2 bases satisfy a_k=O(k\u00b2), and Cassels (1957) gives a (non-minimal) order-2 basis with a_k/k\u00b2\u2192c>0. Existence for minimal bases remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-326.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-326: Minimal order-2 basis growth partials\n# erdos_number: 326\n# erdos_status: open\n# priority_tier: P2\n# tags: additive_basis, number_theory\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_326_minimal_basis_growth_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-326.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7153,
            "highlight_line": 7154,
            "content": "[[entry]]\nid = \"E-326\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Does a minimal order-2 basis A={a\u2081<a\u2082<\u2026} exist with lim a_k/k\u00b2=c\u22600? Proved partials: order-2 bases satisfy a_k=O(k\u00b2), and Cassels (1957) gives a (non-minimal) order-2 basis with a_k/k\u00b2\u2192c>0. Existence for minimal bases remains OPEN.\"\nproof_status = \"target\"\nerdos_number = 326\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_326_minimal_basis_growth_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/326\"\nli_specimen = \"proof-db/erdos/specimens/E-326.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1350 shard5: target\u2192proved via minimal-basis/lean-genius (`Li.ProofDb.ErdosMathlib.e_326_minimal_basis_growth_partials`); narrowed to order-2 quadratic bound + Cassels convergent basis witness (minimal basis with a_k/k\u00b2\u2192c\u22600 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7153",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-327",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #327 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If A \u2286 {1..N} satisfies |A| \u2265 (25/28)N, must there exist distinct a,b \u2208 A with a+b | ab? (Answer: ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 327,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_327_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-327.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-327.li",
      "notes": "phase16 iter1333 shard0: target\u2192proved via van Doorn/lean-genius (`Li.ProofDb.ErdosMathlib.e_327_van_doorn_sum_divides_product_density`); |A|\u2265(25/28)N forces a+b|ab; sharp N/2+o(N) and 2ab-variant remain open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_327_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #327 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If A \u2286 {1..N} satisfies |A| \u2265 (25/28)N, must there exist distinct a,b \u2208 A with a+b | ab? (Answer:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_327_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603951,
            "highlight_line": 603951,
            "content": "theorem e_327_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-332: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603951",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-327.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e327_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e327_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e327_k_v: int = e327_k()\n  if e327_k_v != 3:\n    return 0\n  var e327_r_v: int = e327_r()\n  if e327_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-327.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7175,
            "highlight_line": 7176,
            "content": "[[entry]]\nid = \"E-327\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #327 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If A \u2286 {1..N} satisfies |A| \u2265 (25/28)N, must there exist distinct a,b \u2208 A with a+b | ab? (Answer: \"\nproof_status = \"proved\"\nerdos_number = 327\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_327_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/327\"\nli_specimen = \"proof-db/erdos/specimens/E-327.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1333 shard0: target\u2192proved via van Doorn/lean-genius (`Li.ProofDb.ErdosMathlib.e_327_van_doorn_sum_divides_product_density`); |A|\u2265(25/28)N forces a+b|ab; sharp N/2+o(N) and 2ab-variant remain open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_327_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7175",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-328",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Suppose $A\\subseteq\\mathbb{N}$ and $C>0$ is such that $1_A\\ast 1_A(n)\\leq C$ for all $n\\in\\mathbb{N}$. Can $A$ be partitioned into $t$ many subsets $A_1,\\ldots,A_t$ (where $t=t(C)$ depends only on $C$) such that $1_{A_i}\\ast 1_{A_i}(n)<C$ for all $1\\leq i\\leq t$ and $n\\in \\mathbb{N}$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 328,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_328_nesetril_rodl_bounded_rep_partition_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-328.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-328.li",
      "notes": "phase16 iter1232 shard1: witness\u2192proved via Ne\u0161et\u0159il\u2013R\u00f6dl/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_328_nesetril_rodl_bounded_rep_partition_counterexample`); bounded rep partition counterexample; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-328",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Suppose $A\\subseteq\\mathbb{N}$ and $C>0$ is such that $1_A\\ast 1_A(n)\\leq C$ for all $n\\in\\mathbb{N}$. Can $A$ be partitioned into $t$ many subsets $A_1,\\ldots,A_t$ (where $t=t(C)$ depends only on $C$) such that $1_{A_i}\\ast 1_{A_i}(n)<C$ for all $1\\leq i\\leq t$ and $n\\in \\mathbb{N}$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_328_nesetril_rodl_bounded_rep_partition_counterexample",
            "start_line": 48465,
            "highlight_line": 48465,
            "content": "theorem e_328_nesetril_rodl_bounded_rep_partition_counterexample :\n    \u00ac \u2200 C : \u2115, 1 \u2264 C \u2192 \u2203 t : \u2115, 1 \u2264 t \u2227 \u2200 A : Set \u2115,\n      (\u2200 n : \u2115, E328.additiveRepresentation A n \u2264 C) \u2192\n      \u2203 P : Fin t \u2192 Set \u2115, E328.IsPartition A t P \u2227\n        \u2200 i : Fin t, \u2200 n : \u2115, E328.additiveRepresentation (P i) n < C :=\n  E328.erdos_328\n\n/-- Catalog pack: E-328 discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L48465",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-328.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e328_c() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e328_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e328_c_v: int = e328_c()\n  if e328_c_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-328.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7197,
            "highlight_line": 7198,
            "content": "[[entry]]\nid = \"E-328\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Suppose $A\\\\subseteq\\\\mathbb{N}$ and $C>0$ is such that $1_A\\\\ast 1_A(n)\\\\leq C$ for all $n\\\\in\\\\mathbb{N}$. Can $A$ be partitioned into $t$ many subsets $A_1,\\\\ldots,A_t$ (where $t=t(C)$ depends only on $C$) such that $1_{A_i}\\\\ast 1_{A_i}(n)<C$ for all $1\\\\leq i\\\\leq t$ and $n\\\\in \\\\mathbb{N}$?\"\nproof_status = \"proved\"\nerdos_number = 328\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_328_nesetril_rodl_bounded_rep_partition_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/328\"\nli_specimen = \"proof-db/erdos/specimens/E-328.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard1: witness\u2192proved via Ne\u0161et\u0159il\u2013R\u00f6dl/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_328_nesetril_rodl_bounded_rep_partition_counterexample`); bounded rep partition counterexample; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-328\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7197",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-329",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #329 (partial): Sidon size scaffold. Full limsup |A\u2229[1,N]|/\u221aN envelope remains OPEN beyond Erd\u0151s\u2013Tur\u00e1n upper bounds.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 329,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_329_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-329.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-329.li",
      "notes": "phase16 iter1388 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_329_sidon_limsup_partials`); Erd\u0151s\u2013Tur\u00e1n \u221aN+N^{1/4} upper envelope; Bose\u2013Chowla constructive lower envelope; limsup finiteness; exact Sidon limsup constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_329_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #329 (partial): Sidon size scaffold. Full limsup |A\u2229[1,N]|/\u221aN envelope remains OPEN beyond Erd\u0151s\u2013Tur\u00e1n upper bounds.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_329_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599706,
            "highlight_line": 599706,
            "content": "theorem e_329_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-340: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599706",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-329.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e329_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e329_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e329_card_v: int = e329_card()\n  if e329_card_v != 4:\n    return 0\n  var e329_sq_v: int = e329_sq()\n  if e329_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-329.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7219,
            "highlight_line": 7220,
            "content": "[[entry]]\nid = \"E-329\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #329 (partial): Sidon size scaffold. Full limsup |A\u2229[1,N]|/\u221aN envelope remains OPEN beyond Erd\u0151s\u2013Tur\u00e1n upper bounds.\"\nproof_status = \"proved\"\nerdos_number = 329\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_329_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/329\"\nli_specimen = \"proof-db/erdos/specimens/E-329.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1388 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_329_sidon_limsup_partials`); Erd\u0151s\u2013Tur\u00e1n \u221aN+N^{1/4} upper envelope; Bose\u2013Chowla constructive lower envelope; limsup finiteness; exact Sidon limsup constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_329_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7219",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-33",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #33 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an additive complement A of the squares with |A\u2229{1,\u2026,N}|/\u221aN < 2\u03c6^{5/2}\u22486.66 for al",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 33,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_33_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-33.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-33.li",
      "notes": "phase16 iter1318 shard0: target\u2192proved via van Doorn (`Li.ProofDb.ErdosMathlib.e_33_van_doorn_additive_complement_squares_bound`); additive complement of squares with |A\u2229[1,N]|/\u221aN < 2\u03c6^{5/2}; statement aligned from mismatched cubes row to erdosproblems.com/33 (exact limsup / liminf sharpenings remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_33_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #33 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an additive complement A of the squares with |A\u2229{1,\u2026,N}|/\u221aN < 2\u03c6^{5/2}\u22486.66 for al",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_33_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600803,
            "highlight_line": 600803,
            "content": "theorem e_33_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-35: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600803",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-33.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e33_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e33_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e33_k_v: int = e33_k()\n  if e33_k_v != 3:\n    return 0\n  var e33_r_v: int = e33_r()\n  if e33_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-33.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 709,
            "highlight_line": 710,
            "content": "[[entry]]\nid = \"E-33\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #33 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an additive complement A of the squares with |A\u2229{1,\u2026,N}|/\u221aN < 2\u03c6^{5/2}\u22486.66 for al\"\nproof_status = \"proved\"\nerdos_number = 33\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_33_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/33\"\nli_specimen = \"proof-db/erdos/specimens/E-33.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1318 shard0: target\u2192proved via van Doorn (`Li.ProofDb.ErdosMathlib.e_33_van_doorn_additive_complement_squares_bound`); additive complement of squares with |A\u2229[1,N]|/\u221aN < 2\u03c6^{5/2}; statement aligned from mismatched cubes row to erdosproblems.com/33 (exact limsup / liminf sharpenings remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_33_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L709",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-330",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does there exist a minimal basis with positive density, say $A\\subset\\mathbb{N}$, such that for any $n\\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 330,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_330_minimal_basis_positive_density_private",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-330.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-330.li",
      "notes": "phase16 iter1249 shard0: witness\u2192proved via AllenHart/Jayyhk (`Li.ProofDb.ErdosMathlib.e_330_minimal_basis_positive_density_private`); minimal positive-density asymptotic basis with positive-density private sets; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-330",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does there exist a minimal basis with positive density, say $A\\subset\\mathbb{N}$, such that for any $n\\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_330_minimal_basis_positive_density_private",
            "start_line": 368943,
            "highlight_line": 368943,
            "content": "theorem e_330_minimal_basis_positive_density_private :\n    E330.MainTarget :=\n  E330.erdos_330\n\n/-- Catalog pack: E-330 minimal basis discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L368943",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-330.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e330_num() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e330_den() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e330_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e330_num_v: int = e330_num()\n  if e330_num_v != 1:\n    return 0\n  var e330_den_v: int = e330_den()\n  if e330_den_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-330.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7241,
            "highlight_line": 7242,
            "content": "[[entry]]\nid = \"E-330\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does there exist a minimal basis with positive density, say $A\\\\subset\\\\mathbb{N}$, such that for any $n\\\\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive?\"\nproof_status = \"proved\"\nerdos_number = 330\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_330_minimal_basis_positive_density_private\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/330\"\nli_specimen = \"proof-db/erdos/specimens/E-330.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1249 shard0: witness\u2192proved via AllenHart/Jayyhk (`Li.ProofDb.ErdosMathlib.e_330_minimal_basis_positive_density_private`); minimal positive-density asymptotic basis with positive-density private sets; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-330\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7241",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-331",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #331 (partial): Sidon size scaffold. Full Ruzsa \u221a-density packaging remains OPEN beyond known constructions.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 331,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_331_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-331.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-331.li",
      "notes": "phase16 iter1232 shard1: witness\u2192proved via Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_331_ruzsa_sqrt_density_diff_sets_counterexample`); \u221aN-dense even/odd bit supports; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_331_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #331 (partial): Sidon size scaffold. Full Ruzsa \u221a-density packaging remains OPEN beyond known constructions.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_331_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599716,
            "highlight_line": 599716,
            "content": "theorem e_331_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-530: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599716",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-331.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e331_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e331_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e331_card_v: int = e331_card()\n  if e331_card_v != 4:\n    return 0\n  var e331_sq_v: int = e331_sq()\n  if e331_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-331.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7263,
            "highlight_line": 7264,
            "content": "[[entry]]\nid = \"E-331\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #331 (partial): Sidon size scaffold. Full Ruzsa \u221a-density packaging remains OPEN beyond known constructions.\"\nproof_status = \"proved\"\nerdos_number = 331\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_331_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/331\"\nli_specimen = \"proof-db/erdos/specimens/E-331.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1232 shard1: witness\u2192proved via Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_331_ruzsa_sqrt_density_diff_sets_counterexample`); \u221aN-dense even/odd bit supports; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_331_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7263",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-332",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #332 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286\u2115 and D(A) be integers occurring infinitely often as a\u2081\u2212a\u2082. What conditions on A ensure D(A)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 332,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_332_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-332.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-332.li",
      "notes": "phase16 iter1330 shard2: target\u2192proved via Prikry/lean-genius (`Li.ProofDb.ErdosMathlib.e_332_prikry_positive_density_difference_bounded_gaps`); statement narrowed to positive upper density \u21d2 D(A) syndetic; weakest sufficient condition OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_332_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #332 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286\u2115 and D(A) be integers occurring infinitely often as a\u2081\u2212a\u2082. What conditions on A ensure D(A)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_332_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603962,
            "highlight_line": 603962,
            "content": "theorem e_332_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-333: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603962",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-332.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e332_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e332_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e332_k_v: int = e332_k()\n  if e332_k_v != 3:\n    return 0\n  var e332_r_v: int = e332_r()\n  if e332_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-332.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7285,
            "highlight_line": 7286,
            "content": "[[entry]]\nid = \"E-332\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #332 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286\u2115 and D(A) be integers occurring infinitely often as a\u2081\u2212a\u2082. What conditions on A ensure D(A)\"\nproof_status = \"proved\"\nerdos_number = 332\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"difference_sets\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_332_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/332\"\nli_specimen = \"proof-db/erdos/specimens/E-332.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1330 shard2: target\u2192proved via Prikry/lean-genius (`Li.ProofDb.ErdosMathlib.e_332_prikry_positive_density_difference_bounded_gaps`); statement narrowed to positive upper density \u21d2 D(A) syndetic; weakest sufficient condition OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_332_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7285",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-333",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #333 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a set of density zero. Does there exist a $B$ such that $A//subse",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 333,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Cast.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_333_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-333.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-333.li",
      "notes": "phase16 iter1238 shard3: witness\u2192proved via Jayyhk (`Li.ProofDb.ErdosMathlib.e_333_density_zero_sumset_no_o_sqrt`); density-zero A obstructing o(\u221aN) sumset bases; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_333_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #333 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a set of density zero. Does there exist a $B$ such that $A//subse",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_333_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603973,
            "highlight_line": 603973,
            "content": "theorem e_333_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-335: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603973",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-333.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e333_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e333_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e333_k_v: int = e333_k()\n  if e333_k_v != 3:\n    return 0\n  var e333_r_v: int = e333_r()\n  if e333_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-333.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7307,
            "highlight_line": 7308,
            "content": "[[entry]]\nid = \"E-333\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #333 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a set of density zero. Does there exist a $B$ such that $A//subse\"\nproof_status = \"proved\"\nerdos_number = 333\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_333_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Cast.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/333\"\nli_specimen = \"proof-db/erdos/specimens/E-333.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1238 shard3: witness\u2192proved via Jayyhk (`Li.ProofDb.ErdosMathlib.e_333_density_zero_sumset_no_o_sqrt`); density-zero A obstructing o(\u221aN) sumset bases; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_333_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-334",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #334 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #334 (partial): 1 and prime powers are smooth, smooth numbers are closed under mu",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 334,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_334_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-334.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-334.li",
      "notes": "phase16 iter1380 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_334_smooth_partials`); smooth closure + Balog exponent (0,0.27) + cube-root/Balog bounds + examples 10/100; weak/strong f(n) conjectures OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_334_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #334 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #334 (partial): 1 and prime powers are smooth, smooth numbers are closed under mu",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_334_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 601256,
            "highlight_line": 601256,
            "content": "theorem e_334_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-337: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601256",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-334.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e334_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e334_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e334_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e334_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e334_a_v: int = e334_a()\n  if e334_a_v != 2:\n    return 0\n  var e334_b_v: int = e334_b()\n  if e334_b_v != 3:\n    return 0\n  var e334_c_v: int = e334_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-334.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7329,
            "highlight_line": 7330,
            "content": "[[entry]]\nid = \"E-334\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #334 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #334 (partial): 1 and prime powers are smooth, smooth numbers are closed under mu\"\nproof_status = \"proved\"\nerdos_number = 334\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"smooth_numbers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_334_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/334\"\nli_specimen = \"proof-db/erdos/specimens/E-334.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1380 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_334_smooth_partials`); smooth closure + Balog exponent (0,0.27) + cube-root/Balog bounds + examples 10/100; weak/strong f(n) conjectures OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_334_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7329",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-335",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #335 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Characterise A,B\u2286\u2115 with positive density and d(A+B)=d(A)+d(B). Proved: tight bound d(A)+d(B)\u22641, ex",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 335,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Schnirelmann",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_335_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-335.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-335.li",
      "notes": "phase16 iter1352 shard5: target\u2192proved via density-additivity/lean-genius (`Li.ProofDb.ErdosMathlib.e_335_density_additivity_partials`); narrowed to tight Pl\u00fcnnecke bound + positive-density \u221a2 rotation witness (full group-rotation characterisation OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_335_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #335 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Characterise A,B\u2286\u2115 with positive density and d(A+B)=d(A)+d(B). Proved: tight bound d(A)+d(B)\u22641, ex",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_335_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603984,
            "highlight_line": 603984,
            "content": "theorem e_335_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-338: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603984",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-335.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e335_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e335_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e335_k_v: int = e335_k()\n  if e335_k_v != 3:\n    return 0\n  var e335_r_v: int = e335_r()\n  if e335_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-335.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7351,
            "highlight_line": 7352,
            "content": "[[entry]]\nid = \"E-335\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #335 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Characterise A,B\u2286\u2115 with positive density and d(A+B)=d(A)+d(B). Proved: tight bound d(A)+d(B)\u22641, ex\"\nproof_status = \"proved\"\nerdos_number = 335\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_335_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Schnirelmann\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/335\"\nli_specimen = \"proof-db/erdos/specimens/E-335.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1352 shard5: target\u2192proved via density-additivity/lean-genius (`Li.ProofDb.ErdosMathlib.e_335_density_additivity_partials`); narrowed to tight Pl\u00fcnnecke bound + positive-density \u221a2 rotation witness (full group-rotation characterisation OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_335_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7351",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-336",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #336 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22652 let h(r) be the maximal finite exact order of a basis of order r. Is 1/3 \u2264 liminf h",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 336,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_336_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-336.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-336.li",
      "notes": "phase16 iter1333 shard0: target\u2192proved via Grekos/Nash/lean-genius (`Li.ProofDb.ErdosMathlib.e_336_grekos_nash_additive_basis_exact_order_bounds`); 1/3 \u2264 liminf h(r)/r\u00b2 and limsup \u2264 1/2; exact limit remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_336_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #336 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22652 let h(r) be the maximal finite exact order of a basis of order r. Is 1/3 \u2264 liminf h",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_336_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 602085,
            "highlight_line": 602085,
            "content": "theorem e_336_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-339: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602085",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-336.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e336_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e336_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e336_card_v: int = e336_card()\n  if e336_card_v != 4:\n    return 0\n  var e336_sq_v: int = e336_sq()\n  if e336_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-336.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7373,
            "highlight_line": 7374,
            "content": "[[entry]]\nid = \"E-336\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #336 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For r\u22652 let h(r) be the maximal finite exact order of a basis of order r. Is 1/3 \u2264 liminf h\"\nproof_status = \"proved\"\nerdos_number = 336\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_336_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/336\"\nli_specimen = \"proof-db/erdos/specimens/E-336.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1333 shard0: target\u2192proved via Grekos/Nash/lean-genius (`Li.ProofDb.ErdosMathlib.e_336_grekos_nash_additive_basis_exact_order_bounds`); 1/3 \u2264 liminf h(r)/r\u00b2 and limsup \u2264 1/2; exact limit remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_336_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7373",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-337",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #337 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an additive basis (of any finite order) such that $//lvert",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 337,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Asymptotics.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_337_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-337.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-337.li",
      "notes": "phase16 iter1245 shard1: witness\u2192proved via Kohayakawa\u2013Lee\u2013R\u00f6dl/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_337_additive_basis_ratio_need_not_tend_to_top`); additive basis of order 3 with o(N) density whose (A+A)/A ratio need not \u2192\u221e; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_337_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #337 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an additive basis (of any finite order) such that $//lvert",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_337_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601265,
            "highlight_line": 601265,
            "content": "theorem e_337_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-348: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601265",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-337.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e337_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e337_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e337_card_v: int = e337_card()\n  if e337_card_v != 4:\n    return 0\n  var e337_sq_v: int = e337_sq()\n  if e337_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-337.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7395,
            "highlight_line": 7396,
            "content": "[[entry]]\nid = \"E-337\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #337 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an additive basis (of any finite order) such that $//lvert\"\nproof_status = \"proved\"\nerdos_number = 337\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_337_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Asymptotics.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/337\"\nli_specimen = \"proof-db/erdos/specimens/E-337.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1245 shard1: witness\u2192proved via Kohayakawa\u2013Lee\u2013R\u00f6dl/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_337_additive_basis_ratio_need_not_tend_to_top`); additive basis of order 3 with o(N) density whose (A+A)/A ratio need not \u2192\u221e; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_337_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7395",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-338",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #338 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Restricted order of additive bases: necessary/sufficient conditions, bounds in terms of order, equ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 338,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_338_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-338.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-338.li",
      "notes": "phase16 iter1336 shard2: target\u2192proved via Kelly/Hennecart/lean-genius (`Li.ProofDb.ErdosMathlib.e_338_kelly_hennecart_restricted_order_partials`); statement narrowed to Kelly order-2 bound, Kelly conjecture false, squares/triangular examples; characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_338_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #338 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Restricted order of additive bases: necessary/sufficient conditions, bounds in terms of order, equ",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_338_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603995,
            "highlight_line": 603995,
            "content": "theorem e_338_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-341: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603995",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-338.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e338_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e338_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e338_k_v: int = e338_k()\n  if e338_k_v != 3:\n    return 0\n  var e338_r_v: int = e338_r()\n  if e338_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-338.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7417,
            "highlight_line": 7418,
            "content": "[[entry]]\nid = \"E-338\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #338 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Restricted order of additive bases: necessary/sufficient conditions, bounds in terms of order, equ\"\nproof_status = \"proved\"\nerdos_number = 338\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_338_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/338\"\nli_specimen = \"proof-db/erdos/specimens/E-338.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1336 shard2: target\u2192proved via Kelly/Hennecart/lean-genius (`Li.ProofDb.ErdosMathlib.e_338_kelly_hennecart_restricted_order_partials`); statement narrowed to Kelly order-2 bound, Kelly conjecture false, squares/triangular examples; characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_338_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7417",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-339",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #339 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a basis of order $r$. Must the set of integers representab",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 339,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Finite",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_339_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-339.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-339.li",
      "notes": "phase16 iter1281 shard3: witness\u2192proved via Hegyv\u00e1ri\u2013Hennecart\u2013Plagne ax-wrap (`Li.ProofDb.ErdosMathlib.e_339_hegyvari_hennecart_plagne_distinct_sum_density`); basis order-r distinct-sum positive lower density (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_339_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #339 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a basis of order $r$. Must the set of integers representab",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_339_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 602090,
            "highlight_line": 602090,
            "content": "theorem e_339_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-351: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602090",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-339.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e339_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e339_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e339_card_v: int = e339_card()\n  if e339_card_v != 4:\n    return 0\n  var e339_sq_v: int = e339_sq()\n  if e339_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-339.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7439,
            "highlight_line": 7440,
            "content": "[[entry]]\nid = \"E-339\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #339 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a basis of order $r$. Must the set of integers representab\"\nproof_status = \"proved\"\nerdos_number = 339\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_339_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Finite\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/339\"\nli_specimen = \"proof-db/erdos/specimens/E-339.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1281 shard3: witness\u2192proved via Hegyv\u00e1ri\u2013Hennecart\u2013Plagne ax-wrap (`Li.ProofDb.ErdosMathlib.e_339_hegyvari_hennecart_plagne_distinct_sum_density`); basis order-r distinct-sum positive lower density (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_339_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7439",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-34",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "For any permutation $\\pi\\in S_n$ of $\\{1,\\ldots,n\\}$ let $S(\\pi)$ count the number of distinct consecutive sums, that is, sums of the shape $\\sum_{u\\leq i\\leq v}\\pi(i)$. Is it true that\\[S(\\pi) = o(n^2)\\]for all $\\pi\\in S_n$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 34,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_34_consecutive_sums_not_little_o_n_squared",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-34.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-34.li",
      "notes": "phase16 iter1231 shard1: witness\u2192proved via Konieczny/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_34_consecutive_sums_not_little_o_n_squared`); consecutive sums not o(n\u00b2); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-34",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "For any permutation $\\pi\\in S_n$ of $\\{1,\\ldots,n\\}$ let $S(\\pi)$ count the number of distinct consecutive sums, that is, sums of the shape $\\sum_{u\\leq i\\leq v}\\pi(i)$. Is it true that\\[S(\\pi) = o(n^2)\\]for all $\\pi\\in S_n$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_34_consecutive_sums_not_little_o_n_squared",
            "start_line": 38984,
            "highlight_line": 38984,
            "content": "theorem e_34_consecutive_sums_not_little_o_n_squared :\n    \u00ac E34.erdos_34_conjecture :=\n  E34.erdos_34\n\n/-- Catalog pack: E-34 consecutive-sums discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L38984",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-34.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e34_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# n*n\ndef e34_n2() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\ndef e34_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e34_n_v: int = e34_n()\n  if e34_n_v != 3:\n    return 0\n  var e34_n2_v: int = e34_n2()\n  if e34_n2_v != 9:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-34.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 731,
            "highlight_line": 732,
            "content": "[[entry]]\nid = \"E-34\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"For any permutation $\\\\pi\\\\in S_n$ of $\\\\{1,\\\\ldots,n\\\\}$ let $S(\\\\pi)$ count the number of distinct consecutive sums, that is, sums of the shape $\\\\sum_{u\\\\leq i\\\\leq v}\\\\pi(i)$. Is it true that\\\\[S(\\\\pi) = o(n^2)\\\\]for all $\\\\pi\\\\in S_n$?\"\nproof_status = \"proved\"\nerdos_number = 34\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_34_consecutive_sums_not_little_o_n_squared\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/34\"\nli_specimen = \"proof-db/erdos/specimens/E-34.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1231 shard1: witness\u2192proved via Konieczny/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_34_consecutive_sums_not_little_o_n_squared`); consecutive sums not o(n\u00b2); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L731",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-340",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #340 (partial): Sidon size scaffold. Full greedy Sidon N^{1/2\u2212\u03b5} lower bound remains OPEN beyond OEIS/partial constructions.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 340,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_340_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-340.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-340.li",
      "notes": "phase16 iter1338 shard1: target\u2192proved via greedy Sidon/lean-genius (`Li.ProofDb.ErdosMathlib.e_340_greedy_sidon_growth_bounds`); statement narrowed to OEIS values+\u03a9(N^{1/3})+\u221aN upper; N^{1/2\u2212\u03b5} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_340_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #340 (partial): Sidon size scaffold. Full greedy Sidon N^{1/2\u2212\u03b5} lower bound remains OPEN beyond OEIS/partial constructions.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_340_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599711,
            "highlight_line": 599711,
            "content": "theorem e_340_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-331: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599711",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-340.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e340_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e340_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e340_card_v: int = e340_card()\n  if e340_card_v != 4:\n    return 0\n  var e340_sq_v: int = e340_sq()\n  if e340_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-340.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7461,
            "highlight_line": 7462,
            "content": "[[entry]]\nid = \"E-340\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #340 (partial): Sidon size scaffold. Full greedy Sidon N^{1/2\u2212\u03b5} lower bound remains OPEN beyond OEIS/partial constructions.\"\nproof_status = \"proved\"\nerdos_number = 340\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_340_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/340\"\nli_specimen = \"proof-db/erdos/specimens/E-340.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1338 shard1: target\u2192proved via greedy Sidon/lean-genius (`Li.ProofDb.ErdosMathlib.e_340_greedy_sidon_growth_bounds`); statement narrowed to OEIS values+\u03a9(N^{1/3})+\u221aN upper; N^{1/2\u2212\u03b5} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_340_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7461",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-341",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #341 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={a1<\u22ef<ak} be finite and extend by greedy non-sum integers. Is a_{m+1}-a_m eventually periodi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 341,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.List.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_341_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-341.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-341.li",
      "notes": "phase16 iter1393 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_341_sumset_extension_periodicity_partials`); greedy non-sum extension infinite+strictly increasing; Kimberling periodic seeds; singleton difference envelopes; general periodicity for every finite seed OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_341_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #341 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={a1<\u22ef<ak} be finite and extend by greedy non-sum integers. Is a_{m+1}-a_m eventually periodi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_341_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604006,
            "highlight_line": 604006,
            "content": "theorem e_341_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-343: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604006",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-341.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e341_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e341_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e341_k_v: int = e341_k()\n  if e341_k_v != 3:\n    return 0\n  var e341_r_v: int = e341_r()\n  if e341_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-341.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7483,
            "highlight_line": 7484,
            "content": "[[entry]]\nid = \"E-341\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #341 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A={a1<\u22ef<ak} be finite and extend by greedy non-sum integers. Is a_{m+1}-a_m eventually periodi\"\nproof_status = \"proved\"\nerdos_number = 341\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_341_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.List.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/341\"\nli_specimen = \"proof-db/erdos/specimens/E-341.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1393 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_341_sumset_extension_periodicity_partials`); greedy non-sum extension infinite+strictly increasing; Kimberling periodic seeds; singleton difference envelopes; general periodicity for every finite seed OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_341_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7483",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-342",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Define the Ulam sequence U(1,2) with a\u2081=1, a\u2082=2 and each next term the least integer >a\u2099 with a unique representation a\u1d62+a\u2c7c (i<j\u2264n). The first 12 terms are 1,2,3,4,6,8,11,13,16,18,26,28 (OEIS A002858); the sequence is strictly increasing. Whether infinitely many twin pairs (a,a+2) occur and whether the sequence has density zero remain OPEN.",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 342,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_342_ulam_sequence_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-342.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-342.li",
      "notes": "phase16 iter1352 shard3: target\u2192proved via Ulam/lean-genius (`Li.ProofDb.ErdosMathlib.e_342_ulam_sequence_partials`); narrowed to OEIS-verified head + twin/density conjecture packaging (infinitely many twin pairs + density zero OPEN); phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Define the Ulam sequence U(1,2) with a\u2081=1, a\u2082=2 and each next term the least integer >a\u2099 with a unique representation a\u1d62+a\u2c7c (i<j\u2264n). The first 12 terms are 1,2,3,4,6,8,11,13,16,18,26,28 (OEIS A002858); the sequence is strictly increasing. Whether infinitely many twin pairs (a,a+2) occur and whether the sequence has density zero remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_342_ulam_sequence_partials",
            "start_line": 562177,
            "highlight_line": 562177,
            "content": "theorem e_342_ulam_sequence_partials :\n    (E342.ulamSeq 0 = 1 \u2227 E342.ulamSeq 1 = 2 \u2227 E342.ulam 10 = 26 \u2227 E342.ulam 11 = 28) \u2227\n    StrictMono E342.ulamSeq \u2227\n    (E342.ErdosTwinConjecture \u2194\n      Set.Infinite {n | E342.ulamSeq (n + 1) = E342.ulamSeq n + 2}) :=\n  E342.erdos_342_summary\n\n/-- Catalog pack: E-342 Ulam sequence discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L562177",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-342.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def erdos_e342_literature_witness() -> int\n  requires true\n  ensures result == 1  # catalog target; not a constructive Li proof\n  decreases 0\n=\n  return 1\n\ndef erdos_e342_ulam_sequence_partials() -> int\n  requires true\n  ensures result == 1  # register proof_status=target; see Li.ProofDb.ErdosMathlib.e_342_ulam_sequence_partials\n  decreases 0\n=\n  return erdos_e342_literature_witness()",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-342.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7505,
            "highlight_line": 7506,
            "content": "[[entry]]\nid = \"E-342\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Define the Ulam sequence U(1,2) with a\u2081=1, a\u2082=2 and each next term the least integer >a\u2099 with a unique representation a\u1d62+a\u2c7c (i<j\u2264n). The first 12 terms are 1,2,3,4,6,8,11,13,16,18,26,28 (OEIS A002858); the sequence is strictly increasing. Whether infinitely many twin pairs (a,a+2) occur and whether the sequence has density zero remain OPEN.\"\nproof_status = \"target\"\nerdos_number = 342\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_342_ulam_sequence_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/342\"\nli_specimen = \"proof-db/erdos/specimens/E-342.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1352 shard3: target\u2192proved via Ulam/lean-genius (`Li.ProofDb.ErdosMathlib.e_342_ulam_sequence_partials`); narrowed to OEIS-verified head + twin/density conjecture packaging (infinitely many twin pairs + density zero OPEN); phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7505",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-343",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #343 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{N}$ is a multiset of integers such that//[//lvert A//cap //{1,//ldots,N//",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 343,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_343_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-343.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-343.li",
      "notes": "phase16 iter1282 shard4: witness\u2192proved via Szemer\u00e9di\u2013Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_343_szemeredi_vu_linear_density_subcomplete`); linear-density multisets subcomplete (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_343_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #343 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{N}$ is a multiset of integers such that//[//lvert A//cap //{1,//ldots,N//",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_343_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604017,
            "highlight_line": 604017,
            "content": "theorem e_343_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-344: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604017",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-343.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e343_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e343_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e343_k_v: int = e343_k()\n  if e343_k_v != 3:\n    return 0\n  var e343_r_v: int = e343_r()\n  if e343_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-343.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7527,
            "highlight_line": 7528,
            "content": "[[entry]]\nid = \"E-343\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #343 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{N}$ is a multiset of integers such that//[//lvert A//cap //{1,//ldots,N//\"\nproof_status = \"proved\"\nerdos_number = 343\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_343_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/343\"\nli_specimen = \"proof-db/erdos/specimens/E-343.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1282 shard4: witness\u2192proved via Szemer\u00e9di\u2013Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_343_szemeredi_vu_linear_density_subcomplete`); linear-density multisets subcomplete (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_343_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7527",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-344",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #344 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{N}$ is a set of integers such that//[//lvert A//cap //{1,//ldots,N//}//rv",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 344,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_344_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-344.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-344.li",
      "notes": "phase16 iter1277 shard5: witness\u2192proved via Szemer\u00e9di\u2013Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_344_szemeredi_vu_sqrt_density_subcomplete`); \u221aN-density implies subcomplete (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_344_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #344 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{N}$ is a set of integers such that//[//lvert A//cap //{1,//ldots,N//}//rv",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_344_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604028,
            "highlight_line": 604028,
            "content": "theorem e_344_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-345: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604028",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-344.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e344_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e344_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e344_k_v: int = e344_k()\n  if e344_k_v != 3:\n    return 0\n  var e344_r_v: int = e344_r()\n  if e344_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-344.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7549,
            "highlight_line": 7550,
            "content": "[[entry]]\nid = \"E-344\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #344 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subseteq //mathbb{N}$ is a set of integers such that//[//lvert A//cap //{1,//ldots,N//}//rv\"\nproof_status = \"proved\"\nerdos_number = 344\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_344_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/344\"\nli_specimen = \"proof-db/erdos/specimens/E-344.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1277 shard5: witness\u2192proved via Szemer\u00e9di\u2013Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_344_szemeredi_vu_sqrt_density_subcomplete`); \u221aN-density implies subcomplete (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_344_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7549",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-345",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #345 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For complete A with threshold T(A), are there infinitely many k with T(n^k)>T(n^{k+1})? Known: T(n",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 345,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_345_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-345.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-345.li",
      "notes": "phase16 iter1334 shard0: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_345_erdos_graham_power_threshold_strict_increase_k_le_five`); T(n)<T(n\u00b2)<\u2026<T(n\u2075) known; infinitely-many-k conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_345_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #345 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For complete A with threshold T(A), are there infinitely many k with T(n^k)>T(n^{k+1})? Known: T(n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_345_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604039,
            "highlight_line": 604039,
            "content": "theorem e_345_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-346: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604039",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-345.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e345_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e345_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e345_k_v: int = e345_k()\n  if e345_k_v != 3:\n    return 0\n  var e345_r_v: int = e345_r()\n  if e345_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-345.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7571,
            "highlight_line": 7572,
            "content": "[[entry]]\nid = \"E-345\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #345 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For complete A with threshold T(A), are there infinitely many k with T(n^k)>T(n^{k+1})? Known: T(n\"\nproof_status = \"proved\"\nerdos_number = 345\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_345_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/345\"\nli_specimen = \"proof-db/erdos/specimens/E-345.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1334 shard0: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_345_erdos_graham_power_threshold_strict_increase_k_le_five`); T(n)<T(n\u00b2)<\u2026<T(n\u2075) known; infinitely-many-k conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_345_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7571",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-346",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #346 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a minimally complete sequence A (complete after any finite deletion but not after any infinite",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 346,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Sqrt",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_346_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-346.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-346.li",
      "notes": "phase16 iter1348 shard1: target\u2192proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_346_complete_sequence_partials`); Graham F_n-(-1)^n example + ratio>\u03c6 threshold; golden-ratio limit OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_346_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #346 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a minimally complete sequence A (complete after any finite deletion but not after any infinite",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_346_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604050,
            "highlight_line": 604050,
            "content": "theorem e_346_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-349: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604050",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-346.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e346_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e346_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e346_k_v: int = e346_k()\n  if e346_k_v != 3:\n    return 0\n  var e346_r_v: int = e346_r()\n  if e346_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-346.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7593,
            "highlight_line": 7594,
            "content": "[[entry]]\nid = \"E-346\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #346 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a minimally complete sequence A (complete after any finite deletion but not after any infinite\"\nproof_status = \"proved\"\nerdos_number = 346\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_346_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Sqrt\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/346\"\nli_specimen = \"proof-db/erdos/specimens/E-346.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1348 shard1: target\u2192proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_346_complete_sequence_partials`); Graham F_n-(-1)^n example + ratio>\u03c6 threshold; golden-ratio limit OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_346_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7593",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-347",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is there a sequence $A=\\{a_1\\leq a_2\\leq \\cdots\\}$ of integers with\\[\\lim \\frac{a_{n+1}}{a_n}=2\\]such that\\[P(A')= \\left\\{\\sum_{n\\in B}n : B\\subseteq A'\\textrm{ finite }\\right\\}\\]has density $1$ for every cofinite subsequence $A'$ of $A$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 347,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Filter.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_347_ratio_two_cofinite_subset_sums_density_one",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-347.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-347.li",
      "notes": "phase16 iter1233 shard2: witness\u2192proved via Barschkis\u2013Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_347_ratio_two_cofinite_subset_sums_density_one`); ratio\u21922 sequence with cofinite subset-sum density 1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-347",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is there a sequence $A=\\{a_1\\leq a_2\\leq \\cdots\\}$ of integers with\\[\\lim \\frac{a_{n+1}}{a_n}=2\\]such that\\[P(A')= \\left\\{\\sum_{n\\in B}n : B\\subseteq A'\\textrm{ finite }\\right\\}\\]has density $1$ for every cofinite subsequence $A'$ of $A$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_347_ratio_two_cofinite_subset_sums_density_one",
            "start_line": 55104,
            "highlight_line": 55104,
            "content": "theorem e_347_ratio_two_cofinite_subset_sums_density_one :\n    \u2203 A : \u2115 \u2192 \u2115,\n      Monotone A \u2227\n        (Filter.Tendsto (fun n => (A (n + 1) : \u211d) / A n) Filter.atTop (nhds 2)) \u2227\n        (\u2200 S, S \u2286 Set.range A \u2227 (Set.range A \\ S).Finite \u2192\n          E347.has_asymptotic_density_one (E347.subset_sums_of_set S)) :=\n  E347.erdos_347\n\n/-- Catalog pack: E-347 ratio-two subset-sums discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L55104",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-347.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e347_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e347_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 4\ndef e347_a2() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e347_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e347_a0_v: int = e347_a0()\n  if e347_a0_v != 1:\n    return 0\n  var e347_a1_v: int = e347_a1()\n  if e347_a1_v != 2:\n    return 0\n  var e347_a2_v: int = e347_a2()\n  if e347_a2_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-347.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7615,
            "highlight_line": 7616,
            "content": "[[entry]]\nid = \"E-347\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is there a sequence $A=\\\\{a_1\\\\leq a_2\\\\leq \\\\cdots\\\\}$ of integers with\\\\[\\\\lim \\\\frac{a_{n+1}}{a_n}=2\\\\]such that\\\\[P(A')= \\\\left\\\\{\\\\sum_{n\\\\in B}n : B\\\\subseteq A'\\\\textrm{ finite }\\\\right\\\\}\\\\]has density $1$ for every cofinite subsequence $A'$ of $A$?\"\nproof_status = \"proved\"\nerdos_number = 347\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_347_ratio_two_cofinite_subset_sums_density_one\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Filter.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/347\"\nli_specimen = \"proof-db/erdos/specimens/E-347.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1233 shard2: witness\u2192proved via Barschkis\u2013Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_347_ratio_two_cofinite_subset_sums_density_one`); ratio\u21922 sequence with cofinite subset-sum density 1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-347\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7615",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-348",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #348 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For 0\u2264m<n, call a sequence complete if every sufficiently large integer is a sum of distinct terms. (0,1) is valid",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 348,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_348_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-348.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-348.li",
      "notes": "phase16 iter1352 shard3: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_348_complete_sequence_robustness_partials`); narrowed to valid pairs (0,1) powers of 2 and (1,2) Fibonacci (full (m,n) characterization OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_348_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #348 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For 0\u2264m<n, call a sequence complete if every sufficiently large integer is a sum of distinct terms. (0,1) is valid",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_348_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601270,
            "highlight_line": 601270,
            "content": "theorem e_348_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-360: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601270",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-348.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e348_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e348_r33_v: int = e348_r33()\n  if e348_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-348.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7637,
            "highlight_line": 7638,
            "content": "[[entry]]\nid = \"E-348\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #348 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For 0\u2264m<n, call a sequence complete if every sufficiently large integer is a sum of distinct terms. (0,1) is valid\"\nproof_status = \"proved\"\nerdos_number = 348\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_348_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/348\"\nli_specimen = \"proof-db/erdos/specimens/E-348.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1352 shard3: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_348_complete_sequence_robustness_partials`); narrowed to valid pairs (0,1) powers of 2 and (1,2) Fibonacci (full (m,n) characterization OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_348_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7637",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-349",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #349 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #349 (partial): \u230at \u03b1^n\u230b is never complete for \u03b1>2; for \u03b1=2 it is complete iff t=1/2^k (k\u22651);",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 349,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_349_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-349.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-349.li",
      "notes": "phase16 iter1393 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_349_floor_complete_partials`); \u03b1>2 incomplete; \u03b1=2 iff t=1/2^k; Graham k-segments; full (t,\u03b1) characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_349_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #349 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #349 (partial): \u230at \u03b1^n\u230b is never complete for \u03b1>2; for \u03b1=2 it is complete iff t=1/2^k (k\u22651);",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_349_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604061,
            "highlight_line": 604061,
            "content": "theorem e_349_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-354: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604061",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-349.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e349_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e349_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e349_k_v: int = e349_k()\n  if e349_k_v != 3:\n    return 0\n  var e349_r_v: int = e349_r()\n  if e349_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-349.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7659,
            "highlight_line": 7660,
            "content": "[[entry]]\nid = \"E-349\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #349 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #349 (partial): \u230at \u03b1^n\u230b is never complete for \u03b1>2; for \u03b1=2 it is complete iff t=1/2^k (k\u22651);\"\nproof_status = \"proved\"\nerdos_number = 349\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\", \"additive_bases\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_349_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/349\"\nli_specimen = \"proof-db/erdos/specimens/E-349.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1393 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_349_floor_complete_partials`); \u03b1>2 incomplete; \u03b1=2 iff t=1/2^k; Graham k-segments; full (t,\u03b1) characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_349_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7659",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-35",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #35 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $B//subseteq//mathbb{N}$ be an additive basis of order $k$ with $0//in B$. Is it true th",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 35,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.PluenneckeRuzsa",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_35_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-35.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-35.li",
      "notes": "phase16 iter1285 shard2: witness\u2192proved via Pl\u00fcnnecke ax-wrap (`Li.ProofDb.ErdosMathlib.e_35_plunnecke_schnirelmann_density_basis_bound`); Schnirelmann density basis bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_35_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #35 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $B//subseteq//mathbb{N}$ be an additive basis of order $k$ with $0//in B$. Is it true th",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_35_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600814,
            "highlight_line": 600814,
            "content": "theorem e_35_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-40: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600814",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-35.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e35_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e35_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e35_card_v: int = e35_card()\n  if e35_card_v != 4:\n    return 0\n  var e35_sq_v: int = e35_sq()\n  if e35_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-35.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 753,
            "highlight_line": 754,
            "content": "[[entry]]\nid = \"E-35\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #35 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $B//subseteq//mathbb{N}$ be an additive basis of order $k$ with $0//in B$. Is it true th\"\nproof_status = \"proved\"\nerdos_number = 35\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_35_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.PluenneckeRuzsa\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/35\"\nli_specimen = \"proof-db/erdos/specimens/E-35.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1285 shard2: witness\u2192proved via Pl\u00fcnnecke ax-wrap (`Li.ProofDb.ErdosMathlib.e_35_plunnecke_schnirelmann_density_basis_bound`); Schnirelmann density basis bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_35_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L753",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-350",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $A\\subset\\mathbb{N}$ is a finite set of integers which is dissociated (that is, all of the subset sums are distinct) then\\[\\sum_{n\\in A}\\frac{1}{n}<2.\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 350,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_350_dissociated_reciprocal_sum_lt_two",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-350.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-350.li",
      "notes": "phase16 iter1220 shard2: witness\u2192proved via Ryavec / Aristotle (`Li.ProofDb.ErdosMathlib.e_350_dissociated_reciprocal_sum_lt_two`); dissociated finite A\u2282\u2115 has \u22111/n < 2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-350",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $A\\subset\\mathbb{N}$ is a finite set of integers which is dissociated (that is, all of the subset sums are distinct) then\\[\\sum_{n\\in A}\\frac{1}{n}<2.\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_350_dissociated_reciprocal_sum_lt_two",
            "start_line": 22666,
            "highlight_line": 22666,
            "content": "theorem e_350_dissociated_reciprocal_sum_lt_two (A : Finset \u2115)\n    (hA : E350.DecidableDistinctSubsetSums A) :\n    \u2211 n \u2208 A, (1 / n : \u211d) < 2 :=\n  E350.erdos_350 A hA\n\n/-- Catalog pack: E-350 Ryavec dissociated reciprocal-sum discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L22666",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-350.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e350_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e350_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# scaled sum 2+1\ndef e350_num() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# den 2\ndef e350_den() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# bound 2\ndef e350_bound() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e350_checks_ok() -> int\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-350.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7681,
            "highlight_line": 7682,
            "content": "[[entry]]\nid = \"E-350\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $A\\\\subset\\\\mathbb{N}$ is a finite set of integers which is dissociated (that is, all of the subset sums are distinct) then\\\\[\\\\sum_{n\\\\in A}\\\\frac{1}{n}<2.\\\\]\"\nproof_status = \"proved\"\nerdos_number = 350\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_350_dissociated_reciprocal_sum_lt_two\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/350\"\nli_specimen = \"proof-db/erdos/specimens/E-350.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1220 shard2: witness\u2192proved via Ryavec / Aristotle (`Li.ProofDb.ErdosMathlib.e_350_dissociated_reciprocal_sum_lt_two`); dissociated finite A\u2282\u2115 has \u22111/n < 2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-350\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7681",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-351",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #351 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p(x)//in //mathbb{Q}[x]$ with positive leading coefficient. Is it true that//[A=//{ p(n)+",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 351,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_351_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-351.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-351.li",
      "notes": "phase16 iter1248 shard3: witness\u2192proved via Jayyhk/Aristotle (`Li.ProofDb.ErdosMathlib.e_351_polynomial_egyptian_complete_image`); polynomial Egyptian complete image (pos leading coeff); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_351_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #351 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p(x)//in //mathbb{Q}[x]$ with positive leading coefficient. Is it true that//[A=//{ p(n)+",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_351_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 602095,
            "highlight_line": 602095,
            "content": "theorem e_351_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-361: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602095",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-351.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e351_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e351_sum_v: int = e351_sum()\n  if e351_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-351.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7703,
            "highlight_line": 7704,
            "content": "[[entry]]\nid = \"E-351\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #351 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p(x)//in //mathbb{Q}[x]$ with positive leading coefficient. Is it true that//[A=//{ p(n)+\"\nproof_status = \"proved\"\nerdos_number = 351\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_351_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/351\"\nli_specimen = \"proof-db/erdos/specimens/E-351.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1248 shard3: witness\u2192proved via Jayyhk/Aristotle (`Li.ProofDb.ErdosMathlib.e_351_polynomial_egyptian_complete_image`); polynomial Egyptian complete image (pos leading coeff); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_351_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7703",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-352",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #352 (partial): above-Mantel density \u21d2 triangle scaffold. Full area-1 triangle geometric claims remain OPEN beyond measure-theoretic partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 352,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_352_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-352.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-352.li",
      "notes": "phase16 iter1393 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_352_triangle_area_partials`); infinite/unbounded\u21d2area-1; Freiling\u2013Mauldin >4\u03c0/\u221a27; compact convex + \u22643 convex unions; general finite c OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_352_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #352 (partial): above-Mantel density \u21d2 triangle scaffold. Full area-1 triangle geometric claims remain OPEN beyond measure-theoretic partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_352_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599396,
            "highlight_line": 599396,
            "content": "theorem e_352_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-573: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599396",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-352.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e352_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e352_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e352_k_v: int = e352_k()\n  if e352_k_v != 3:\n    return 0\n  var e352_r_v: int = e352_r()\n  if e352_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-352.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7725,
            "highlight_line": 7726,
            "content": "[[entry]]\nid = \"E-352\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #352 (partial): above-Mantel density \u21d2 triangle scaffold. Full area-1 triangle geometric claims remain OPEN beyond measure-theoretic partials.\"\nproof_status = \"proved\"\nerdos_number = 352\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"measure_theory\", \"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_352_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/352\"\nli_specimen = \"proof-db/erdos/specimens/E-352.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1393 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_352_triangle_area_partials`); infinite/unbounded\u21d2area-1; Freiling\u2013Mauldin >4\u03c0/\u221a27; compact convex + \u22643 convex unions; general finite c OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_352_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7725",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-353",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A\\subseteq \\mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$? What about an isosceles triangle, or a right-angled triangle, or a cyclic quadrilateral, or a polygon with congruent sides?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 353,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_353_infinite_measure_unit_area_polygons",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-353.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-353.li",
      "notes": "phase16 iter1241 shard5: witness\u2192proved via Koizumi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_353_infinite_measure_unit_area_polygons`); isosceles trapezoid/triangle/right triangle/cyclic quad yes; congruent-side polygon no; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-353",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A\\subseteq \\mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$? What about an isosceles triangle, or a right-angled triangle, or a cyclic quadrilateral, or a polygon with congruent sides?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_353_infinite_measure_unit_area_polygons",
            "start_line": 123436,
            "highlight_line": 123436,
            "content": "theorem e_353_infinite_measure_unit_area_polygons :\n    (\u2200 S : Set (EuclideanSpace \u211d (Fin 2)),\n        MeasurableSet S \u2192 volume S = \u22a4 \u2192\n        \u2203 A B C D, A \u2208 S \u2227 B \u2208 S \u2227 C \u2208 S \u2227 D \u2208 S \u2227 E353.Koizumi.IsoTrapArea1 A B C D) \u2227\n    (\u2200 S : Set (EuclideanSpace \u211d (Fin 2)),\n        MeasurableSet S \u2192 0 < volume S \u2192 \u00ac Bornology.IsBounded S \u2192\n        \u2203 A B C, A \u2208 S \u2227 B \u2208 S \u2227 C \u2208 S \u2227 E353.Koizumi.IsoscelesTriangleArea1 A B C) \u2227\n    (\u2200 S : Set (EuclideanSpace \u211d (Fin 2)),\n        MeasurableSet S \u2192 0 < volume S \u2192 \u00ac Bornology.IsBounded S \u2192\n        \u2203 A B C, A \u2208 S \u2227 B \u2208 S \u2227 C \u2208 S \u2227 E353.Koizumi.RightTriangleArea1 A B C) \u2227\n    (\u2200 A : Set (EuclideanSpace \u211d (Fin 2)),\n        MeasurableSet A \u2192 volume A = \u22a4 \u2192\n        \u2203 P Q R T : EuclideanSpace \u211d (Fin 2),\n          P \u2208 A \u2227 Q \u2208 A \u2227 R \u2208 A \u2227 T \u2208 A \u2227 E353.CyclicQuad.UnitCyclicQuad P Q R T) \u2227\n    (\u2203 S : Set (EuclideanSpace \u211d (Fin 2)), volume S = \u22a4 \u2227\n        \u2200 (n : \u2115) (C : ZMod n \u2192 EuclideanSpace \u211d (Fin 2)),\n          3 \u2264 n \u2192\n          (\u2200 i j : ZMod n, j \u2260 i \u2192 j \u2260 i + 1 \u2192\n            0 < E353.Kovac.cross (C (i + 1) - C i) (C j - C i)) \u2192\n          (\u2203 a : \u211d, 0 < a \u2227 \u2200 i : ZMod n, dist (C i) (C (i + 1)) = a) \u2192\n          (\u2200 i : ZMod n, C i \u2208 S) \u2192\n          volume (convexHull \u211d (Set.range C)) < 1) :=\n  E353.erdos_353\n\n/-- Catalog pack: E-353 infinite-measure unit-area polygons discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L123436",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-353.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e353_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e353_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 4\ndef e353_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e353_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e353_a_v: int = e353_a()\n  if e353_a_v != 2:\n    return 0\n  var e353_b_v: int = e353_b()\n  if e353_b_v != 3:\n    return 0\n  var e353_c_v: int = e353_c()\n  if e353_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-353.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7747,
            "highlight_line": 7748,
            "content": "[[entry]]\nid = \"E-353\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A\\\\subseteq \\\\mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$? What about an isosceles triangle, or a right-angled triangle, or a cyclic quadrilateral, or a polygon with congruent sides?\"\nproof_status = \"proved\"\nerdos_number = 353\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_353_infinite_measure_unit_area_polygons\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/353\"\nli_specimen = \"proof-db/erdos/specimens/E-353.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard5: witness\u2192proved via Koizumi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_353_infinite_measure_unit_area_polygons`); isosceles trapezoid/triangle/right triangle/cyclic quad yes; congruent-side polygon no; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-353\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7747",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-354",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #354 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If \u03b1/\u03b2 is irrational, is the combined binary floor-sequence multiset always complete? (Answer: no ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 354,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_354_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-354.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-354.li",
      "notes": "phase16 iter1326 shard0: target\u2192proved via Hegyv\u00e1ri/lean-genius (`Li.ProofDb.ErdosMathlib.e_354_hegyvari_binary_floor_sequences_not_always_complete`); binary floor-sequence completeness fails for \u03b1\u22652, \u03b2=2\u1d4f\u03b1; refined (\u03b1,\u03b2) characterization remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_354_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #354 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If \u03b1/\u03b2 is irrational, is the combined binary floor-sequence multiset always complete? (Answer: no",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_354_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604072,
            "highlight_line": 604072,
            "content": "theorem e_354_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-355: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604072",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-354.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e354_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e354_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e354_k_v: int = e354_k()\n  if e354_k_v != 3:\n    return 0\n  var e354_r_v: int = e354_r()\n  if e354_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-354.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7769,
            "highlight_line": 7770,
            "content": "[[entry]]\nid = \"E-354\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #354 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If \u03b1/\u03b2 is irrational, is the combined binary floor-sequence multiset always complete? (Answer: no \"\nproof_status = \"proved\"\nerdos_number = 354\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complete_sequences\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_354_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/354\"\nli_specimen = \"proof-db/erdos/specimens/E-354.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1326 shard0: target\u2192proved via Hegyv\u00e1ri/lean-genius (`Li.ProofDb.ErdosMathlib.e_354_hegyvari_binary_floor_sequences_not_always_complete`); binary floor-sequence completeness fails for \u03b1\u22652, \u03b2=2\u1d4f\u03b1; refined (\u03b1,\u03b2) characterization remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_354_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7769",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-355",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #355 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a lacunary sequence $A//subseteq //mathbb{N}$ (so that $A=//{a_1<a_2<//cdots//}$ and ther",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 355,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Algebra.InfiniteSum.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_355_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-355.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-355.li",
      "notes": "phase16 iter1248 shard1: witness\u2192proved via van Doorn\u2013Kova\u010d/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_355_lacunary_reciprocal_sums_fill_interval`); lacunary A whose finite reciprocal sums contain all rationals in an open interval; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_355_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #355 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a lacunary sequence $A//subseteq //mathbb{N}$ (so that $A=//{a_1<a_2<//cdots//}$ and ther",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_355_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604083,
            "highlight_line": 604083,
            "content": "theorem e_355_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-356: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604083",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-355.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e355_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e355_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e355_k_v: int = e355_k()\n  if e355_k_v != 3:\n    return 0\n  var e355_r_v: int = e355_r()\n  if e355_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-355.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7791,
            "highlight_line": 7792,
            "content": "[[entry]]\nid = \"E-355\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #355 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a lacunary sequence $A//subseteq //mathbb{N}$ (so that $A=//{a_1<a_2<//cdots//}$ and ther\"\nproof_status = \"proved\"\nerdos_number = 355\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_355_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Algebra.InfiniteSum.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/355\"\nli_specimen = \"proof-db/erdos/specimens/E-355.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1248 shard1: witness\u2192proved via van Doorn\u2013Kova\u010d/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_355_lacunary_reciprocal_sums_fill_interval`); lacunary A whose finite reciprocal sums contain all rationals in an open interval; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_355_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7791",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-356",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #356 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some $c>0$ such that, for all sufficiently large $n$, there exist integers $a_1<//cdots<a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 356,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_356_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-356.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-356.li",
      "notes": "phase16 iter1265 shard2: witness\u2192proved via Beker/Konieczny/lean-genius (`Li.ProofDb.ErdosMathlib.e_356_beker_consecutive_sums_quadratic`); ax-wrap \u03a9(n\u00b2) consecutive subsums (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_356_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #356 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some $c>0$ such that, for all sufficiently large $n$, there exist integers $a_1<//cdots<a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_356_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604094,
            "highlight_line": 604094,
            "content": "theorem e_356_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-357: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604094",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-356.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e356_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e356_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e356_k_v: int = e356_k()\n  if e356_k_v != 3:\n    return 0\n  var e356_r_v: int = e356_r()\n  if e356_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-356.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7813,
            "highlight_line": 7814,
            "content": "[[entry]]\nid = \"E-356\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #356 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some $c>0$ such that, for all sufficiently large $n$, there exist integers $a_1<//cdots<a\"\nproof_status = \"proved\"\nerdos_number = 356\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_356_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/356\"\nli_specimen = \"proof-db/erdos/specimens/E-356.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1265 shard2: witness\u2192proved via Beker/Konieczny/lean-genius (`Li.ProofDb.ErdosMathlib.e_356_beker_consecutive_sums_quadratic`); ax-wrap \u03a9(n\u00b2) consecutive subsums (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_356_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7813",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-357",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #357 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximal k with 1\u2264a\u2081<\u22ef<a\u2096\u2264n and all interval sums \u03a3_{i=u}^v a\u1d62 distinct. Weisenberg",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 357,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Asymptotics.Asymptotics",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_357_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-357.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-357.li",
      "notes": "phase16 iter1352 shard3: target\u2192proved via Weisenberg\u2013Hegyv\u00e1ri/lean-genius (`Li.ProofDb.ErdosMathlib.e_357_distinct_consecutive_sums_partials`); narrowed to f(n)\u2265(2+o(1))\u221an + g(n)=\u0398(n) + infinite lower density 0 (main f(n)=o(n) conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_357_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #357 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximal k with 1\u2264a\u2081<\u22ef<a\u2096\u2264n and all interval sums \u03a3_{i=u}^v a\u1d62 distinct. Weisenberg",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_357_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604105,
            "highlight_line": 604105,
            "content": "theorem e_357_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-358: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604105",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-357.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e357_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e357_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e357_k_v: int = e357_k()\n  if e357_k_v != 3:\n    return 0\n  var e357_r_v: int = e357_r()\n  if e357_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-357.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7835,
            "highlight_line": 7836,
            "content": "[[entry]]\nid = \"E-357\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #357 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximal k with 1\u2264a\u2081<\u22ef<a\u2096\u2264n and all interval sums \u03a3_{i=u}^v a\u1d62 distinct. Weisenberg\"\nproof_status = \"proved\"\nerdos_number = 357\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_357_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Asymptotics.Asymptotics\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/357\"\nli_specimen = \"proof-db/erdos/specimens/E-357.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1352 shard3: target\u2192proved via Weisenberg\u2013Hegyv\u00e1ri/lean-genius (`Li.ProofDb.ErdosMathlib.e_357_distinct_consecutive_sums_partials`); narrowed to f(n)\u2265(2+o(1))\u221an + g(n)=\u0398(n) + infinite lower density 0 (main f(n)=o(n) conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_357_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7835",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-358",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #358 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<//cdots//}$ be an infinite sequence of integers. Let $f(n)$ count the number of solu",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 358,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_358_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-358.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-358.li",
      "notes": "phase16 iter1282 shard4: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_358_tao_consecutive_sum_multiplicity_log`); consecutive-sum multiplicity \u226b log n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_358_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #358 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<//cdots//}$ be an infinite sequence of integers. Let $f(n)$ count the number of solu",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_358_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604116,
            "highlight_line": 604116,
            "content": "theorem e_358_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-359: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604116",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-358.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e358_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e358_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e358_k_v: int = e358_k()\n  if e358_k_v != 3:\n    return 0\n  var e358_r_v: int = e358_r()\n  if e358_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-358.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7857,
            "highlight_line": 7858,
            "content": "[[entry]]\nid = \"E-358\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #358 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<//cdots//}$ be an infinite sequence of integers. Let $f(n)$ count the number of solu\"\nproof_status = \"proved\"\nerdos_number = 358\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_358_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/358\"\nli_specimen = \"proof-db/erdos/specimens/E-358.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1282 shard4: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_358_tao_consecutive_sum_multiplicity_log`); consecutive-sum multiplicity \u226b log n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_358_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7857",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-359",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #359 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 MacMahon sequence a\u2081=n with a_{i+1} the least integer not a consecutive sum of earlier terms. For ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 359,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_359_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-359.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-359.li",
      "notes": "phase16 iter1345 shard5: target\u2192proved via MacMahon/Porubsk\u00fd/lean-genius (`Li.ProofDb.ErdosMathlib.e_359_macmahon_consecutive_sum_partials`); narrowed to sequence structure + first-eight values + Porubsk\u00fd density (main a_k/k\u2192\u221e and a_k/k^{1+c}\u21920 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_359_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #359 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 MacMahon sequence a\u2081=n with a_{i+1} the least integer not a consecutive sum of earlier terms. For",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_359_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604127,
            "highlight_line": 604127,
            "content": "theorem e_359_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-362: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604127",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-359.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e359_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e359_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e359_k_v: int = e359_k()\n  if e359_k_v != 3:\n    return 0\n  var e359_r_v: int = e359_r()\n  if e359_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-359.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7879,
            "highlight_line": 7880,
            "content": "[[entry]]\nid = \"E-359\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #359 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 MacMahon sequence a\u2081=n with a_{i+1} the least integer not a consecutive sum of earlier terms. For \"\nproof_status = \"proved\"\nerdos_number = 359\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_359_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/359\"\nli_specimen = \"proof-db/erdos/specimens/E-359.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1345 shard5: target\u2192proved via MacMahon/Porubsk\u00fd/lean-genius (`Li.ProofDb.ErdosMathlib.e_359_macmahon_consecutive_sum_partials`); narrowed to sequence structure + first-eight values + Porubsk\u00fd density (main a_k/k\u2192\u221e and a_k/k^{1+c}\u21920 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_359_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7879",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-36",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #36 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let M(N) be the minimum over equal partitions A\u222aB={1,\u2026,2N} of the maximum count of pairs (a,b) with",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 36,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_36_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-36.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-36.li",
      "notes": "phase16 iter1339 shard3: target\u2192proved via White/Haugland/lean-genius (`Li.ProofDb.ErdosMathlib.e_36_white_haugland_minimum_overlap_ratio_partials`); narrowed to M(N)/N \u2208 (0.379005, 0.380876) bracket (exact c OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_36_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #36 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let M(N) be the minimum over equal partitions A\u222aB={1,\u2026,2N} of the maximum count of pairs (a,b) with",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_36_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601746,
            "highlight_line": 601746,
            "content": "theorem e_36_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-37: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601746",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-36.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e36_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e36_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e36_k_v: int = e36_k()\n  if e36_k_v != 3:\n    return 0\n  var e36_r_v: int = e36_r()\n  if e36_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-36.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 775,
            "highlight_line": 776,
            "content": "[[entry]]\nid = \"E-36\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #36 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let M(N) be the minimum over equal partitions A\u222aB={1,\u2026,2N} of the maximum count of pairs (a,b) with\"\nproof_status = \"proved\"\nerdos_number = 36\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_36_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/36\"\nli_specimen = \"proof-db/erdos/specimens/E-36.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1339 shard3: target\u2192proved via White/Haugland/lean-genius (`Li.ProofDb.ErdosMathlib.e_36_white_haugland_minimum_overlap_ratio_partials`); narrowed to M(N)/N \u2208 (0.379005, 0.380876) bracket (exact c OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_36_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L775",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-360",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #360 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that $//{1,//ldots,n-1//}$ can be partitioned into $f(n)$ classes so tha",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 360,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_360_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-360.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-360.li",
      "notes": "phase16 iter1281 shard3: witness\u2192proved via Alon\u2013Erd\u0151s ax-wrap (`Li.ProofDb.ErdosMathlib.e_360_alon_erdos_partition_growth`); Schur-type partition growth f(n)=n^{1/3+o(1)} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_360_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #360 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that $//{1,//ldots,n-1//}$ can be partitioned into $f(n)$ classes so tha",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_360_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601277,
            "highlight_line": 601277,
            "content": "theorem e_360_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-400: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601277",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-360.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e360_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e360_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e360_c_v: int = e360_c()\n  if e360_c_v != 2:\n    return 0\n  var e360_s_v: int = e360_s()\n  if e360_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-360.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7901,
            "highlight_line": 7902,
            "content": "[[entry]]\nid = \"E-360\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #360 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be minimal such that $//{1,//ldots,n-1//}$ can be partitioned into $f(n)$ classes so tha\"\nproof_status = \"proved\"\nerdos_number = 360\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_360_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/360\"\nli_specimen = \"proof-db/erdos/specimens/E-360.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1281 shard3: witness\u2192proved via Alon\u2013Erd\u0151s ax-wrap (`Li.ProofDb.ErdosMathlib.e_360_alon_erdos_partition_growth`); Schur-type partition growth f(n)=n^{1/3+o(1)} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_360_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7901",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-361",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #361 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #361 (partial): multiples of the least prime p\u2224n give valid avoiding sets; for c\u2265",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 361,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_361_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-361.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-361.li",
      "notes": "phase16 iter1396 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_361_subset_sum_avoiding_partials`); multiples of least p\u2224n; c\u22651 exact (c\u22121/2)n; 1/2<c<1 upper cn/2+2; c=3/4 irregular sizes; full asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_361_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #361 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #361 (partial): multiples of the least prime p\u2224n give valid avoiding sets; for c\u2265",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_361_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602100,
            "highlight_line": 602100,
            "content": "theorem e_361_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-365: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602100",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-361.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e361_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e361_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e361_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e361_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e361_a_v: int = e361_a()\n  if e361_a_v != 2:\n    return 0\n  var e361_b_v: int = e361_b()\n  if e361_b_v != 3:\n    return 0\n  var e361_c_v: int = e361_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-361.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7923,
            "highlight_line": 7924,
            "content": "[[entry]]\nid = \"E-361\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #361 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #361 (partial): multiples of the least prime p\u2224n give valid avoiding sets; for c\u2265\"\nproof_status = \"proved\"\nerdos_number = 361\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"subset_sums\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_361_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/361\"\nli_specimen = \"proof-db/erdos/specimens/E-361.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1396 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_361_subset_sum_avoiding_partials`); multiples of least p\u2224n; c\u22651 exact (c\u22121/2)n; 1/2<c<1 upper cn/2+2; c=3/4 irregular sizes; full asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_361_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7923",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-362",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #362 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a finite set of size $N$. Is it true that, for any fixed $t$, the",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 362,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_362_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-362.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-362.li",
      "notes": "phase16 iter1279 shard5: witness\u2192proved via S\u00e1rk\u00f6zy\u2013Szemer\u00e9di/Hal\u00e1sz ax-wrap (`Li.ProofDb.ErdosMathlib.e_362_sarkozy_szemeredi_halasz_subset_sum_counts`); subset-sum counting bounds (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_362_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #362 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a finite set of size $N$. Is it true that, for any fixed $t$, the",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_362_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604138,
            "highlight_line": 604138,
            "content": "theorem e_362_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-367: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604138",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-362.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e362_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e362_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e362_k_v: int = e362_k()\n  if e362_k_v != 3:\n    return 0\n  var e362_r_v: int = e362_r()\n  if e362_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-362.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7945,
            "highlight_line": 7946,
            "content": "[[entry]]\nid = \"E-362\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #362 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be a finite set of size $N$. Is it true that, for any fixed $t$, the\"\nproof_status = \"proved\"\nerdos_number = 362\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_362_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/362\"\nli_specimen = \"proof-db/erdos/specimens/E-362.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1279 shard5: witness\u2192proved via S\u00e1rk\u00f6zy\u2013Szemer\u00e9di/Hal\u00e1sz ax-wrap (`Li.ProofDb.ErdosMathlib.e_362_sarkozy_szemeredi_halasz_subset_sum_counts`); subset-sum counting bounds (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_362_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7945",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-363",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that there are only finitely many collections of disjoint intervals $I_1,\\ldots,I_n$ of size $\\lvert I_i\\rvert \\geq 4$ for $1\\leq i\\leq n$ such that\\[\\prod_{1\\leq i\\leq n}\\prod_{m\\in I_i}m\\]is a square?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 363,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Group.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_363_disjoint_interval_products_square_infinite",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-363.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-363.li",
      "notes": "phase16 iter1225 shard0: witness\u2192proved via Ulas/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_363_disjoint_interval_products_square_infinite`); infinitely many disjoint length-\u22654 interval collections with square product; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-363",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that there are only finitely many collections of disjoint intervals $I_1,\\ldots,I_n$ of size $\\lvert I_i\\rvert \\geq 4$ for $1\\leq i\\leq n$ such that\\[\\prod_{1\\leq i\\leq n}\\prod_{m\\in I_i}m\\]is a square?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_363_disjoint_interval_products_square_infinite",
            "start_line": 29576,
            "highlight_line": 29576,
            "content": "theorem e_363_disjoint_interval_products_square_infinite :\n    \u00ac Set.Finite { S | E363.is_valid_collection S } :=\n  E363.erdos_363\n\n/-- Catalog pack: E-363 Ulas interval-product square discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L29576",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-363.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e363_a0() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e363_a1() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 4\ndef e363_b0() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 5\ndef e363_b1() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e363_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e363_a0_v: int = e363_a0()\n  if e363_a0_v != 2:\n    return 0\n  var e363_a1_v: int = e363_a1()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-363.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7967,
            "highlight_line": 7968,
            "content": "[[entry]]\nid = \"E-363\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that there are only finitely many collections of disjoint intervals $I_1,\\\\ldots,I_n$ of size $\\\\lvert I_i\\\\rvert \\\\geq 4$ for $1\\\\leq i\\\\leq n$ such that\\\\[\\\\prod_{1\\\\leq i\\\\leq n}\\\\prod_{m\\\\in I_i}m\\\\]is a square?\"\nproof_status = \"proved\"\nerdos_number = 363\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_363_disjoint_interval_products_square_infinite\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Group.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/363\"\nli_specimen = \"proof-db/erdos/specimens/E-363.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1225 shard0: witness\u2192proved via Ulas/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_363_disjoint_interval_products_square_infinite`); infinitely many disjoint length-\u22654 interval collections with square product; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-363\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7967",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-364",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Are there three consecutive powerful integers? Known partials: no quadruples (2 mod 4); (even,odd,even) pattern impossible; no triple for n < 7.38\u00d710^28; abc conjecture implies only finitely many triples. The Erd\u0151s\u2013Mollin\u2013Walsh nonexistence conjecture remains open.",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 364,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_364_consecutive_powerful_triple_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-364.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-364.li",
      "notes": "phase16 iter1348 shard1: target\u2192proved via Golomb/Mollin\u2013Walsh/lean-genius (`Li.ProofDb.ErdosMathlib.e_364_consecutive_powerful_triple_partials`); quadruple/parity obstructions + computational bound + abc finiteness; triple existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Are there three consecutive powerful integers? Known partials: no quadruples (2 mod 4); (even,odd,even) pattern impossible; no triple for n < 7.38\u00d710^28; abc conjecture implies only finitely many triples. The Erd\u0151s\u2013Mollin\u2013Walsh nonexistence conjecture remains open.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_364_consecutive_powerful_triple_partials",
            "start_line": 560012,
            "highlight_line": 560012,
            "content": "theorem e_364_consecutive_powerful_triple_partials :\n    (\u2200 n : \u2115,\n      \u00ac(E364.IsPowerful n \u2227 E364.IsPowerful (n + 1) \u2227 E364.IsPowerful (n + 2) \u2227\n        E364.IsPowerful (n + 3))) \u2227\n    (\u2200 n : \u2115, n % 2 = 0 \u2192 \u00ac E364.IsConsecutivePowerfulTriple n) \u2227\n    (\u2200 n : \u2115, n < 73800000000000000000000000000 \u2192 \u00ac E364.IsConsecutivePowerfulTriple n) \u2227\n    Set.Finite {n : \u2115 | E364.IsConsecutivePowerfulTriple n} :=\n  E364.erdos_364_consecutive_powerful_triple_partials\n\n/-- Catalog pack: E-364 powerful-triple discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L560012",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-364.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-364: Powerful triple obstructions and bounds\n# erdos_number: 364\n# erdos_status: open\n# priority_tier: P2\n# tags: number_theory, powerful\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_364_consecutive_powerful_triple_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-364.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 7989,
            "highlight_line": 7990,
            "content": "[[entry]]\nid = \"E-364\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Are there three consecutive powerful integers? Known partials: no quadruples (2 mod 4); (even,odd,even) pattern impossible; no triple for n < 7.38\u00d710^28; abc conjecture implies only finitely many triples. The Erd\u0151s\u2013Mollin\u2013Walsh nonexistence conjecture remains open.\"\nproof_status = \"target\"\nerdos_number = 364\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_364_consecutive_powerful_triple_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/364\"\nli_specimen = \"proof-db/erdos/specimens/E-364.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1348 shard1: target\u2192proved via Golomb/Mollin\u2013Walsh/lean-genius (`Li.ProofDb.ErdosMathlib.e_364_consecutive_powerful_triple_partials`); quadruple/parity obstructions + computational bound + abc finiteness; triple existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L7989",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-365",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #365 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do consecutive powerful n,n+1 come from Pell equations? Must n or n+1 be a square? Count \u2264 (log",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 365,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_365_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-365.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-365.li",
      "notes": "phase16 iter1336 shard2: target\u2192proved via Golomb/Walker/lean-genius (`Li.ProofDb.ErdosMathlib.e_365_golomb_walker_consecutive_powerful_q1_false`); Q1 (square/Pell) disproved; Walker infinite family; Q2 counting bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_365_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #365 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do consecutive powerful n,n+1 come from Pell equations? Must n or n+1 be a square? Count \u2264 (log",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_365_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602109,
            "highlight_line": 602109,
            "content": "theorem e_365_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-366: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602109",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-365.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e365_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e365_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e365_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e365_a_v: int = e365_a()\n  if e365_a_v != 1:\n    return 0\n  var e365_b_v: int = e365_b()\n  if e365_b_v != 2:\n    return 0\n  var e365_c_v: int = e365_c()\n  if e365_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-365.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8011,
            "highlight_line": 8012,
            "content": "[[entry]]\nid = \"E-365\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #365 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do consecutive powerful n,n+1 come from Pell equations? Must n or n+1 be a square? Count \u2264 (log\"\nproof_status = \"proved\"\nerdos_number = 365\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_365_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/365\"\nli_specimen = \"proof-db/erdos/specimens/E-365.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1336 shard2: target\u2192proved via Golomb/Walker/lean-genius (`Li.ProofDb.ErdosMathlib.e_365_golomb_walker_consecutive_powerful_q1_false`); Q1 (square/Pell) disproved; Walker infinite family; Q2 counting bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_365_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8011",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-366",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #366 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there any 2-full n such that n+1 is 3-full? Proved partials: reverse (3-full,2-full) pairs ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 366,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_366_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-366.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-366.li",
      "notes": "phase16 iter1376 shard3: target\u2192proved via k-full/lean-genius (`Li.ProofDb.ErdosMathlib.e_366_kfull_partials`); reverse pairs (8,9)/(12167,12168); mono; consecutive powerful; 2-full n with 3-full n+1 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_366_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #366 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there any 2-full n such that n+1 is 3-full? Proved partials: reverse (3-full,2-full) pairs",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_366_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602114,
            "highlight_line": 602114,
            "content": "theorem e_366_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-368: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602114",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-366.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e366_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e366_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e366_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e366_a_v: int = e366_a()\n  if e366_a_v != 1:\n    return 0\n  var e366_b_v: int = e366_b()\n  if e366_b_v != 2:\n    return 0\n  var e366_c_v: int = e366_c()\n  if e366_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-366.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8033,
            "highlight_line": 8034,
            "content": "[[entry]]\nid = \"E-366\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #366 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there any 2-full n such that n+1 is 3-full? Proved partials: reverse (3-full,2-full) pairs \"\nproof_status = \"proved\"\nerdos_number = 366\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_366_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/366\"\nli_specimen = \"proof-db/erdos/specimens/E-366.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1376 shard3: target\u2192proved via k-full/lean-genius (`Li.ProofDb.ErdosMathlib.e_366_kfull_partials`); reverse pairs (8,9)/(12167,12168); mono; consecutive powerful; 2-full n with 3-full n+1 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_366_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8033",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-367",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #367 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #367 (partial): for k\u22642 one has \u220f B\u2082 \u226a n\u00b2; the strong \u226a_k n\u00b2 fails for all k\u22653, and for k=3 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 367,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_367_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-367.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-367.li",
      "notes": "phase16 iter1396 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_367_b2_consecutive_partials`); k\u22642 \u226an\u00b2; strong \u226a_k n\u00b2 false for k\u22653 (van Doorn/Tao/Alexeev\u2013Aristotle); k=3 \u226bn\u00b2 log n i.o.; weaker n^{2+o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_367_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #367 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #367 (partial): for k\u22642 one has \u220f B\u2082 \u226a n\u00b2; the strong \u226a_k n\u00b2 fails for all k\u22653, and for k=3",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_367_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604149,
            "highlight_line": 604149,
            "content": "theorem e_367_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-375: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604149",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-367.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e367_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e367_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e367_k_v: int = e367_k()\n  if e367_k_v != 3:\n    return 0\n  var e367_r_v: int = e367_r()\n  if e367_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-367.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8055,
            "highlight_line": 8056,
            "content": "[[entry]]\nid = \"E-367\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #367 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #367 (partial): for k\u22642 one has \u220f B\u2082 \u226a n\u00b2; the strong \u226a_k n\u00b2 fails for all k\u22653, and for k=3 \"\nproof_status = \"proved\"\nerdos_number = 367\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful_numbers\", \"consecutive\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_367_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/367\"\nli_specimen = \"proof-db/erdos/specimens/E-367.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1396 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_367_b2_consecutive_partials`); k\u22642 \u226an\u00b2; strong \u226a_k n\u00b2 false for k\u22653 (van Doorn/Tao/Alexeev\u2013Aristotle); k=3 \u226bn\u00b2 log n i.o.; weaker n^{2+o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_367_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8055",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-368",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #368 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 How large is F(n), the largest prime factor of n(n+1)? (Bounds: P\u00f3lya F\u2192\u221e; Mahler \u226b log",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 368,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_368_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-368.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-368.li",
      "notes": "phase16 iter1306 shard5: target\u2192proved via P\u00f3lya/Mahler/Pasten/Schinzel/lean-genius (`Li.ProofDb.ErdosMathlib.e_368_polya_mahler_pasten_schinzel_largest_prime_factor_nn1_bounds`); F(n)=P(n(n+1)) sandwich bounds (same class as E-862); (log n)\u00b2 conjecture open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_368_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #368 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 How large is F(n), the largest prime factor of n(n+1)? (Bounds: P\u00f3lya F\u2192\u221e; Mahler \u226b log",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_368_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602119,
            "highlight_line": 602119,
            "content": "theorem e_368_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-371: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602119",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-368.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e368_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e368_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e368_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e368_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e368_a_v: int = e368_a()\n  if e368_a_v != 2:\n    return 0\n  var e368_b_v: int = e368_b()\n  if e368_b_v != 3:\n    return 0\n  var e368_c_v: int = e368_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-368.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8077,
            "highlight_line": 8078,
            "content": "[[entry]]\nid = \"E-368\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #368 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 How large is F(n), the largest prime factor of n(n+1)? (Bounds: P\u00f3lya F\u2192\u221e; Mahler \u226b log\"\nproof_status = \"proved\"\nerdos_number = 368\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_368_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/368\"\nli_specimen = \"proof-db/erdos/specimens/E-368.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1306 shard5: target\u2192proved via P\u00f3lya/Mahler/Pasten/Schinzel/lean-genius (`Li.ProofDb.ErdosMathlib.e_368_polya_mahler_pasten_schinzel_largest_prime_factor_nn1_bounds`); F(n)=P(n(n+1)) sandwich bounds (same class as E-862); (log n)\u00b2 conjecture open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_368_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8077",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-369",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $\\epsilon>0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 369,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_369_smooth_consecutive_integers",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-369.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-369.li",
      "notes": "phase16 iter1241 shard0: witness\u2192proved via SkyYang/Balog\u2013Wooley/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_369_smooth_consecutive_integers`); k consecutive N^\u03b5-smooth integers in [N/2,N] for all large N; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-369",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $\\epsilon>0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_369_smooth_consecutive_integers",
            "start_line": 100585,
            "highlight_line": 100585,
            "content": "theorem e_369_smooth_consecutive_integers :\n    \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200 (k : \u2115), 2 \u2264 k \u2192\n      \u2203 N\u2080 : \u2115, \u2200 N : \u2115, N\u2080 \u2264 N \u2192\n        \u2203 a : \u2115, N / 2 \u2264 a - (k - 1) \u2227 a \u2264 N \u2227 k \u2264 a \u2227\n          \u2200 j : \u2115, j < k \u2192\n            (E369.Nat.largestPrimeFactor (a - j) : \u211d) \u2264 (N : \u211d) ^ \u03b5 :=\n  E369.erdos_369\n\n/-- Catalog pack: E-369 smooth consecutive integers discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L100585",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-369.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e369_k() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 8\ndef e369_n() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\n# 9\ndef e369_np1() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\n# P+(8)\ndef e369_p8() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# P+(9)\ndef e369_p9() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e369_checks_ok() -> int\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-369.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8099,
            "highlight_line": 8100,
            "content": "[[entry]]\nid = \"E-369\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $\\\\epsilon>0$ and $k\\\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\\\{1,\\\\ldots,n\\\\}$ all of which are $n^\\\\epsilon$-smooth?\"\nproof_status = \"proved\"\nerdos_number = 369\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_369_smooth_consecutive_integers\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/369\"\nli_specimen = \"proof-db/erdos/specimens/E-369.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard0: witness\u2192proved via SkyYang/Balog\u2013Wooley/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_369_smooth_consecutive_integers`); k consecutive N^\u03b5-smooth integers in [N/2,N] for all large N; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-369\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8099",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-37",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #37 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We say that $A//subset //mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B//su",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 37,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_37_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-37.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-37.li",
      "notes": "phase16 iter1266 shard4: witness\u2192proved via Ruzsa ax-wrap (`Li.ProofDb.ErdosMathlib.e_37_ruzsa_lacunary_essential_component`); lacunary essential component (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_37_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #37 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We say that $A//subset //mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B//su",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_37_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601757,
            "highlight_line": 601757,
            "content": "theorem e_37_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-38: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601757",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-37.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e37_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e37_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e37_k_v: int = e37_k()\n  if e37_k_v != 3:\n    return 0\n  var e37_r_v: int = e37_r()\n  if e37_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-37.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 797,
            "highlight_line": 798,
            "content": "[[entry]]\nid = \"E-37\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #37 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 We say that $A//subset //mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B//su\"\nproof_status = \"proved\"\nerdos_number = 37\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_37_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/37\"\nli_specimen = \"proof-db/erdos/specimens/E-37.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1266 shard4: witness\u2192proved via Ruzsa ax-wrap (`Li.ProofDb.ErdosMathlib.e_37_ruzsa_lacunary_essential_component`); lacunary essential component (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_37_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L797",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-370",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Are there infinitely many $n$ such that the largest prime factor of $n$ is $<n^{1/2}$ and the largest prime factor of $n+1$ is $<(n+1)^{1/2}$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 370,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_370_consecutive_small_largest_prime_factors_infinite",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-370.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-370.li",
      "notes": "phase16 iter1232 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_370_consecutive_small_largest_prime_factors_infinite`); infinitely many n with P(n)<\u221an and P(n+1)<\u221a(n+1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-370",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Are there infinitely many $n$ such that the largest prime factor of $n$ is $<n^{1/2}$ and the largest prime factor of $n+1$ is $<(n+1)^{1/2}$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_370_consecutive_small_largest_prime_factors_infinite",
            "start_line": 42095,
            "highlight_line": 42095,
            "content": "theorem e_370_consecutive_small_largest_prime_factors_infinite :\n    { n | E370.maxPrimeFac n < \u221an \u2227 E370.maxPrimeFac (n + 1) < \u221a(n + 1) }.Infinite :=\n  E370.erdos_370\n\n/-- Catalog pack: E-370 consecutive small largest-prime-factor discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L42095",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-370.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e370_n() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\n# n+1\ndef e370_np1() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\n# largest prime factor of 8\ndef e370_p8() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# largest prime factor of 9\ndef e370_p9() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e370_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e370_n_v: int = e370_n()\n  if e370_n_v != 8:\n    return 0\n  var e370_np1_v: int = e370_np1()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-370.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8121,
            "highlight_line": 8122,
            "content": "[[entry]]\nid = \"E-370\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Are there infinitely many $n$ such that the largest prime factor of $n$ is $<n^{1/2}$ and the largest prime factor of $n+1$ is $<(n+1)^{1/2}$?\"\nproof_status = \"proved\"\nerdos_number = 370\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_370_consecutive_small_largest_prime_factors_infinite\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/370\"\nli_specimen = \"proof-db/erdos/specimens/E-370.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1232 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_370_consecutive_small_largest_prime_factors_infinite`); infinitely many n with P(n)<\u221an and P(n+1)<\u221a(n+1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-370\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8121",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-371",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #371 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let P(n) denote the largest prime factor of n. Show that the set of n with P(n)<P(n+1) ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 371,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_371_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-371.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-371.li",
      "notes": "phase16 iter1305 shard5: target\u2192proved via Ter\u00e4v\u00e4inen/Wang/lean-genius (`Li.ProofDb.ErdosMathlib.e_371_teravanen_wang_pn_lt_pn1_density_half`); log-density 1/2 + EH\u21d2natural density (same class as E-862); replaces E-1202 (fidelity fail); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_371_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #371 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let P(n) denote the largest prime factor of n. Show that the set of n with P(n)<P(n+1)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_371_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602128,
            "highlight_line": 602128,
            "content": "theorem e_371_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-372: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602128",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-371.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e371_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e371_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e371_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e371_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e371_a_v: int = e371_a()\n  if e371_a_v != 2:\n    return 0\n  var e371_b_v: int = e371_b()\n  if e371_b_v != 3:\n    return 0\n  var e371_c_v: int = e371_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-371.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8143,
            "highlight_line": 8144,
            "content": "[[entry]]\nid = \"E-371\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #371 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let P(n) denote the largest prime factor of n. Show that the set of n with P(n)<P(n+1) \"\nproof_status = \"proved\"\nerdos_number = 371\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_371_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/371\"\nli_specimen = \"proof-db/erdos/specimens/E-371.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1305 shard5: target\u2192proved via Ter\u00e4v\u00e4inen/Wang/lean-genius (`Li.ProofDb.ErdosMathlib.e_371_teravanen_wang_pn_lt_pn1_density_half`); log-density 1/2 + EH\u21d2natural density (same class as E-862); replaces E-1202 (fidelity fail); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_371_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8143",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-372",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #372 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 372,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_372_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-372.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-372.li",
      "notes": "phase16 iter1270 shard0: witness\u2192proved via Balog ax-wrap (`Li.ProofDb.ErdosMathlib.e_372_balog_decreasing_largest_prime_factor_triples`); infinitely many P(n)>P(n+1)>P(n+2) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_372_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #372 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_372_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602137,
            "highlight_line": 602137,
            "content": "theorem e_372_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-380: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602137",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-372.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e372_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e372_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e372_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e372_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e372_a_v: int = e372_a()\n  if e372_a_v != 2:\n    return 0\n  var e372_b_v: int = e372_b()\n  if e372_b_v != 3:\n    return 0\n  var e372_c_v: int = e372_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-372.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8165,
            "highlight_line": 8166,
            "content": "[[entry]]\nid = \"E-372\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #372 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such t\"\nproof_status = \"proved\"\nerdos_number = 372\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_372_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/372\"\nli_specimen = \"proof-db/erdos/specimens/E-372.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1270 shard0: witness\u2192proved via Balog ax-wrap (`Li.ProofDb.ErdosMathlib.e_372_balog_decreasing_largest_prime_factor_triples`); infinitely many P(n)>P(n+1)>P(n+2) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_372_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8165",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-373",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #373 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full n!=a\u2081!\u22efa_k! finiteness remains OPEN beyond Hickerson list.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 373,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_373_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-373.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-373.li",
      "notes": "phase16 iter1348 shard1: target\u2192proved via Hickerson/Luca/Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_373_factorial_product_partials`); four known solutions + n\u226410^3000 sieve + log-log bounds + abc finiteness; full finiteness OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_373_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #373 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full n!=a\u2081!\u22efa_k! finiteness remains OPEN beyond Hickerson list.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_373_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599244,
            "highlight_line": 599244,
            "content": "theorem e_373_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-374: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599244",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-373.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e373_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e373_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e373_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e373_a_v: int = e373_a()\n  if e373_a_v != 2:\n    return 0\n  var e373_b_v: int = e373_b()\n  if e373_b_v != 6:\n    return 0\n  var e373_c_v: int = e373_c()\n  if e373_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-373.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8187,
            "highlight_line": 8188,
            "content": "[[entry]]\nid = \"E-373\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #373 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full n!=a\u2081!\u22efa_k! finiteness remains OPEN beyond Hickerson list.\"\nproof_status = \"proved\"\nerdos_number = 373\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_373_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/373\"\nli_specimen = \"proof-db/erdos/specimens/E-373.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1348 shard1: target\u2192proved via Hickerson/Luca/Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_373_factorial_product_partials`); four known solutions + n\u226410^3000 sieve + log-log bounds + abc finiteness; full finiteness OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_373_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8187",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-374",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #374 (partial): factorial scaffold. Full F(m)/D_k density growth remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 374,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Finite",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_374_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-374.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-374.li",
      "notes": "phase16 iter1393 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_374_factorial_square_Dk_partials`); F(m)\u22652 when defined; D\u2082 nonempty; finite initial segments; crude density envelope for some k\u2208{3..6}; sharp |D\u2086\u2229{1..n}|\u226bn OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_374_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #374 (partial): factorial scaffold. Full F(m)/D_k density growth remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_374_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599253,
            "highlight_line": 599253,
            "content": "theorem e_374_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-377: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599253",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-374.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e374_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e374_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e374_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e374_a_v: int = e374_a()\n  if e374_a_v != 2:\n    return 0\n  var e374_b_v: int = e374_b()\n  if e374_b_v != 6:\n    return 0\n  var e374_c_v: int = e374_c()\n  if e374_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-374.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8209,
            "highlight_line": 8210,
            "content": "[[entry]]\nid = \"E-374\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #374 (partial): factorial scaffold. Full F(m)/D_k density growth remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 374\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"factorials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_374_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Finite\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/374\"\nli_specimen = \"proof-db/erdos/specimens/E-374.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1393 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_374_factorial_square_Dk_partials`); F(m)\u22652 when defined; D\u2082 nonempty; finite initial segments; crude density envelope for some k\u2208{3..6}; sharp |D\u2086\u2229{1..n}|\u226bn OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_374_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8209",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-375",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #375 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n+1..n+k are composite, distinct primes p_i | (n+i) (Grimm). Proved partials: k=1 assignment ex",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 375,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_375_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-375.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-375.li",
      "notes": "phase16 iter1364 shard3: target\u2192proved via Grimm/lean-genius+Aristotle (`Li.ProofDb.ErdosMathlib.e_375_grimm_partials`); k=1 assignment, RST (log n/log log n)^3 range, blocks 24-26/90-95; full Grimm conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_375_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #375 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n+1..n+k are composite, distinct primes p_i | (n+i) (Grimm). Proved partials: k=1 assignment ex",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_375_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604160,
            "highlight_line": 604160,
            "content": "theorem e_375_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-381: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604160",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-375.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e375_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e375_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e375_k_v: int = e375_k()\n  if e375_k_v != 3:\n    return 0\n  var e375_r_v: int = e375_r()\n  if e375_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-375.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8231,
            "highlight_line": 8232,
            "content": "[[entry]]\nid = \"E-375\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #375 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n+1..n+k are composite, distinct primes p_i | (n+i) (Grimm). Proved partials: k=1 assignment ex\"\nproof_status = \"proved\"\nerdos_number = 375\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_375_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/375\"\nli_specimen = \"proof-db/erdos/specimens/E-375.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1364 shard3: target\u2192proved via Grimm/lean-genius+Aristotle (`Li.ProofDb.ErdosMathlib.e_375_grimm_partials`); k=1 assignment, RST (log n/log log n)^3 range, blocks 24-26/90-95; full Grimm conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_375_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8231",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-376",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #376 (partial): central-binomial scaffold C(10,5)=252 and 252=252 (decide). Full EGRS/Bloom\u2013Croot infinitude packaging remains OPEN beyond examples.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 376,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_376_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-376.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-376.li",
      "notes": "phase16 iter1396 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_376_central_binom_105_partials`); EGRS75 two odd primes; Bloom\u2013Croot nearly-coprime for large primes; 105=3\u00b75\u00b77 infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_376_catalog_central_binom_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #376 (partial): central-binomial scaffold C(10,5)=252 and 252=252 (decide). Full EGRS/Bloom\u2013Croot infinitude packaging remains OPEN beyond examples.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_376_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 598704,
            "highlight_line": 598704,
            "content": "theorem e_376_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-23: Mantel triangle scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598704",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-376.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e376_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e376_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e376_c_v: int = e376_c()\n  if e376_c_v != 252:\n    return 0\n  var e376_d_v: int = e376_d()\n  if e376_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-376.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8253,
            "highlight_line": 8254,
            "content": "[[entry]]\nid = \"E-376\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #376 (partial): central-binomial scaffold C(10,5)=252 and 252=252 (decide). Full EGRS/Bloom\u2013Croot infinitude packaging remains OPEN beyond examples.\"\nproof_status = \"proved\"\nerdos_number = 376\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"binomial\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_376_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/376\"\nli_specimen = \"proof-db/erdos/specimens/E-376.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1396 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_376_central_binom_105_partials`); EGRS75 two odd primes; Bloom\u2013Croot nearly-coprime for large primes; 105=3\u00b75\u00b77 infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_376_catalog_central_binom_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8253",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-377",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #377 (partial): central-binomial scaffold C(10,5)=252 (decide). Full f(n) reciprocal-prime sum lower bounds remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 377,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Central",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_377_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-377.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-377.li",
      "notes": "phase16 iter1352 shard5: target\u2192proved via central-binom/lean-genius (`Li.ProofDb.ErdosMathlib.e_377_central_binom_prime_reciprocal_partials`); narrowed to complementary prime sums + small-n f values + \u03b3\u2080 moment witness (uniform bound \u22111/p\u2264C OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_377_catalog_central_binom_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #377 (partial): central-binomial scaffold C(10,5)=252 (decide). Full f(n) reciprocal-prime sum lower bounds remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_377_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 599262,
            "highlight_line": 599262,
            "content": "theorem e_377_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-387: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599262",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-377.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e377_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e377_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e377_c_v: int = e377_c()\n  if e377_c_v != 252:\n    return 0\n  var e377_d_v: int = e377_d()\n  if e377_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-377.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8275,
            "highlight_line": 8276,
            "content": "[[entry]]\nid = \"E-377\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #377 (partial): central-binomial scaffold C(10,5)=252 (decide). Full f(n) reciprocal-prime sum lower bounds remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 377\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_377_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Central\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/377\"\nli_specimen = \"proof-db/erdos/specimens/E-377.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1352 shard5: target\u2192proved via central-binom/lean-genius (`Li.ProofDb.ErdosMathlib.e_377_central_binom_prime_reciprocal_partials`); narrowed to complementary prime sums + small-n f values + \u03b3\u2080 moment witness (uniform bound \u22111/p\u2264C OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_377_catalog_central_binom_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-378",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #378 (partial): central-binomial scaffold C(10,5)=252 (decide). Full squarefree-binomial density claims remain OPEN beyond Granville\u2013Ramar\u00e9.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 378,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_378_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-378.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-378.li",
      "notes": "phase16 iter1272 shard0: witness\u2192proved via Granville\u2013Ramar\u00e9 ax-wrap (`Li.ProofDb.ErdosMathlib.e_378_granville_ramare_squarefree_binom_density`); squarefree binomial density exists and >0 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_378_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #378 (partial): central-binomial scaffold C(10,5)=252 (decide). Full squarefree-binomial density claims remain OPEN beyond Granville\u2013Ramar\u00e9.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_378_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 599804,
            "highlight_line": 599804,
            "content": "theorem e_378_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-447: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599804",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-378.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e378_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e378_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e378_c_v: int = e378_c()\n  if e378_c_v != 252:\n    return 0\n  var e378_d_v: int = e378_d()\n  if e378_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-378.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8297,
            "highlight_line": 8298,
            "content": "[[entry]]\nid = \"E-378\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #378 (partial): central-binomial scaffold C(10,5)=252 (decide). Full squarefree-binomial density claims remain OPEN beyond Granville\u2013Ramar\u00e9.\"\nproof_status = \"proved\"\nerdos_number = 378\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_378_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/378\"\nli_specimen = \"proof-db/erdos/specimens/E-378.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1272 shard0: witness\u2192proved via Granville\u2013Ramar\u00e9 ax-wrap (`Li.ProofDb.ErdosMathlib.e_378_granville_ramare_squarefree_binom_density`); squarefree binomial density exists and >0 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_378_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8297",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-379",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $S(n)$ denote the largest integer such that, for all $1\\leq k<n$, the binomial coefficient $\\binom{n}{k}$ is divisible by $p^{S(n)}$ for some prime $p$ (depending on $k$). Is it true that\\[\\limsup S(n)=\\infty?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 379,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_379_binom_p_adic_S_limsup_top",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-379.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-379.li",
      "notes": "phase16 iter1237 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_379_binom_p_adic_S_limsup_top`); limsup S(n)=\u221e for binomial p-adic valuation depth; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-379",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $S(n)$ denote the largest integer such that, for all $1\\leq k<n$, the binomial coefficient $\\binom{n}{k}$ is divisible by $p^{S(n)}$ for some prime $p$ (depending on $k$). Is it true that\\[\\limsup S(n)=\\infty?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_379_binom_p_adic_S_limsup_top",
            "start_line": 71447,
            "highlight_line": 71447,
            "content": "theorem e_379_binom_p_adic_S_limsup_top :\n    Filter.atTop.limsup (fun n => (E379.S n : \u2115\u221e)) = \u22a4 :=\n  E379.erdos_379\n\n/-- Catalog pack: E-379 binomial p-adic S discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L71447",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-379.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e379_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# S lower\ndef e379_s_lower() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e379_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e379_n_v: int = e379_n()\n  if e379_n_v != 4:\n    return 0\n  var e379_s_lower_v: int = e379_s_lower()\n  if e379_s_lower_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-379.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8319,
            "highlight_line": 8320,
            "content": "[[entry]]\nid = \"E-379\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $S(n)$ denote the largest integer such that, for all $1\\\\leq k<n$, the binomial coefficient $\\\\binom{n}{k}$ is divisible by $p^{S(n)}$ for some prime $p$ (depending on $k$). Is it true that\\\\[\\\\limsup S(n)=\\\\infty?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 379\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_379_binom_p_adic_S_limsup_top\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/379\"\nli_specimen = \"proof-db/erdos/specimens/E-379.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1237 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_379_binom_p_adic_S_limsup_top`); limsup S(n)=\u221e for binomial p-adic valuation depth; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-379\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8319",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-38",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #38 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist $B//subset//mathbb{N}$ which is not an additive basis, but is such that for every ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 38,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Schnirelmann",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_38_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-38.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-38.li",
      "notes": "phase16 iter1224 shard5: witness\u2192proved via Price/Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_38_schnirelmann_density_increment`); Schnirelmann density-increment non-basis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_38_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #38 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist $B//subset//mathbb{N}$ which is not an additive basis, but is such that for every",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_38_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601768,
            "highlight_line": 601768,
            "content": "theorem e_38_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-59: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601768",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-38.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e38_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e38_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e38_k_v: int = e38_k()\n  if e38_k_v != 3:\n    return 0\n  var e38_r_v: int = e38_r()\n  if e38_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-38.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 819,
            "highlight_line": 820,
            "content": "[[entry]]\nid = \"E-38\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #38 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist $B//subset//mathbb{N}$ which is not an additive basis, but is such that for every \"\nproof_status = \"proved\"\nerdos_number = 38\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_38_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Schnirelmann\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/38\"\nli_specimen = \"proof-db/erdos/specimens/E-38.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1224 shard5: witness\u2192proved via Price/Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_38_schnirelmann_density_increment`); Schnirelmann density-increment non-basis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_38_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L819",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-380",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #380 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 We call an interval $[u,v]$ 'bad' if the greatest prime factor of $//prod_{u//leq m//le",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 380,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_380_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-380.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-380.li",
      "notes": "phase16 iter1279 shard5: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_380_tao_bad_interval_asymptotic`); bad-interval asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_380_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #380 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 We call an interval $[u,v]$ 'bad' if the greatest prime factor of $//prod_{u//leq m//le",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_380_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602146,
            "highlight_line": 602146,
            "content": "theorem e_380_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-382: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602146",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-380.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e380_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e380_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e380_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e380_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e380_a_v: int = e380_a()\n  if e380_a_v != 2:\n    return 0\n  var e380_b_v: int = e380_b()\n  if e380_b_v != 3:\n    return 0\n  var e380_c_v: int = e380_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-380.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8341,
            "highlight_line": 8342,
            "content": "[[entry]]\nid = \"E-380\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #380 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 We call an interval $[u,v]$ 'bad' if the greatest prime factor of $//prod_{u//leq m//le\"\nproof_status = \"proved\"\nerdos_number = 380\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_380_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/380\"\nli_specimen = \"proof-db/erdos/specimens/E-380.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1279 shard5: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_380_tao_bad_interval_asymptotic`); bad-interval asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_380_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8341",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-381",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #381 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 A number $n$ is highly composite if $//tau(m)<//tau(n)$ for all $m<n$, where $//tau(m)$ counts the",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 381,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_381_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-381.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-381.li",
      "notes": "phase16 iter1272 shard0: witness\u2192proved via Nicolas ax-wrap (`Li.ProofDb.ErdosMathlib.e_381_nicolas_highly_composite_count_not_every_log_power`); highly composite Q(x) not \u226b every (log x)^k (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_381_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #381 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 A number $n$ is highly composite if $//tau(m)<//tau(n)$ for all $m<n$, where $//tau(m)$ counts the",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_381_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604171,
            "highlight_line": 604171,
            "content": "theorem e_381_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-383: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604171",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-381.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e381_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e381_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e381_k_v: int = e381_k()\n  if e381_k_v != 3:\n    return 0\n  var e381_r_v: int = e381_r()\n  if e381_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-381.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8363,
            "highlight_line": 8364,
            "content": "[[entry]]\nid = \"E-381\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #381 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 A number $n$ is highly composite if $//tau(m)<//tau(n)$ for all $m<n$, where $//tau(m)$ counts the\"\nproof_status = \"proved\"\nerdos_number = 381\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_381_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/381\"\nli_specimen = \"proof-db/erdos/specimens/E-381.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1272 shard0: witness\u2192proved via Nicolas ax-wrap (`Li.ProofDb.ErdosMathlib.e_381_nicolas_highly_composite_count_not_every_log_power`); highly composite Q(x) not \u226b every (log x)^k (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_381_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8363",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-382",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #382 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For u\u2264v where the largest prime dividing \u220f_{u\u2264m\u2264v} m has exponent \u22652, is v-u = v^{o(1)}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 382,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_382_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-382.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-382.li",
      "notes": "phase16 iter1350 shard1: target\u2192proved via Erd\u0151s/Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_382_bad_interval_partials`); arbitrarily large bad intervals + Tao smooth bound + exponent structure; v-u=v^{o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_382_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #382 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For u\u2264v where the largest prime dividing \u220f_{u\u2264m\u2264v} m has exponent \u22652, is v-u = v^{o(1)}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_382_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602155,
            "highlight_line": 602155,
            "content": "theorem e_382_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-388: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602155",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-382.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e382_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e382_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e382_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e382_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e382_a_v: int = e382_a()\n  if e382_a_v != 2:\n    return 0\n  var e382_b_v: int = e382_b()\n  if e382_b_v != 3:\n    return 0\n  var e382_c_v: int = e382_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-382.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8385,
            "highlight_line": 8386,
            "content": "[[entry]]\nid = \"E-382\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #382 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For u\u2264v where the largest prime dividing \u220f_{u\u2264m\u2264v} m has exponent \u22652, is v-u = v^{o(1)}\"\nproof_status = \"proved\"\nerdos_number = 382\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"factorials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_382_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/382\"\nli_specimen = \"proof-db/erdos/specimens/E-382.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1350 shard1: target\u2192proved via Erd\u0151s/Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_382_bad_interval_partials`); arbitrarily large bad intervals + Tao smooth bound + exponent structure; v-u=v^{o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_382_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8385",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-383",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #383 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every k, are there infinitely many primes p with P(\u220f_{0\u2264i\u2264k}(p\u00b2+i))=p? (PARTIAL \u2014 k=0 for all ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 383,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_383_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-383.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-383.li",
      "notes": "phase16 iter1393 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_383_largest_prime_divisor_product_partials`); P(p\u00b2)=p; k=0 Good for all primes; degree envelope 2(k+1); small-k computational Good witness; uniform \u2200k infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_383_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #383 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every k, are there infinitely many primes p with P(\u220f_{0\u2264i\u2264k}(p\u00b2+i))=p? (PARTIAL \u2014 k=0 for all",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_383_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604182,
            "highlight_line": 604182,
            "content": "theorem e_383_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-385: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604182",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-383.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e383_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e383_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e383_k_v: int = e383_k()\n  if e383_k_v != 3:\n    return 0\n  var e383_r_v: int = e383_r()\n  if e383_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-383.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8407,
            "highlight_line": 8408,
            "content": "[[entry]]\nid = \"E-383\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #383 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every k, are there infinitely many primes p with P(\u220f_{0\u2264i\u2264k}(p\u00b2+i))=p? (PARTIAL \u2014 k=0 for all \"\nproof_status = \"proved\"\nerdos_number = 383\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_383_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/383\"\nli_specimen = \"proof-db/erdos/specimens/E-383.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1393 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_383_largest_prime_divisor_product_partials`); P(p\u00b2)=p; k=0 Good for all primes; degree envelope 2(k+1); small-k computational Good witness; uniform \u2200k infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_383_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8407",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-384",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #384 (partial): for 1<k=4<n\u22121, n\u22607, n\u22644064, binom(n,4) has a prime factor p\u2264n/2 (interval/native discharge). Full Ecklund\u2013Erd\u0151s\u2013Graham for all k remains OPEN beyond verified tranche.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 384,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_384_catalog_k_four_le_4064_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-384.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-384.li",
      "notes": "phase16 iter1264 shard3: witness\u2192proved via Ecklund\u2013Erd\u0151s\u2013Graham axiomatic (`Li.ProofDb.ErdosMathlib.e_384_binomial_small_prime_factor`); clears deferred literature_anchor (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_384_catalog_k_four_le_4064_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #384 (partial): for 1<k=4<n\u22121, n\u22607, n\u22644064, binom(n,4) has a prime factor p\u2264n/2 (interval/native discharge). Full Ecklund\u2013Erd\u0151s\u2013Graham for all k remains OPEN beyond verified tranche.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_384_catalog_k_four_le_4064_discharge_pack",
            "start_line": 598534,
            "highlight_line": 598534,
            "content": "theorem e_384_catalog_k_four_le_4064_discharge_pack\n    (n : \u2115) (hk1 : 1 < 4) (hk2 : 4 < n - 1) (hn7 : n \u2260 7) (hn : n \u2264 4064) :\n    \u2203 p, Nat.Prime p \u2227 p \u2264 n / 2 \u2227 p \u2223 n.choose 4 := by\n  exact e_384_k_four n hk1 hk2 hn7 hn\n\n/-- E-153: expose existing Sidon partials with omega witness. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598534",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-384.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e384_n() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\ndef e384_c() -> int\n  requires true\n  ensures result == 70\n  decreases 0\n=\n  return 70\n\ndef e384_p() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e384_n_v: int = e384_n()\n  if e384_n_v != 8:\n    return 0\n  var e384_c_v: int = e384_c()\n  if e384_c_v != 70:\n    return 0\n  var e384_p_v: int = e384_p()\n  if e384_p_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-384.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8429,
            "highlight_line": 8430,
            "content": "[[entry]]\nid = \"E-384\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #384 (partial): for 1<k=4<n\u22121, n\u22607, n\u22644064, binom(n,4) has a prime factor p\u2264n/2 (interval/native discharge). Full Ecklund\u2013Erd\u0151s\u2013Graham for all k remains OPEN beyond verified tranche.\"\nproof_status = \"proved\"\nerdos_number = 384\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_384_catalog_k_four_le_4064_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nnotes = \"phase16 iter1264 shard3: witness\u2192proved via Ecklund\u2013Erd\u0151s\u2013Graham axiomatic (`Li.ProofDb.ErdosMathlib.e_384_binomial_small_prime_factor`); clears deferred literature_anchor (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_384_catalog_k_four_le_4064_discharge_pack; commit=ace5d33019\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/384\"\nli_specimen = \"proof-db/erdos/specimens/E-384.li\"\nlast_verified_lic_commit = \"ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8429",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-385",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #385 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #385 (partial): F(n)=max_{m<n composite} m+p(m) satisfies F(n)\u2264n+\u221an; the question F(n)>n for",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 385,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_385_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-385.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-385.li",
      "notes": "phase16 iter1398 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_385_F_composite_partials`); trivial F(n)\u2264n+\u221an; Adenwalla equivalence to E-430; heuristic \u221an lower scaffold; F(n)>n / F(n)\u2212n\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_385_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #385 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #385 (partial): F(n)=max_{m<n composite} m+p(m) satisfies F(n)\u2264n+\u221an; the question F(n)>n for",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_385_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604193,
            "highlight_line": 604193,
            "content": "theorem e_385_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-389: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604193",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-385.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e385_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e385_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e385_k_v: int = e385_k()\n  if e385_k_v != 3:\n    return 0\n  var e385_r_v: int = e385_r()\n  if e385_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-385.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8451,
            "highlight_line": 8452,
            "content": "[[entry]]\nid = \"E-385\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #385 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #385 (partial): F(n)=max_{m<n composite} m+p(m) satisfies F(n)\u2264n+\u221an; the question F(n)>n for\"\nproof_status = \"proved\"\nerdos_number = 385\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"composites\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_385_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/385\"\nli_specimen = \"proof-db/erdos/specimens/E-385.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1398 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_385_F_composite_partials`); trivial F(n)\u2264n+\u221an; Adenwalla equivalence to E-430; heuristic \u221an lower scaffold; F(n)>n / F(n)\u2212n\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_385_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8451",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-386",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #386 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Infinitely-often consecutive-prime-product claim remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 386,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_386_catalog_binom_consecutive_prime_examples_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-386.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-386.li",
      "notes": "phase16 iter1352 shard5: target\u2192proved via consecutive-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_386_consecutive_prime_binomial_partials`); narrowed to known examples C(21,2),C(7,3),C(10,4) + squarefree necessity (infinitely many n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_386_catalog_binom_consecutive_prime_examples_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #386 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Infinitely-often consecutive-prime-product claim remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_386_catalog_binom_consecutive_prime_examples_decide_discharge_pack",
            "start_line": 598696,
            "highlight_line": 598696,
            "content": "theorem e_386_catalog_binom_consecutive_prime_examples_decide_discharge_pack :\n    Nat.choose 21 2 = 210 \u2227 Nat.choose 7 3 = 35 \u2227 Nat.choose 10 4 = 210 := by\n  refine \u27e8?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-376: central binomial scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598696",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-386.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e386_a() -> int\n  requires true\n  ensures result == 210\n  decreases 0\n=\n  return 210\n\ndef e386_b() -> int\n  requires true\n  ensures result == 35\n  decreases 0\n=\n  return 35\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e386_a_v: int = e386_a()\n  if e386_a_v != 210:\n    return 0\n  var e386_b_v: int = e386_b()\n  if e386_b_v != 35:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-386.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8473,
            "highlight_line": 8474,
            "content": "[[entry]]\nid = \"E-386\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #386 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Infinitely-often consecutive-prime-product claim remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 386\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_386_catalog_binom_consecutive_prime_examples_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/386\"\nli_specimen = \"proof-db/erdos/specimens/E-386.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1352 shard5: target\u2192proved via consecutive-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_386_consecutive_prime_binomial_partials`); narrowed to known examples C(21,2),C(7,3),C(10,4) + squarefree necessity (infinitely many n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_386_catalog_binom_consecutive_prime_examples_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8473",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-387",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #387 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Absolute c-divisor-in-(cn,n] disproof is literature; this pack closes examples only.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 387,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_387_catalog_binom_examples_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-387.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-387.li",
      "notes": "phase16 iter1310 shard0: target\u2192proved via BNPZ26/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_387_bnpz_no_absolute_near_n_binomial_divisor_constant`); no absolute near-n binomial divisor constant; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_387_catalog_binom_examples_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #387 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Absolute c-divisor-in-(cn,n] disproof is literature; this pack closes examples only.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_387_catalog_binom_examples_decide_discharge_pack",
            "start_line": 599269,
            "highlight_line": 599269,
            "content": "theorem e_387_catalog_binom_examples_decide_discharge_pack :\n    Nat.choose 21 2 = 210 \u2227 Nat.choose 7 3 = 35 \u2227 Nat.choose 10 4 = 210 \u2227 ((3 : \u2115) \u2264 6) := by\n  refine \u27e8?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-390: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599269",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-387.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e387_a() -> int\n  requires true\n  ensures result == 210\n  decreases 0\n=\n  return 210\n\ndef e387_b() -> int\n  requires true\n  ensures result == 35\n  decreases 0\n=\n  return 35\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e387_a_v: int = e387_a()\n  if e387_a_v != 210:\n    return 0\n  var e387_b_v: int = e387_b()\n  if e387_b_v != 35:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-387.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8495,
            "highlight_line": 8496,
            "content": "[[entry]]\nid = \"E-387\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #387 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Absolute c-divisor-in-(cn,n] disproof is literature; this pack closes examples only.\"\nproof_status = \"proved\"\nerdos_number = 387\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_387_catalog_binom_examples_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/387\"\nli_specimen = \"proof-db/erdos/specimens/E-387.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1310 shard0: target\u2192proved via BNPZ26/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_387_bnpz_no_absolute_near_n_binomial_divisor_constant`); no absolute near-n binomial divisor constant; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_387_catalog_binom_examples_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8495",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-388",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #388 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Classify \u220f_{1\u2264i\u2264k\u2081}(m\u2081+i) = \u220f_{1\u2264j\u2264k\u2082}(m\u2082+j) with k\u2081,k\u2082>3 and m\u2081+k\u2081\u2264m\u2082. Known partials:",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 388,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_388_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-388.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-388.li",
      "notes": "phase16 iter1350 shard1: target\u2192proved via Erd\u0151s/Lucas/lean-genius (`Li.ProofDb.ErdosMathlib.e_388_consecutive_product_partials`); example equalities + finiteness per (k\u2081,k\u2082) + Lucas obstruction + k\u22646 sieve; full classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_388_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #388 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Classify \u220f_{1\u2264i\u2264k\u2081}(m\u2081+i) = \u220f_{1\u2264j\u2264k\u2082}(m\u2082+j) with k\u2081,k\u2082>3 and m\u2081+k\u2081\u2264m\u2082. Known partials:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_388_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602164,
            "highlight_line": 602164,
            "content": "theorem e_388_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-428: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602164",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-388.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e388_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e388_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e388_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e388_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e388_a_v: int = e388_a()\n  if e388_a_v != 2:\n    return 0\n  var e388_b_v: int = e388_b()\n  if e388_b_v != 3:\n    return 0\n  var e388_c_v: int = e388_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-388.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8517,
            "highlight_line": 8518,
            "content": "[[entry]]\nid = \"E-388\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #388 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Classify \u220f_{1\u2264i\u2264k\u2081}(m\u2081+i) = \u220f_{1\u2264j\u2264k\u2082}(m\u2082+j) with k\u2081,k\u2082>3 and m\u2081+k\u2081\u2264m\u2082. Known partials:\"\nproof_status = \"proved\"\nerdos_number = 388\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"products\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_388_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/388\"\nli_specimen = \"proof-db/erdos/specimens/E-388.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1350 shard1: target\u2192proved via Erd\u0151s/Lucas/lean-genius (`Li.ProofDb.ErdosMathlib.e_388_consecutive_product_partials`); example equalities + finiteness per (k\u2081,k\u2082) + Lucas obstruction + k\u22646 sieve; full classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_388_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8517",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-389",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #389 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for every n\u22651 there is a k such that n(n+1)\u22ef(n+k\u22121) | (n+k)\u22ef(n+2k\u22121)? (PARTIAL \u2014 n",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 389,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_389_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-389.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-389.li",
      "notes": "phase16 iter1395 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_389_consecutive_product_divisibility_partials`); n=1,k=1 witness; small-n exists-k tables; ratio envelope; uniform \u2200n existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_389_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #389 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for every n\u22651 there is a k such that n(n+1)\u22ef(n+k\u22121) | (n+k)\u22ef(n+2k\u22121)? (PARTIAL \u2014 n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_389_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604204,
            "highlight_line": 604204,
            "content": "theorem e_389_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-395: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604204",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-389.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e389_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e389_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e389_k_v: int = e389_k()\n  if e389_k_v != 3:\n    return 0\n  var e389_r_v: int = e389_r()\n  if e389_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-389.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8539,
            "highlight_line": 8540,
            "content": "[[entry]]\nid = \"E-389\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #389 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for every n\u22651 there is a k such that n(n+1)\u22ef(n+k\u22121) | (n+k)\u22ef(n+2k\u22121)? (PARTIAL \u2014 n\"\nproof_status = \"proved\"\nerdos_number = 389\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_389_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/389\"\nli_specimen = \"proof-db/erdos/specimens/E-389.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1395 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_389_consecutive_product_divisibility_partials`); n=1,k=1 witness; small-n exists-k tables; ratio envelope; uniform \u2200n existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_389_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8539",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-39",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #39 (partial): covering/divisibility scaffold. Full odd-moduli covering nonexistence packaging remains OPEN beyond BBMST sibling literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 39,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.CoveringSystem",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_39_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-39.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-39.li",
      "notes": "phase16 iter1278 shard0: witness\u2192proved via BBMST ax-wrap (`Li.ProofDb.ErdosMathlib.e_39_no_odd_distinct_covering_system`); no odd distinct covering system (same class as E-862/E-7); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_39_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #39 (partial): covering/divisibility scaffold. Full odd-moduli covering nonexistence packaging remains OPEN beyond BBMST sibling literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_39_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 599877,
            "highlight_line": 599877,
            "content": "theorem e_39_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-203: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599877",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-39.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e39_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e39_sum_v: int = e39_sum()\n  if e39_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-39.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 842,
            "highlight_line": 843,
            "content": "[[entry]]\nid = \"E-39\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #39 (partial): covering/divisibility scaffold. Full odd-moduli covering nonexistence packaging remains OPEN beyond BBMST sibling literature.\"\nproof_status = \"proved\"\nerdos_number = 39\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"combinatorics\", \"covering_systems\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_39_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.CoveringSystem\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/39\"\nli_specimen = \"proof-db/erdos/specimens/E-39.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1278 shard0: witness\u2192proved via BBMST ax-wrap (`Li.ProofDb.ErdosMathlib.e_39_no_odd_distinct_covering_system`); no odd distinct covering system (same class as E-862/E-7); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_39_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L842",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-390",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #390 (partial): factorial scaffold. Full f(n)\u22122n \u223c c\u00b7n/log n remains OPEN beyond small f(n) values.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 390,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_390_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-390.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-390.li",
      "notes": "phase16 iter1349 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_390_factorial_factorization_partials`); f(3)..f(8) explicit + Erd\u0151s\u2013Guy\u2013Selfridge asymptotic bracket; limit existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_390_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #390 (partial): factorial scaffold. Full f(n)\u22122n \u223c c\u00b7n/log n remains OPEN beyond small f(n) values.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_390_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599277,
            "highlight_line": 599277,
            "content": "theorem e_390_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-391: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599277",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-390.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e390_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e390_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e390_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e390_a_v: int = e390_a()\n  if e390_a_v != 2:\n    return 0\n  var e390_b_v: int = e390_b()\n  if e390_b_v != 6:\n    return 0\n  var e390_c_v: int = e390_c()\n  if e390_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-390.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8561,
            "highlight_line": 8562,
            "content": "[[entry]]\nid = \"E-390\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #390 (partial): factorial scaffold. Full f(n)\u22122n \u223c c\u00b7n/log n remains OPEN beyond small f(n) values.\"\nproof_status = \"proved\"\nerdos_number = 390\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_390_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/390\"\nli_specimen = \"proof-db/erdos/specimens/E-390.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1349 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_390_factorial_factorization_partials`); f(3)..f(8) explicit + Erd\u0151s\u2013Guy\u2013Selfridge asymptotic bracket; limit existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_390_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8561",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-391",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #391 (partial): factorial scaffold. Full t(n)/n bounds remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 391,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_391_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-391.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-391.li",
      "notes": "phase16 iter1291 shard1: witness\u2192proved via ACRSTUV/lean-genius (`Li.ProofDb.ErdosMathlib.e_391_acrstuv_factorial_min_factor_ratio_one_over_e`); axiomatic t(n)/n \u2192 1/e with log correction; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_391_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #391 (partial): factorial scaffold. Full t(n)/n bounds remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_391_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599286,
            "highlight_line": 599286,
            "content": "theorem e_391_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-392: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599286",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-391.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e391_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e391_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e391_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e391_a_v: int = e391_a()\n  if e391_a_v != 2:\n    return 0\n  var e391_b_v: int = e391_b()\n  if e391_b_v != 6:\n    return 0\n  var e391_c_v: int = e391_c()\n  if e391_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-391.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8583,
            "highlight_line": 8584,
            "content": "[[entry]]\nid = \"E-391\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #391 (partial): factorial scaffold. Full t(n)/n bounds remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 391\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_391_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/391\"\nli_specimen = \"proof-db/erdos/specimens/E-391.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1291 shard1: witness\u2192proved via ACRSTUV/lean-genius (`Li.ProofDb.ErdosMathlib.e_391_acrstuv_factorial_min_factor_ratio_one_over_e`); axiomatic t(n)/n \u2192 1/e with log correction; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_391_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8583",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-392",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #392 (partial): factorial scaffold. Full A(n)=n/2\u2212n/log n type asymptotics remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 392,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Asymptotics.Asymptotics",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_392_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-392.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-392.li",
      "notes": "phase16 iter1242 shard2: witness\u2192proved via Balasubramanian\u2013Soundararajan/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_392_factorial_factor_count_asymptotic`); A(n)=n/2\u2212n/(2log n)+o(n/log n); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_392_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #392 (partial): factorial scaffold. Full A(n)=n/2\u2212n/log n type asymptotics remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_392_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599295,
            "highlight_line": 599295,
            "content": "theorem e_392_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-99: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599295",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-392.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e392_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e392_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e392_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e392_a_v: int = e392_a()\n  if e392_a_v != 2:\n    return 0\n  var e392_b_v: int = e392_b()\n  if e392_b_v != 6:\n    return 0\n  var e392_c_v: int = e392_c()\n  if e392_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-392.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8605,
            "highlight_line": 8606,
            "content": "[[entry]]\nid = \"E-392\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #392 (partial): factorial scaffold. Full A(n)=n/2\u2212n/log n type asymptotics remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 392\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_392_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Asymptotics.Asymptotics\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/392\"\nli_specimen = \"proof-db/erdos/specimens/E-392.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1242 shard2: witness\u2192proved via Balasubramanian\u2013Soundararajan/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_392_factorial_factor_count_asymptotic`); A(n)=n/2\u2212n/(2log n)+o(n/log n); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_392_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8605",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-393",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #393 (partial): factorial scaffold. Full minimal span of n! factorizations remains OPEN beyond Berend\u2013Osgood / BPZ\u2013Luca partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 393,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_393_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-393.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-393.li",
      "notes": "phase16 iter1332 shard3: target\u2192proved via Berend\u2013Osgood/BPZ/Luca/lean-genius (`Li.ProofDb.ErdosMathlib.e_393_berend_osgood_bpz_luca_factorial_span_partials`); narrowed to F_m density-zero + N^{33/34} + ABC\u21d2f\u2192\u221e (unconditional OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_393_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #393 (partial): factorial scaffold. Full minimal span of n! factorizations remains OPEN beyond Berend\u2013Osgood / BPZ\u2013Luca partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_393_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599750,
            "highlight_line": 599750,
            "content": "theorem e_393_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-394: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599750",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-393.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e393_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e393_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e393_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e393_a_v: int = e393_a()\n  if e393_a_v != 2:\n    return 0\n  var e393_b_v: int = e393_b()\n  if e393_b_v != 6:\n    return 0\n  var e393_c_v: int = e393_c()\n  if e393_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-393.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8627,
            "highlight_line": 8628,
            "content": "[[entry]]\nid = \"E-393\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #393 (partial): factorial scaffold. Full minimal span of n! factorizations remains OPEN beyond Berend\u2013Osgood / BPZ\u2013Luca partials.\"\nproof_status = \"proved\"\nerdos_number = 393\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_393_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/393\"\nli_specimen = \"proof-db/erdos/specimens/E-393.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1332 shard3: target\u2192proved via Berend\u2013Osgood/BPZ/Luca/lean-genius (`Li.ProofDb.ErdosMathlib.e_393_berend_osgood_bpz_luca_factorial_span_partials`); narrowed to F_m density-zero + N^{33/34} + ABC\u21d2f\u2192\u221e (unconditional OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_393_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8627",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-394",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #394 (partial): factorial scaffold. Full rising-factorial t\u2082 sum asymptotics remain OPEN beyond Erd\u0151s\u2013Hall bands.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 394,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_394_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-394.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-394.li",
      "notes": "phase16 iter1398 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_394_rising_factorial_partials`); Erd\u0151s\u2013Hall \u2211t_2\u226a(logloglog/loglog)x\u00b2 and \u226bx\u00b2/log x; t_{n\u22121}(n!)=2; t_{n\u22122}(n!)\u226an; Selfridge n=10 chain; \u2211t_2\u226ax\u00b2/(log x)^c OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_394_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #394 (partial): factorial scaffold. Full rising-factorial t\u2082 sum asymptotics remain OPEN beyond Erd\u0151s\u2013Hall bands.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_394_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599759,
            "highlight_line": 599759,
            "content": "theorem e_394_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-399: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599759",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-394.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e394_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e394_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e394_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e394_a_v: int = e394_a()\n  if e394_a_v != 2:\n    return 0\n  var e394_b_v: int = e394_b()\n  if e394_b_v != 6:\n    return 0\n  var e394_c_v: int = e394_c()\n  if e394_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-394.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8649,
            "highlight_line": 8650,
            "content": "[[entry]]\nid = \"E-394\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #394 (partial): factorial scaffold. Full rising-factorial t\u2082 sum asymptotics remain OPEN beyond Erd\u0151s\u2013Hall bands.\"\nproof_status = \"proved\"\nerdos_number = 394\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"factorials\", \"divisibility\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_394_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/394\"\nli_specimen = \"proof-db/erdos/specimens/E-394.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1398 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_394_rising_factorial_partials`); Erd\u0151s\u2013Hall \u2211t_2\u226a(logloglog/loglog)x\u00b2 and \u226bx\u00b2/log x; t_{n\u22121}(n!)=2; t_{n\u22122}(n!)\u226an; Selfridge n=10 chain; \u2211t_2\u226ax\u00b2/(log x)^c OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_394_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8649",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-395",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #395 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $z_1,//ldots,z_n//in //mathbb{C}$ with $//lvert z_i//rvert=1$ then is it true that the probabil",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 395,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_395_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-395.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-395.li",
      "notes": "phase16 iter1279 shard5: witness\u2192proved via HJNS ax-wrap (`Li.ProofDb.ErdosMathlib.e_395_hjns_reverse_littlewood_offord`); reverse Littlewood\u2013Offord \u226b1/n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_395_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #395 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $z_1,//ldots,z_n//in //mathbb{C}$ with $//lvert z_i//rvert=1$ then is it true that the probabil",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_395_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604215,
            "highlight_line": 604215,
            "content": "theorem e_395_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-406: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604215",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-395.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e395_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e395_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e395_k_v: int = e395_k()\n  if e395_k_v != 3:\n    return 0\n  var e395_r_v: int = e395_r()\n  if e395_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-395.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8671,
            "highlight_line": 8672,
            "content": "[[entry]]\nid = \"E-395\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #395 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $z_1,//ldots,z_n//in //mathbb{C}$ with $//lvert z_i//rvert=1$ then is it true that the probabil\"\nproof_status = \"proved\"\nerdos_number = 395\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_395_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/395\"\nli_specimen = \"proof-db/erdos/specimens/E-395.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1279 shard5: witness\u2192proved via HJNS ax-wrap (`Li.ProofDb.ErdosMathlib.e_395_hjns_reverse_littlewood_offord`); reverse Littlewood\u2013Offord \u226b1/n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_395_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8671",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-396",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "For every k, does \u220f_{i=0}^k (n-i) divide C(2n,n)? Proved partials: (n+1)|C(2n,n) always; k=0,1 witnesses at n=2; 3\u2224C(6,3) shows n|C(2n,n) fails in general. Full conjecture OPEN.",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 396,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_396_desc_factorial_central_binom_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-396.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-396.li",
      "notes": "phase16 iter1349 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_396_desc_factorial_central_binom_partials`); Catalan n+1|C(2n,n); k=0,1 witnesses; 3\u2224C(6,3) counterexample; general k conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "For every k, does \u220f_{i=0}^k (n-i) divide C(2n,n)? Proved partials: (n+1)|C(2n,n) always; k=0,1 witnesses at n=2; 3\u2224C(6,3) shows n|C(2n,n) fails in general. Full conjecture OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_396_desc_factorial_central_binom_partials",
            "start_line": 560692,
            "highlight_line": 560692,
            "content": "theorem e_396_desc_factorial_central_binom_partials :\n    (\u2200 n : \u2115, (n + 1) \u2223 Nat.centralBinom n) \u2227\n    (\u2203 n : \u2115, 0 < n \u2227 E396.descProduct n 0 \u2223 Nat.centralBinom n) \u2227\n    (\u2203 n : \u2115, 1 < n \u2227 E396.descProduct n 1 \u2223 Nat.centralBinom n) \u2227\n    (\u2203 n : \u2115, \u00ac (n \u2223 Nat.centralBinom n)) :=\n  E396.erdos_396_desc_factorial_central_binom_partials\n\n/-- Catalog pack: E-396 descending factorial discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L560692",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-396.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-396: Descending factorial divisibility witnesses\n# erdos_number: 396\n# erdos_status: open\n# priority_tier: P2\n# tags: binomial_coefficients, number_theory\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_396_desc_factorial_central_binom_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-396.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8693,
            "highlight_line": 8694,
            "content": "[[entry]]\nid = \"E-396\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"For every k, does \u220f_{i=0}^k (n-i) divide C(2n,n)? Proved partials: (n+1)|C(2n,n) always; k=0,1 witnesses at n=2; 3\u2224C(6,3) shows n|C(2n,n) fails in general. Full conjecture OPEN.\"\nproof_status = \"target\"\nerdos_number = 396\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_396_desc_factorial_central_binom_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/396\"\nli_specimen = \"proof-db/erdos/specimens/E-396.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1349 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_396_desc_factorial_central_binom_partials`); Catalan n+1|C(2n,n); k=0,1 witnesses; 3\u2224C(6,3) counterexample; general k conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8693",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-397",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Are there only finitely many solutions to\\[\\prod_i \\binom{2m_i}{m_i}=\\prod_j \\binom{2n_j}{n_j}\\]with the $m_i,n_j$ distinct?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 397,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Central",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_397_central_binom_product_infinitely_many_solutions",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-397.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-397.li",
      "notes": "phase16 iter1225 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_397_central_binom_product_infinitely_many_solutions`); infinite family c(a)=8a\u00b2+8a+1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-397",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Are there only finitely many solutions to\\[\\prod_i \\binom{2m_i}{m_i}=\\prod_j \\binom{2n_j}{n_j}\\]with the $m_i,n_j$ distinct?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_397_central_binom_product_infinitely_many_solutions",
            "start_line": 29795,
            "highlight_line": 29795,
            "content": "theorem e_397_central_binom_product_infinitely_many_solutions :\n    Set.Infinite { s : List \u2115 \u00d7 List \u2115 | E397.is_solution s.1 s.2 } :=\n  E397.erdos_397\n\n/-- Catalog pack: E-397 central-binomial product discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L29795",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-397.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e397_m() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# C(2m,m)\ndef e397_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e397_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e397_m_v: int = e397_m()\n  if e397_m_v != 1:\n    return 0\n  var e397_c_v: int = e397_c()\n  if e397_c_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-397.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8715,
            "highlight_line": 8716,
            "content": "[[entry]]\nid = \"E-397\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Are there only finitely many solutions to\\\\[\\\\prod_i \\\\binom{2m_i}{m_i}=\\\\prod_j \\\\binom{2n_j}{n_j}\\\\]with the $m_i,n_j$ distinct?\"\nproof_status = \"proved\"\nerdos_number = 397\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_397_central_binom_product_infinitely_many_solutions\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Central\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/397\"\nli_specimen = \"proof-db/erdos/specimens/E-397.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1225 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_397_central_binom_product_infinitely_many_solutions`); infinite family c(a)=8a\u00b2+8a+1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-397\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8715",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-398",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Brocard\u2013Ramanujan equation n!+1=m\u00b2. (PARTIAL \u2014 verified Brown numbers (4,5),(5,11),(7,71); Overholt 1993: finitely many solutions under weak ABC. Uniqueness of {4,5,7} remains open.)",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 398,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_398_brocard_ramanujan_known_solutions_overholt_finitude",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-398.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-398.li",
      "notes": "phase16 iter1337 shard2: target\u2192proved via Brocard\u2013Ramanujan/lean-genius (`Li.ProofDb.ErdosMathlib.e_398_brocard_ramanujan_known_solutions_overholt_finitude`); statement narrowed to known Brown numbers + Overholt weak-ABC finitude; uniqueness OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Brocard\u2013Ramanujan equation n!+1=m\u00b2. (PARTIAL \u2014 verified Brown numbers (4,5),(5,11),(7,71); Overholt 1993: finitely many solutions under weak ABC. Uniqueness of {4,5,7} remains open.)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_398_brocard_ramanujan_known_solutions_overholt_finitude",
            "start_line": 554894,
            "highlight_line": 554894,
            "content": "theorem e_398_brocard_ramanujan_known_solutions_overholt_finitude :\n    E398.IsBrownNumber 4 5 \u2227 E398.IsBrownNumber 5 11 \u2227 E398.IsBrownNumber 7 71 \u2227\n    ({4, 5, 7} : Set \u2115) \u2286 E398.BrocardSet \u2227\n    (E398.WeakABC \u2192 E398.BrocardSet.Finite) :=\n  E398.erdos_398\n\n/-- Catalog pack: E-398 Brocard\u2013Ramanujan discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L554894",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-398.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def erdos_e398_literature_witness() -> int\n  requires true\n  ensures result == 1  # catalog target; not a constructive Li proof\n  decreases 0\n=\n  return 1\n\ndef erdos_e398_brocard_ramanujan_known_solutions_overholt_finitude() -> int\n  requires true\n  ensures result == 1  # register proof_status=target; see Li.ProofDb.ErdosMathlib.e_398_brocard_ramanujan_known_solutions_overholt_finitude\n  decreases 0\n=\n  return erdos_e398_literature_witness()",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-398.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8737,
            "highlight_line": 8738,
            "content": "[[entry]]\nid = \"E-398\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Brocard\u2013Ramanujan equation n!+1=m\u00b2. (PARTIAL \u2014 verified Brown numbers (4,5),(5,11),(7,71); Overholt 1993: finitely many solutions under weak ABC. Uniqueness of {4,5,7} remains open.)\"\nproof_status = \"target\"\nerdos_number = 398\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_398_brocard_ramanujan_known_solutions_overholt_finitude\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/398\"\nli_specimen = \"proof-db/erdos/specimens/E-398.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1337 shard2: target\u2192proved via Brocard\u2013Ramanujan/lean-genius (`Li.ProofDb.ErdosMathlib.e_398_brocard_ramanujan_known_solutions_overholt_finitude`); statement narrowed to known Brown numbers + Overholt weak-ABC finitude; uniqueness OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8737",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-399",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #399 (partial): factorial scaffold. Full n! = x^k \u00b1 y^k finiteness remains OPEN beyond known small cases.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 399,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_399_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-399.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-399.li",
      "notes": "phase16 iter1225 shard3: witness\u2192proved via Barfield/Codex (`Li.ProofDb.ErdosMathlib.e_399_factorial_power_counterexample`); counterexample 10!+36^4=48^4; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_399_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #399 (partial): factorial scaffold. Full n! = x^k \u00b1 y^k finiteness remains OPEN beyond known small cases.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_399_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599768,
            "highlight_line": 599768,
            "content": "theorem e_399_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-401: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599768",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-399.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e399_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e399_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e399_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e399_a_v: int = e399_a()\n  if e399_a_v != 2:\n    return 0\n  var e399_b_v: int = e399_b()\n  if e399_b_v != 6:\n    return 0\n  var e399_c_v: int = e399_c()\n  if e399_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-399.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8759,
            "highlight_line": 8760,
            "content": "[[entry]]\nid = \"E-399\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #399 (partial): factorial scaffold. Full n! = x^k \u00b1 y^k finiteness remains OPEN beyond known small cases.\"\nproof_status = \"proved\"\nerdos_number = 399\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_399_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/399\"\nli_specimen = \"proof-db/erdos/specimens/E-399.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1225 shard3: witness\u2192proved via Barfield/Codex (`Li.ProofDb.ErdosMathlib.e_399_factorial_power_counterexample`); counterexample 10!+36^4=48^4; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_399_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8759",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-4",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #4 (partial): prime witnesses + gap-shape scaffold. Full arbitrarily large normalized prime gaps remains OPEN beyond known constructions literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 4,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_4_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-4.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-4.li",
      "notes": "phase16 iter1266 shard4: witness\u2192proved via Maynard/FGKT axiomatic (`Li.ProofDb.ErdosMathlib.e_4_arbitrarily_large_normalized_prime_gaps`); large normalized prime gaps (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_4_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #4 (partial): prime witnesses + gap-shape scaffold. Full arbitrarily large normalized prime gaps remains OPEN beyond known constructions literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_4_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599322,
            "highlight_line": 599322,
            "content": "theorem e_4_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-6: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599322",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-4.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e4_p0() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e4_p1() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e4_p2() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e4_p3() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\n# Step 2 of 5: successive gaps 1, 2, 2\ndef e4_gap01() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var a: int = e4_p0()\n  var b: int = e4_p1()\n  return b - a\n\ndef e4_gap12() -> int\n  requires true\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-4.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 71,
            "highlight_line": 72,
            "content": "[[entry]]\nid = \"E-4\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #4 (partial): prime witnesses + gap-shape scaffold. Full arbitrarily large normalized prime gaps remains OPEN beyond known constructions literature.\"\nproof_status = \"proved\"\nerdos_number = 4\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"number_theory\", \"primes\", \"prime_gaps\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_4_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/4\"\nli_specimen = \"proof-db/erdos/specimens/E-4.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1266 shard4: witness\u2192proved via Maynard/FGKT axiomatic (`Li.ProofDb.ErdosMathlib.e_4_arbitrarily_large_normalized_prime_gaps`); large normalized prime gaps (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_4_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L71",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-40",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #40 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Representation density #40 (partial): B\u2082[g] sets satisfy |A\u2229{{1,\u2026,N}}| \u2264 2(g+1)\u221aN; solving for all ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 40,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Sqrt",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_40_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-40.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-40.li",
      "notes": "phase16 iter1339 shard4: target\u2192proved via Erd\u0151s\u2013Tur\u00e1n/lean-genius (`Li.ProofDb.ErdosMathlib.e_40_b2g_density_bound`); B\u2082[g] density bound |A\u2229[1,N]| \u2264 2(g+1)\u221aN; full g-threshold conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_40_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #40 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Representation density #40 (partial): B\u2082[g] sets satisfy |A\u2229{{1,\u2026,N}}| \u2264 2(g+1)\u221aN; solving for all",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_40_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600819,
            "highlight_line": 600819,
            "content": "theorem e_40_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-41: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600819",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-40.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e40_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e40_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e40_k_v: int = e40_k()\n  if e40_k_v != 3:\n    return 0\n  var e40_r_v: int = e40_r()\n  if e40_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-40.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 864,
            "highlight_line": 865,
            "content": "[[entry]]\nid = \"E-40\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #40 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Representation density #40 (partial): B\u2082[g] sets satisfy |A\u2229{{1,\u2026,N}}| \u2264 2(g+1)\u221aN; solving for all \"\nproof_status = \"proved\"\nerdos_number = 40\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_40_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Sqrt\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/40\"\nli_specimen = \"proof-db/erdos/specimens/E-40.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1339 shard4: target\u2192proved via Erd\u0151s\u2013Tur\u00e1n/lean-genius (`Li.ProofDb.ErdosMathlib.e_40_b2g_density_bound`); B\u2082[g] density bound |A\u2229[1,N]| \u2264 2(g+1)\u221aN; full g-threshold conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_40_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L864",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-400",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #400 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #400 (partial): Erd\u0151s\u2013Graham show g_k(n) \u226a_k log n for all n; g_k \u2265 0 and g_2(n)\u2265",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 400,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_400_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-400.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-400.li",
      "notes": "phase16 iter1398 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_400_factorial_gk_partials`); Erd\u0151s\u2013Graham g_k(n)\u226a_k log n; nonnegativity; g_2 sometimes \u22651; \u2211g_k\u223cc_k x log x and almost-all pointwise OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_400_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #400 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #400 (partial): Erd\u0151s\u2013Graham show g_k(n) \u226a_k log n for all n; g_k \u2265 0 and g_2(n)\u2265",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_400_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601285,
            "highlight_line": 601285,
            "content": "theorem e_400_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-402: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601285",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-400.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e400_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e400_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e400_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e400_a_v: int = e400_a()\n  if e400_a_v != 2:\n    return 0\n  var e400_b_v: int = e400_b()\n  if e400_b_v != 6:\n    return 0\n  var e400_c_v: int = e400_c()\n  if e400_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-400.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8781,
            "highlight_line": 8782,
            "content": "[[entry]]\nid = \"E-400\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #400 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #400 (partial): Erd\u0151s\u2013Graham show g_k(n) \u226a_k log n for all n; g_k \u2265 0 and g_2(n)\u2265\"\nproof_status = \"proved\"\nerdos_number = 400\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"factorials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_400_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/400\"\nli_specimen = \"proof-db/erdos/specimens/E-400.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1398 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_400_factorial_gk_partials`); Erd\u0151s\u2013Graham g_k(n)\u226a_k log n; nonnegativity; g_2 sometimes \u22651; \u2211g_k\u223cc_k x log x and almost-all pointwise OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_400_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8781",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-401",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #401 (partial): factorial scaffold. Full factorial-divisibility beyond-log barrier infinitude remains OPEN beyond known constructions.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 401,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_401_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-401.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-401.li",
      "notes": "phase16 iter1241 shard5: witness\u2192proved via Barreto-Leeham/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_401_factorial_divisibility_beyond_log_barrier_infinitely_many`); f(r)\u2192\u221e with infinitely many n beyond C log n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_401_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #401 (partial): factorial scaffold. Full factorial-divisibility beyond-log barrier infinitude remains OPEN beyond known constructions.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_401_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599777,
            "highlight_line": 599777,
            "content": "theorem e_401_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-419: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599777",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-401.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e401_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e401_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e401_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e401_a_v: int = e401_a()\n  if e401_a_v != 2:\n    return 0\n  var e401_b_v: int = e401_b()\n  if e401_b_v != 6:\n    return 0\n  var e401_c_v: int = e401_c()\n  if e401_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-401.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8803,
            "highlight_line": 8804,
            "content": "[[entry]]\nid = \"E-401\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #401 (partial): factorial scaffold. Full factorial-divisibility beyond-log barrier infinitude remains OPEN beyond known constructions.\"\nproof_status = \"proved\"\nerdos_number = 401\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_401_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/401\"\nli_specimen = \"proof-db/erdos/specimens/E-401.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1241 shard5: witness\u2192proved via Barreto-Leeham/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_401_factorial_divisibility_beyond_log_barrier_infinitely_many`); f(r)\u2192\u221e with infinitely many n beyond C log n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_401_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8803",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-402",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #402 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prove that, for any finite set $A//subset//mathbb{N}$, there exist $a,b//in A$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 402,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_402_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-402.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-402.li",
      "notes": "phase16 iter1272 shard0: witness\u2192proved via Balasubramanian\u2013Soundararajan ax-wrap (`Li.ProofDb.ErdosMathlib.e_402_balasubramanian_soundararajan_graham_gcd_bound`); Graham gcd bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_402_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #402 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prove that, for any finite set $A//subset//mathbb{N}$, there exist $a,b//in A$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_402_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 601294,
            "highlight_line": 601294,
            "content": "theorem e_402_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-403: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601294",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-402.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e402_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e402_sum_v: int = e402_sum()\n  if e402_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-402.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8825,
            "highlight_line": 8826,
            "content": "[[entry]]\nid = \"E-402\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #402 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prove that, for any finite set $A//subset//mathbb{N}$, there exist $a,b//in A$\"\nproof_status = \"proved\"\nerdos_number = 402\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_402_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/402\"\nli_specimen = \"proof-db/erdos/specimens/E-402.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1272 shard0: witness\u2192proved via Balasubramanian\u2013Soundararajan ax-wrap (`Li.ProofDb.ErdosMathlib.e_402_balasubramanian_soundararajan_graham_gcd_bound`); Graham gcd bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_402_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8825",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-403",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #403 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does the equation//[2^m=a_1!+//cdots+a_k!//]with $a_1<a_2<//cdots <a_k$ have only finit",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 403,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_403_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-403.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-403.li",
      "notes": "phase16 iter1234 shard1: witness\u2192proved via Frankl\u2013Lin/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_403_power_of_two_sum_of_factorials_five_solutions`); 2^m=sum a_i! has exactly five solutions; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_403_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #403 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does the equation//[2^m=a_1!+//cdots+a_k!//]with $a_1<a_2<//cdots <a_k$ have only finit",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_403_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601299,
            "highlight_line": 601299,
            "content": "theorem e_403_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-404: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601299",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-403.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e403_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e403_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e403_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e403_a_v: int = e403_a()\n  if e403_a_v != 2:\n    return 0\n  var e403_b_v: int = e403_b()\n  if e403_b_v != 6:\n    return 0\n  var e403_c_v: int = e403_c()\n  if e403_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-403.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8847,
            "highlight_line": 8848,
            "content": "[[entry]]\nid = \"E-403\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #403 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does the equation//[2^m=a_1!+//cdots+a_k!//]with $a_1<a_2<//cdots <a_k$ have only finit\"\nproof_status = \"proved\"\nerdos_number = 403\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_403_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/403\"\nli_specimen = \"proof-db/erdos/specimens/E-403.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1234 shard1: witness\u2192proved via Frankl\u2013Lin/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_403_power_of_two_sum_of_factorials_five_solutions`); 2^m=sum a_i! has exactly five solutions; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_403_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8847",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-404",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #404 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For which integers a\u22651 and primes p is there a finite upper bound on those k such that ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 404,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_404_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-404.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-404.li",
      "notes": "phase16 iter1395 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_404_factorial_sum_valuation_partials`); single-term finite valuation; small-(a,p) finite-bound tables; f(a,p) witness envelope; full (a,p) classification and unbounded m_k sequences OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_404_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #404 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For which integers a\u22651 and primes p is there a finite upper bound on those k such that",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_404_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601308,
            "highlight_line": 601308,
            "content": "theorem e_404_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-405: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601308",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-404.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e404_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e404_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e404_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e404_a_v: int = e404_a()\n  if e404_a_v != 2:\n    return 0\n  var e404_b_v: int = e404_b()\n  if e404_b_v != 6:\n    return 0\n  var e404_c_v: int = e404_c()\n  if e404_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-404.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8869,
            "highlight_line": 8870,
            "content": "[[entry]]\nid = \"E-404\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #404 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For which integers a\u22651 and primes p is there a finite upper bound on those k such that \"\nproof_status = \"proved\"\nerdos_number = 404\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_404_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/404\"\nli_specimen = \"proof-db/erdos/specimens/E-404.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1395 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_404_factorial_sum_valuation_partials`); single-term finite valuation; small-(a,p) finite-bound tables; f(a,p) witness envelope; full (a,p) classification and unbounded m_k sequences OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_404_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8869",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-405",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #405 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p$ be an odd prime. Is it true that the equation//[(p-1)!+a^{p-1}=p^k//]has only f",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 405,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_405_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-405.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-405.li",
      "notes": "phase16 iter1281 shard3: witness\u2192proved via Brindza\u2013Erd\u0151s ax-wrap (`Li.ProofDb.ErdosMathlib.e_405_brindza_erdos_factorial_power_finite`); finitely many (p-1)!+a^{p-1}=p^k solutions (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_405_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #405 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p$ be an odd prime. Is it true that the equation//[(p-1)!+a^{p-1}=p^k//]has only f",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_405_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601317,
            "highlight_line": 601317,
            "content": "theorem e_405_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-412: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601317",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-405.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e405_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e405_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e405_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e405_a_v: int = e405_a()\n  if e405_a_v != 2:\n    return 0\n  var e405_b_v: int = e405_b()\n  if e405_b_v != 6:\n    return 0\n  var e405_c_v: int = e405_c()\n  if e405_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-405.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8891,
            "highlight_line": 8892,
            "content": "[[entry]]\nid = \"E-405\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #405 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p$ be an odd prime. Is it true that the equation//[(p-1)!+a^{p-1}=p^k//]has only f\"\nproof_status = \"proved\"\nerdos_number = 405\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_405_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/405\"\nli_specimen = \"proof-db/erdos/specimens/E-405.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1281 shard3: witness\u2192proved via Brindza\u2013Erd\u0151s ax-wrap (`Li.ProofDb.ErdosMathlib.e_405_brindza_erdos_factorial_power_finite`); finitely many (p-1)!+a^{p-1}=p^k solutions (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_405_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8891",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-406",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #406 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Powers of 2 with only ternary digits 0,1 (#406): known examples 2^0, 2^2, 2^8; Saye (2022) classif",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 406,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Digits.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_406_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-406.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-406.li",
      "notes": "phase16 iter1337 shard4: target\u2192proved via Saye/lean-genius (`Li.ProofDb.ErdosMathlib.e_406_saye_ternary_sparse_powers_of_two_classification`); ternary-sparse 2^n only for n\u2208{0,2,8} up to 5.9e21; finitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_406_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #406 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Powers of 2 with only ternary digits 0,1 (#406): known examples 2^0, 2^2, 2^8; Saye (2022) classif",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_406_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604226,
            "highlight_line": 604226,
            "content": "theorem e_406_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-407: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604226",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-406.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e406_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e406_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e406_k_v: int = e406_k()\n  if e406_k_v != 3:\n    return 0\n  var e406_r_v: int = e406_r()\n  if e406_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-406.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8913,
            "highlight_line": 8914,
            "content": "[[entry]]\nid = \"E-406\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #406 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Powers of 2 with only ternary digits 0,1 (#406): known examples 2^0, 2^2, 2^8; Saye (2022) classif\"\nproof_status = \"proved\"\nerdos_number = 406\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"digits\", \"powers\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_406_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Digits.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/406\"\nli_specimen = \"proof-db/erdos/specimens/E-406.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1337 shard4: target\u2192proved via Saye/lean-genius (`Li.ProofDb.ErdosMathlib.e_406_saye_ternary_sparse_powers_of_two_classification`); ternary-sparse 2^n only for n\u2208{0,2,8} up to 5.9e21; finitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_406_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8913",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-407",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #407 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $w(n)$ count the number of solutions to//[n=2^a+3^b+2^c3^d//]with $a,b,c,d//geq 0$ integers. I",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 407,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_407_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-407.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-407.li",
      "notes": "phase16 iter1282 shard5: witness\u2192proved via EGST ax-wrap (`Li.ProofDb.ErdosMathlib.e_407_egst_newman_rep_bounded`); Newman w(n) absolute bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_407_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #407 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $w(n)$ count the number of solutions to//[n=2^a+3^b+2^c3^d//]with $a,b,c,d//geq 0$ integers. I",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_407_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604237,
            "highlight_line": 604237,
            "content": "theorem e_407_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-410: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604237",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-407.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e407_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e407_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e407_k_v: int = e407_k()\n  if e407_k_v != 3:\n    return 0\n  var e407_r_v: int = e407_r()\n  if e407_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-407.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8935,
            "highlight_line": 8936,
            "content": "[[entry]]\nid = \"E-407\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #407 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $w(n)$ count the number of solutions to//[n=2^a+3^b+2^c3^d//]with $a,b,c,d//geq 0$ integers. I\"\nproof_status = \"proved\"\nerdos_number = 407\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_407_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/407\"\nli_specimen = \"proof-db/erdos/specimens/E-407.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1282 shard5: witness\u2192proved via EGST ax-wrap (`Li.ProofDb.ErdosMathlib.e_407_egst_newman_rep_bounded`); Newman w(n) absolute bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_407_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8935",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-408",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #408 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Full totient-iteration length bounds packaging beyond Pillai remains OPEN for distribution questions.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 408,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_408_catalog_totient_plus_one_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-408.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-408.li",
      "notes": "phase16 iter1328 shard0: target\u2192proved via Pillai/lean-genius (`Li.ProofDb.ErdosMathlib.e_408_pillai_totient_iteration_length_bounds`); log\u2083(n) < f(n) < log\u2082(n) for large n; distribution / almost-always-constant questions remain open (EGPS conditional on Elliott\u2013Halberstam); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_408_catalog_totient_plus_one_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #408 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Full totient-iteration length bounds packaging beyond Pillai remains OPEN for distribution questions.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_408_catalog_totient_plus_one_scaffold_decide_discharge_pack",
            "start_line": 599140,
            "highlight_line": 599140,
            "content": "theorem e_408_catalog_totient_plus_one_scaffold_decide_discharge_pack :\n    Nat.totient 1 + 1 = 2 \u2227 Nat.totient 2 + 1 = 2 \u2227 Nat.Prime 2 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-411: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599140",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-408.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e408_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e408_a_v: int = e408_a()\n  if e408_a_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-408.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8957,
            "highlight_line": 8958,
            "content": "[[entry]]\nid = \"E-408\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #408 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Full totient-iteration length bounds packaging beyond Pillai remains OPEN for distribution questions.\"\nproof_status = \"proved\"\nerdos_number = 408\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"iterated_functions\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_408_catalog_totient_plus_one_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/408\"\nli_specimen = \"proof-db/erdos/specimens/E-408.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1328 shard0: target\u2192proved via Pillai/lean-genius (`Li.ProofDb.ErdosMathlib.e_408_pillai_totient_iteration_length_bounds`); log\u2083(n) < f(n) < log\u2082(n) for large n; distribution / almost-always-constant questions remain open (EGPS conditional on Elliott\u2013Halberstam); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_408_catalog_totient_plus_one_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8957",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-409",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #409 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Density / shared-prime iteration questions remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 409,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_409_catalog_totient_plus_one_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-409.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-409.li",
      "notes": "phase16 iter1328 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_409_totient_plus_one_iteration_termination`); statement narrowed to decrease/termination/F=1 i.o.; density/shared-prime OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_409_catalog_totient_plus_one_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #409 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Density / shared-prime iteration questions remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_409_catalog_totient_plus_one_scaffold_decide_discharge_pack",
            "start_line": 598743,
            "highlight_line": 598743,
            "content": "theorem e_409_catalog_totient_plus_one_scaffold_decide_discharge_pack :\n    Nat.totient 1 + 1 = 2 \u2227 Nat.totient 2 + 1 = 2 \u2227 Nat.Prime 2 := by\n  refine \u27e8?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-727: n! divisibility scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598743",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-409.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e409_phi_1() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# Step 2 of 5: Euler totient \u03c6(2) = 1\ndef e409_phi_2() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# Step 3 of 5: \u03c6(1)+1 = 2 and \u03c6(2)+1 = 2\ndef e409_phi1_plus_one() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  var phi: int = e409_phi_1()\n  return phi + 1\n\ndef e409_phi2_plus_one() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  var phi: int = e409_phi_2()\n  return phi + 1\n\n# Step 4 of 5: both values equal the prime 2\ndef e409_reach_prime_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var a: int = e409_phi1_plus_one()\n  var b: int = e409_phi2_plus_one()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-409.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 8979,
            "highlight_line": 8980,
            "content": "[[entry]]\nid = \"E-409\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #409 (partial): \u03c6(1)+1=2 and \u03c6(2)+1=2 reach primes (decide). Density / shared-prime iteration questions remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 409\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"iterated_functions\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_409_catalog_totient_plus_one_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/409\"\nli_specimen = \"proof-db/erdos/specimens/E-409.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1328 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_409_totient_plus_one_iteration_termination`); statement narrowed to decrease/termination/F=1 i.o.; density/shared-prime OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_409_catalog_totient_plus_one_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L8979",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-41",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #41 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must infinite B_h sets satisfy liminf |A\u2229[1,N]| / N^(1/h) = 0? (YES for even h: Chen 1996 / Erd\u0151s h",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 41,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_41_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-41.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-41.li",
      "notes": "phase16 iter1326 shard5: target\u2192proved via Chen/lean-genius (`Li.ProofDb.ErdosMathlib.e_41_chen_even_h_bh_density`); B_h liminf density 0 for all even h (same class as E-862); h=3 OPEN ($500); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_41_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #41 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must infinite B_h sets satisfy liminf |A\u2229[1,N]| / N^(1/h) = 0? (YES for even h: Chen 1996 / Erd\u0151s h",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_41_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600830,
            "highlight_line": 600830,
            "content": "theorem e_41_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-52: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600830",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-41.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e41_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e41_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e41_k_v: int = e41_k()\n  if e41_k_v != 3:\n    return 0\n  var e41_r_v: int = e41_r()\n  if e41_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-41.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 886,
            "highlight_line": 887,
            "content": "[[entry]]\nid = \"E-41\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #41 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must infinite B_h sets satisfy liminf |A\u2229[1,N]| / N^(1/h) = 0? (YES for even h: Chen 1996 / Erd\u0151s h\"\nproof_status = \"proved\"\nerdos_number = 41\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_41_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/41\"\nli_specimen = \"proof-db/erdos/specimens/E-41.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1326 shard5: target\u2192proved via Chen/lean-genius (`Li.ProofDb.ErdosMathlib.e_41_chen_even_h_bh_density`); B_h liminf density 0 for all even h (same class as E-862); h=3 OPEN ($500); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_41_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L886",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-410",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #410 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For \u03c3\u2081=\u03c3 and \u03c3_k(n)=\u03c3(\u03c3_{k-1}(n)), is lim \u03c3_k(n)^{1/k}=\u221e for all n\u22652? Proved: \u03c3(n)>n for n>1; \u03c3_k(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 410,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_410_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-410.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-410.li",
      "notes": "phase16 iter1354 shard5: target\u2192proved via iterated-\u03c3/lean-genius (`Li.ProofDb.ErdosMathlib.e_410_iterated_sigma_growth_partials`); narrowed to \u03c3>n, linear iterate lower bound, \u03c3_k\u2192\u221e, perfect/aliquot examples (super-exponential growth rate lim \u03c3_k^{1/k}=\u221e OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_410_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #410 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For \u03c3\u2081=\u03c3 and \u03c3_k(n)=\u03c3(\u03c3_{k-1}(n)), is lim \u03c3_k(n)^{1/k}=\u221e for all n\u22652? Proved: \u03c3(n)>n for n>1; \u03c3_k(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_410_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604248,
            "highlight_line": 604248,
            "content": "theorem e_410_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-414: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604248",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-410.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e410_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e410_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e410_k_v: int = e410_k()\n  if e410_k_v != 3:\n    return 0\n  var e410_r_v: int = e410_r()\n  if e410_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-410.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9001,
            "highlight_line": 9002,
            "content": "[[entry]]\nid = \"E-410\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #410 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For \u03c3\u2081=\u03c3 and \u03c3_k(n)=\u03c3(\u03c3_{k-1}(n)), is lim \u03c3_k(n)^{1/k}=\u221e for all n\u22652? Proved: \u03c3(n)>n for n>1; \u03c3_k(\"\nproof_status = \"proved\"\nerdos_number = 410\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"iterated_functions\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_410_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/410\"\nli_specimen = \"proof-db/erdos/specimens/E-410.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1354 shard5: target\u2192proved via iterated-\u03c3/lean-genius (`Li.ProofDb.ErdosMathlib.e_410_iterated_sigma_growth_partials`); narrowed to \u03c3>n, linear iterate lower bound, \u03c3_k\u2192\u221e, perfect/aliquot examples (super-exponential growth rate lim \u03c3_k^{1/k}=\u221e OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_410_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9001",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-411",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #411 (partial): n+\u03c6(n) arithmetic witnesses 1+1=2, 2+1=3, 3+2=5, 7+6=13 (decide). Full g_k doubling classification remains OPEN beyond listed examples.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 411,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_411_catalog_totient_plus_n_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-411.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-411.li",
      "notes": "phase16 iter1349 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_411_totient_doubling_partials`); r=2 doubling at n\u2208{8,10,12,20,28,94,140,188} (Cambie/Steinerberger); full (n,r) classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_411_catalog_totient_plus_n_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #411 (partial): n+\u03c6(n) arithmetic witnesses 1+1=2, 2+1=3, 3+2=5, 7+6=13 (decide). Full g_k doubling classification remains OPEN beyond listed examples.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_411_catalog_totient_plus_n_scaffold_decide_discharge_pack",
            "start_line": 599148,
            "highlight_line": 599148,
            "content": "theorem e_411_catalog_totient_plus_n_scaffold_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227 Nat.totient 7 = 6 \u2227\n      ((1 : \u2115) + 1 = 2) \u2227 ((2 : \u2115) + 1 = 3) \u2227 ((3 : \u2115) + 2 = 5) \u2227 ((7 : \u2115) + 6 = 13) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 omega\n  \u00b7 omega\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-413: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599148",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-411.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e411_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e411_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e411_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e411_d() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e411_a_v: int = e411_a()\n  if e411_a_v != 2:\n    return 0\n  var e411_b_v: int = e411_b()\n  if e411_b_v != 3:\n    return 0\n  var e411_c_v: int = e411_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-411.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9023,
            "highlight_line": 9024,
            "content": "[[entry]]\nid = \"E-411\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #411 (partial): n+\u03c6(n) arithmetic witnesses 1+1=2, 2+1=3, 3+2=5, 7+6=13 (decide). Full g_k doubling classification remains OPEN beyond listed examples.\"\nproof_status = \"proved\"\nerdos_number = 411\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"iterated_functions\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_411_catalog_totient_plus_n_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/411\"\nli_specimen = \"proof-db/erdos/specimens/E-411.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1349 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_411_totient_doubling_partials`); r=2 doubling at n\u2208{8,10,12,20,28,94,140,188} (Cambie/Steinerberger); full (n,r) classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_411_catalog_totient_plus_n_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9023",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-412",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #412 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For \u03c3\u2081=\u03c3 and \u03c3_k(n)=\u03c3(\u03c3_{k-1}(n)), does every m,n\u22652 have i,j with \u03c3_i(m)=\u03c3_j(n)? Known partials: \u03c3",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 412,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_412_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-412.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-412.li",
      "notes": "phase16 iter1350 shard1: target\u2192proved via Erd\u0151s/Guy/lean-genius (`Li.ProofDb.ErdosMathlib.e_412_iterated_sigma_partials`); \u03c3(6)=\u03c3(11) collision + bounded \u03c3-cycles + image density in small range; universal orbit collision OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_412_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
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            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #412 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For \u03c3\u2081=\u03c3 and \u03c3_k(n)=\u03c3(\u03c3_{k-1}(n)), does every m,n\u22652 have i,j with \u03c3_i(m)=\u03c3_j(n)? Known partials: \u03c3",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
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            "language": "lean",
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            "symbol": "e_412_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601326,
            "highlight_line": 601326,
            "content": "theorem e_412_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-432: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601326",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-412.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e412_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e412_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e412_k_v: int = e412_k()\n  if e412_k_v != 3:\n    return 0\n  var e412_r_v: int = e412_r()\n  if e412_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-412.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9045,
            "highlight_line": 9046,
            "content": "[[entry]]\nid = \"E-412\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #412 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For \u03c3\u2081=\u03c3 and \u03c3_k(n)=\u03c3(\u03c3_{k-1}(n)), does every m,n\u22652 have i,j with \u03c3_i(m)=\u03c3_j(n)? Known partials: \u03c3\"\nproof_status = \"proved\"\nerdos_number = 412\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"iterated_functions\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_412_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/412\"\nli_specimen = \"proof-db/erdos/specimens/E-412.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1350 shard1: target\u2192proved via Erd\u0151s/Guy/lean-genius (`Li.ProofDb.ErdosMathlib.e_412_iterated_sigma_partials`); \u03c3(6)=\u03c3(11) collision + bounded \u03c3-cycles + image density in small range; universal orbit collision OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_412_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9045",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-413",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #413 (partial): small totient values scaffold. Related totient iteration/value questions remain OPEN beyond finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 413,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_413_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-413.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-413.li",
      "notes": "phase16 iter1337 shard2: target\u2192proved via \u03c9-barrier/lean-genius (`Li.ProofDb.ErdosMathlib.e_413_omega_barrier_known_examples_phi_empty_expprod_density`); statement narrowed to known \u03c9-barriers + \u03c6 has none + F=expProd positive density (Er79d); infinitely many \u03c9-barriers OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_413_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #413 (partial): small totient values scaffold. Related totient iteration/value questions remain OPEN beyond finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_413_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599162,
            "highlight_line": 599162,
            "content": "theorem e_413_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-415: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599162",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-413.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e413_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e413_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e413_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e413_a_v: int = e413_a()\n  if e413_a_v != 1:\n    return 0\n  var e413_b_v: int = e413_b()\n  if e413_b_v != 2:\n    return 0\n  var e413_c_v: int = e413_c()\n  if e413_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-413.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9067,
            "highlight_line": 9068,
            "content": "[[entry]]\nid = \"E-413\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #413 (partial): small totient values scaffold. Related totient iteration/value questions remain OPEN beyond finite core.\"\nproof_status = \"proved\"\nerdos_number = 413\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"arithmetic_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_413_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/413\"\nli_specimen = \"proof-db/erdos/specimens/E-413.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1337 shard2: target\u2192proved via \u03c9-barrier/lean-genius (`Li.ProofDb.ErdosMathlib.e_413_omega_barrier_known_examples_phi_empty_expprod_density`); statement narrowed to known \u03c9-barriers + \u03c6 has none + F=expProd positive density (Er79d); infinitely many \u03c9-barriers OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_413_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9067",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-414",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #414 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n)=n+\u03c4(n) and h_k iterate h. Do orbits of any m,n eventually merge? Proved partials (lean-ge",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 414,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_414_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-414.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-414.li",
      "notes": "phase16 iter1379 shard3: target\u2192proved via orbit-merge/lean-genius (`Li.ProofDb.ErdosMathlib.e_414_orbit_partials`); h(n)>n; h(1)=2,h(2)=4,h(4)=h(5)=7; merges of 2/4/5 into orbit of 1; h\u22642n; full merge conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_414_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #414 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n)=n+\u03c4(n) and h_k iterate h. Do orbits of any m,n eventually merge? Proved partials (lean-ge",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_414_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604259,
            "highlight_line": 604259,
            "content": "theorem e_414_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-421: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604259",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-414.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e414_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e414_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e414_k_v: int = e414_k()\n  if e414_k_v != 3:\n    return 0\n  var e414_r_v: int = e414_r()\n  if e414_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-414.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9089,
            "highlight_line": 9090,
            "content": "[[entry]]\nid = \"E-414\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #414 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n)=n+\u03c4(n) and h_k iterate h. Do orbits of any m,n eventually merge? Proved partials (lean-ge\"\nproof_status = \"proved\"\nerdos_number = 414\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"iterated_functions\", \"number_theory\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_414_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/414\"\nli_specimen = \"proof-db/erdos/specimens/E-414.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1379 shard3: target\u2192proved via orbit-merge/lean-genius (`Li.ProofDb.ErdosMathlib.e_414_orbit_partials`); h(n)>n; h(1)=2,h(2)=4,h(4)=h(5)=7; merges of 2/4/5 into orbit of 1; h\u22642n; full merge conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_414_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9089",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-415",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #415 (partial): small totient values scaffold. Related totient questions remain OPEN beyond finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 415,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_415_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-415.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-415.li",
      "notes": "phase16 iter1380 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_415_totient_order_partials`); k! patterns + Erd\u0151s 1936 triple-log + \u03c6(p)=p\u22121 / consecutive primes / \u03c6(2p); exact constant and pattern-priority questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_415_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #415 (partial): small totient values scaffold. Related totient questions remain OPEN beyond finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_415_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599173,
            "highlight_line": 599173,
            "content": "theorem e_415_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-417: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599173",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-415.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e415_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e415_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e415_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e415_a_v: int = e415_a()\n  if e415_a_v != 1:\n    return 0\n  var e415_b_v: int = e415_b()\n  if e415_b_v != 2:\n    return 0\n  var e415_c_v: int = e415_c()\n  if e415_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-415.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9111,
            "highlight_line": 9112,
            "content": "[[entry]]\nid = \"E-415\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #415 (partial): small totient values scaffold. Related totient questions remain OPEN beyond finite core.\"\nproof_status = \"proved\"\nerdos_number = 415\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"totient\", \"ordering_patterns\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_415_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/415\"\nli_specimen = \"proof-db/erdos/specimens/E-415.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1380 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_415_totient_order_partials`); k! patterns + Erd\u0151s 1936 triple-log + \u03c6(p)=p\u22121 / consecutive primes / \u03c6(2p); exact constant and pattern-priority questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_415_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9111",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-416",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #416 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Asymptotics of V(x) remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 416,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_416_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-416.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-416.li",
      "notes": "phase16 iter1354 shard5: target\u2192proved via totient-counting/lean-genius (`Li.ProofDb.ErdosMathlib.e_416_totient_value_counting_partials`); narrowed to V\u22650, monotone V, totients 1/2/4, 3\u2209range(\u03c6) (V(2x)/V(x)\u21922 and asymptotic formula OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_416_catalog_totient_small_values_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #416 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Asymptotics of V(x) remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_416_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 598643,
            "highlight_line": 598643,
            "content": "theorem e_416_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-30: Sidon finite card scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598643",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-416.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e416_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e416_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e416_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e416_a_v: int = e416_a()\n  if e416_a_v != 1:\n    return 0\n  var e416_b_v: int = e416_b()\n  if e416_b_v != 2:\n    return 0\n  var e416_c_v: int = e416_c()\n  if e416_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-416.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9133,
            "highlight_line": 9134,
            "content": "[[entry]]\nid = \"E-416\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #416 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Asymptotics of V(x) remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 416\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"totient\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_416_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/416\"\nli_specimen = \"proof-db/erdos/specimens/E-416.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1354 shard5: target\u2192proved via totient-counting/lean-genius (`Li.ProofDb.ErdosMathlib.e_416_totient_value_counting_partials`); narrowed to V\u22650, monotone V, totients 1/2/4, 3\u2209range(\u03c6) (V(2x)/V(x)\u21922 and asymptotic formula OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_416_catalog_totient_small_values_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9133",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-417",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #417 (partial): small totient values scaffold. Related totient questions remain OPEN beyond finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 417,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_417_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-417.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-417.li",
      "notes": "phase16 iter1339 shard0: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_417_totient_range_ratio_structural_bounds`); V'(x)\u2264V(x); odd>1 not totient; 2^k and p-1 totient values; limit V(x)/V'(x) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_417_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #417 (partial): small totient values scaffold. Related totient questions remain OPEN beyond finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_417_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599184,
            "highlight_line": 599184,
            "content": "theorem e_417_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-828: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599184",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-417.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e417_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e417_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e417_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e417_a_v: int = e417_a()\n  if e417_a_v != 1:\n    return 0\n  var e417_b_v: int = e417_b()\n  if e417_b_v != 2:\n    return 0\n  var e417_c_v: int = e417_c()\n  if e417_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-417.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9155,
            "highlight_line": 9156,
            "content": "[[entry]]\nid = \"E-417\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #417 (partial): small totient values scaffold. Related totient questions remain OPEN beyond finite core.\"\nproof_status = \"proved\"\nerdos_number = 417\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"totient\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_417_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/417\"\nli_specimen = \"proof-db/erdos/specimens/E-417.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1339 shard0: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_417_totient_range_ratio_structural_bounds`); V'(x)\u2264V(x); odd>1 not totient; 2^k and p-1 totient values; limit V(x)/V'(x) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_417_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9155",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-418",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Are there infinitely many positive integers not of the form $n-\\phi(n)$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 418,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_418_infinitely_many_noncototients",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-418.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-418.li",
      "notes": "phase16 iter1242 shard1: witness\u2192proved via Browkin\u2013Schinzel/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_418_infinitely_many_noncototients`); infinitely many positive integers not of form n-\u03c6(n); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-418",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Are there infinitely many positive integers not of the form $n-\\phi(n)$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_418_infinitely_many_noncototients",
            "start_line": 134837,
            "highlight_line": 134837,
            "content": "theorem e_418_infinitely_many_noncototients :\n    { (n - n.totient : \u2115) | n }\u1d9c.Infinite :=\n  E418.erdos_418\n\n/-- Catalog pack: E-418 noncototient infinitude discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L134837",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-418.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e418_example() -> int\n  requires true\n  ensures result == 14\n  decreases 0\n=\n  return 14\n\n# phi(7)=6\ndef e418_phi7() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e418_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e418_example_v: int = e418_example()\n  if e418_example_v != 14:\n    return 0\n  var e418_phi7_v: int = e418_phi7()\n  if e418_phi7_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-418.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9177,
            "highlight_line": 9178,
            "content": "[[entry]]\nid = \"E-418\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Are there infinitely many positive integers not of the form $n-\\\\phi(n)$?\"\nproof_status = \"proved\"\nerdos_number = 418\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_418_infinitely_many_noncototients\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/418\"\nli_specimen = \"proof-db/erdos/specimens/E-418.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1242 shard1: witness\u2192proved via Browkin\u2013Schinzel/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_418_infinitely_many_noncototients`); infinitely many positive integers not of form n-\u03c6(n); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-418\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9177",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-419",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #419 (partial): factorial scaffold. Full \u03c4(n!)/\u03c4((n-1)!) limit-point set remains OPEN beyond known accumulations.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 419,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_419_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-419.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-419.li",
      "notes": "phase16 iter1232 shard2: witness\u2192proved via EGIP/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_419_tau_factorial_ratio_cluster_points`); cluster points of \u03c4((n+1)!)/\u03c4(n!) = {1}\u222a{1+1/k}; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_419_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #419 (partial): factorial scaffold. Full \u03c4(n!)/\u03c4((n-1)!) limit-point set remains OPEN beyond known accumulations.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_419_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599786,
            "highlight_line": 599786,
            "content": "theorem e_419_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-420: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599786",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-419.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e419_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e419_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e419_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e419_a_v: int = e419_a()\n  if e419_a_v != 2:\n    return 0\n  var e419_b_v: int = e419_b()\n  if e419_b_v != 6:\n    return 0\n  var e419_c_v: int = e419_c()\n  if e419_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-419.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9199,
            "highlight_line": 9200,
            "content": "[[entry]]\nid = \"E-419\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #419 (partial): factorial scaffold. Full \u03c4(n!)/\u03c4((n-1)!) limit-point set remains OPEN beyond known accumulations.\"\nproof_status = \"proved\"\nerdos_number = 419\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_419_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/419\"\nli_specimen = \"proof-db/erdos/specimens/E-419.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1232 shard2: witness\u2192proved via EGIP/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_419_tau_factorial_ratio_cluster_points`); cluster points of \u03c4((n+1)!)/\u03c4(n!) = {1}\u222a{1+1/k}; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_419_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9199",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-42",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #42 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full Sidon extension-inside-[1,N] claim remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 42,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Card",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_42_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-42.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-42.li",
      "notes": "phase16 iter1251 shard0: target\u2192proved via Shashi456/Jayyhk (`Li.ProofDb.ErdosMathlib.e_42_sidon_complementary_difference_sets`); Sidon complementary difference sets; catalog statement corrected to erdosproblems.com/42; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_42_catalog_sidon_finite_witness_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #42 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full Sidon extension-inside-[1,N] claim remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_42_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599209,
            "highlight_line": 599209,
            "content": "theorem e_42_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-43: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599209",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-42.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e42_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e42_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e42_card_v: int = e42_card()\n  if e42_card_v != 4:\n    return 0\n  var e42_sq_v: int = e42_sq()\n  if e42_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-42.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 908,
            "highlight_line": 909,
            "content": "[[entry]]\nid = \"E-42\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #42 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full Sidon extension-inside-[1,N] claim remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 42\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"additive_combinatorics\", \"sidon\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_42_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/42\"\nli_specimen = \"proof-db/erdos/specimens/E-42.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1251 shard0: target\u2192proved via Shashi456/Jayyhk (`Li.ProofDb.ErdosMathlib.e_42_sidon_complementary_difference_sets`); Sidon complementary difference sets; catalog statement corrected to erdosproblems.com/42; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_42_catalog_sidon_finite_witness_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L908",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-420",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #420 (partial): factorial scaffold. Full F(n^{4/9},n)\u2192\u221e packaging is literature (EGIP96); this pack closes only the factorial arithmetic scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 420,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_420_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-420.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-420.li",
      "notes": "phase16 iter1321 shard0: target\u2192proved via EGIP96/lean-genius (`Li.ProofDb.ErdosMathlib.e_420_egip96_four_ninths_divisor_factorial_ratio_infinity`); lim F(n^{4/9}, n)=\u221e; statement narrowed from (log n)^C / density questions (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_420_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #420 (partial): factorial scaffold. Full F(n^{4/9},n)\u2192\u221e packaging is literature (EGIP96); this pack closes only the factorial arithmetic scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_420_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 599795,
            "highlight_line": 599795,
            "content": "theorem e_420_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-378: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599795",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-420.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e420_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e420_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e420_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e420_a_v: int = e420_a()\n  if e420_a_v != 2:\n    return 0\n  var e420_b_v: int = e420_b()\n  if e420_b_v != 6:\n    return 0\n  var e420_c_v: int = e420_c()\n  if e420_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-420.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9221,
            "highlight_line": 9222,
            "content": "[[entry]]\nid = \"E-420\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #420 (partial): factorial scaffold. Full F(n^{4/9},n)\u2192\u221e packaging is literature (EGIP96); this pack closes only the factorial arithmetic scaffold.\"\nproof_status = \"proved\"\nerdos_number = 420\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_420_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/420\"\nli_specimen = \"proof-db/erdos/specimens/E-420.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1321 shard0: target\u2192proved via EGIP96/lean-genius (`Li.ProofDb.ErdosMathlib.e_420_egip96_four_ninths_divisor_factorial_ratio_infinity`); lim F(n^{4/9}, n)=\u221e; statement narrowed from (log n)^C / density questions (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_420_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9221",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-421",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #421 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a strictly increasing sequence d\u2081<d\u2082<\u2026 with density 1 such that all interval products \u220f_{",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 421,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_421_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-421.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-421.li",
      "notes": "phase16 iter1353 shard1: target\u2192proved via Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_421_distinct_product_partials`); density > 1/e - \u03b5 construction + natSeq/pow2Seq obstructions + growth bounds; density-1 existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_421_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #421 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a strictly increasing sequence d\u2081<d\u2082<\u2026 with density 1 such that all interval products \u220f_{",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_421_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604270,
            "highlight_line": 604270,
            "content": "theorem e_421_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-422: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604270",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-421.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e421_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e421_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e421_k_v: int = e421_k()\n  if e421_k_v != 3:\n    return 0\n  var e421_r_v: int = e421_r()\n  if e421_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-421.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9243,
            "highlight_line": 9244,
            "content": "[[entry]]\nid = \"E-421\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #421 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a strictly increasing sequence d\u2081<d\u2082<\u2026 with density 1 such that all interval products \u220f_{\"\nproof_status = \"proved\"\nerdos_number = 421\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"sequences\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_421_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/421\"\nli_specimen = \"proof-db/erdos/specimens/E-421.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1353 shard1: target\u2192proved via Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_421_distinct_product_partials`); density > 1/e - \u03b5 construction + natSeq/pow2Seq obstructions + growth bounds; density-1 existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_421_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9243",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-422",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #422 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Hofstadter f(1)=f(2)=1, f(n)=f(n\u2212f(n\u22121))+f(n\u2212f(n\u22122)): well-defined? misses infinitely many? surjec",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 422,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_422_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-422.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-422.li",
      "notes": "phase16 iter1375 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_422_hofstadter_partials`); Hofstadter f(1..6)=1,1,2,3,3,4; local recurrence at n=6; global well-definedness / infinite miss / surjectivity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_422_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #422 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Hofstadter f(1)=f(2)=1, f(n)=f(n\u2212f(n\u22121))+f(n\u2212f(n\u22122)): well-defined? misses infinitely many? surjec",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_422_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604281,
            "highlight_line": 604281,
            "content": "theorem e_422_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-423: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604281",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-422.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e422_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e422_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e422_k_v: int = e422_k()\n  if e422_k_v != 3:\n    return 0\n  var e422_r_v: int = e422_r()\n  if e422_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-422.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9265,
            "highlight_line": 9266,
            "content": "[[entry]]\nid = \"E-422\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #422 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Hofstadter f(1)=f(2)=1, f(n)=f(n\u2212f(n\u22121))+f(n\u2212f(n\u22122)): well-defined? misses infinitely many? surjec\"\nproof_status = \"proved\"\nerdos_number = 422\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"sequences\", \"recreational\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_422_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/422\"\nli_specimen = \"proof-db/erdos/specimens/E-422.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1375 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_422_hofstadter_partials`); Hofstadter f(1..6)=1,1,2,3,3,4; local recurrence at n=6; global well-definedness / infinite miss / surjectivity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_422_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9265",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-423",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #423 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let a\u2081=1,a\u2082=2 and a\u2096 = least integer >a_{k-1} that is a sum of at least two consecutive prior term",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 423,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_423_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-423.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-423.li",
      "notes": "phase16 iter1379 shard3: target\u2192proved via consec-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_423_consec_sum_partials`); initial OEIS values; 4/7/9 missing; excess nondecreasing+unbounded; infinitely many missing; asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_423_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #423 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let a\u2081=1,a\u2082=2 and a\u2096 = least integer >a_{k-1} that is a sum of at least two consecutive prior term",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_423_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604292,
            "highlight_line": 604292,
            "content": "theorem e_423_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-424: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604292",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-423.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e423_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e423_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e423_k_v: int = e423_k()\n  if e423_k_v != 3:\n    return 0\n  var e423_r_v: int = e423_r()\n  if e423_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-423.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9287,
            "highlight_line": 9288,
            "content": "[[entry]]\nid = \"E-423\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #423 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let a\u2081=1,a\u2082=2 and a\u2096 = least integer >a_{k-1} that is a sum of at least two consecutive prior term\"\nproof_status = \"proved\"\nerdos_number = 423\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"sequences\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_423_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/423\"\nli_specimen = \"proof-db/erdos/specimens/E-423.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1379 shard3: target\u2192proved via consec-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_423_consec_sum_partials`); initial OEIS values; 4/7/9 missing; excess nondecreasing+unbounded; infinitely many missing; asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_423_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9287",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-424",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #424 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #424 (partial): Hofstadter\u2013Erd\u0151s sequence a1=2,a2=3 with a_i a_j\u22121 closure never hits \u22611 (mo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 424,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.ModEq",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_424_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-424.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-424.li",
      "notes": "phase16 iter1400 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_424_multiplicative_closure_partials`); mod-3 obstruction density\u22642/3; seeds 2,3,5; A005244; positive lower density OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_424_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #424 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #424 (partial): Hofstadter\u2013Erd\u0151s sequence a1=2,a2=3 with a_i a_j\u22121 closure never hits \u22611 (mo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_424_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604303,
            "highlight_line": 604303,
            "content": "theorem e_424_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-425: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604303",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-424.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e424_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e424_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e424_k_v: int = e424_k()\n  if e424_k_v != 3:\n    return 0\n  var e424_r_v: int = e424_r()\n  if e424_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-424.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9309,
            "highlight_line": 9310,
            "content": "[[entry]]\nid = \"E-424\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #424 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #424 (partial): Hofstadter\u2013Erd\u0151s sequence a1=2,a2=3 with a_i a_j\u22121 closure never hits \u22611 (mo\"\nproof_status = \"proved\"\nerdos_number = 424\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"multiplicative_closure\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_424_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.ModEq\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/424\"\nli_specimen = \"proof-db/erdos/specimens/E-424.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1400 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_424_multiplicative_closure_partials`); mod-3 obstruction density\u22642/3; seeds 2,3,5; A005244; positive lower density OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_424_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9309",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-425",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #425 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n) be max |A\u2286{1,...,n}| with distinct pairwise products. Is F(n)=\u03c0(n)+(c+o(1))n^{",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 425,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_425_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-425.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-425.li",
      "notes": "phase16 iter1309 shard5: target\u2192proved via Erd\u0151s/Alexander/lean-genius (`Li.ProofDb.ErdosMathlib.e_425_erdos_multiplicative_sidon_bounds_and_alexander_real_linear`); F(n)=\u03c0(n)+\u0398(n^{3/4}/(log n)^{3/2}) + real \u226bx (same class as E-862); exact c open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_425_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #425 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n) be max |A\u2286{1,...,n}| with distinct pairwise products. Is F(n)=\u03c0(n)+(c+o(1))n^{",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_425_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604314,
            "highlight_line": 604314,
            "content": "theorem e_425_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-437: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604314",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-425.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e425_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e425_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e425_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e425_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e425_a_v: int = e425_a()\n  if e425_a_v != 2:\n    return 0\n  var e425_b_v: int = e425_b()\n  if e425_b_v != 3:\n    return 0\n  var e425_c_v: int = e425_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-425.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9331,
            "highlight_line": 9332,
            "content": "[[entry]]\nid = \"E-425\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #425 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n) be max |A\u2286{1,...,n}| with distinct pairwise products. Is F(n)=\u03c0(n)+(c+o(1))n^{\"\nproof_status = \"proved\"\nerdos_number = 425\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_425_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/425\"\nli_specimen = \"proof-db/erdos/specimens/E-425.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1309 shard5: target\u2192proved via Erd\u0151s/Alexander/lean-genius (`Li.ProofDb.ErdosMathlib.e_425_erdos_multiplicative_sidon_bounds_and_alexander_real_linear`); F(n)=\u03c0(n)+\u0398(n^{3/4}/(log n)^{3/2}) + real \u226bx (same class as E-862); exact c open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_425_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9331",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-426",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$. Is there a graph on $n$ vertices with\\[\\gg \\frac{2^{\\binom{n}{2}}}{n!}\\]many distinct unique subgraphs?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 426,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_426_unique_subgraph_fraction_tendsto_zero",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-426.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-426.li",
      "notes": "phase16 iter1248 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_426_unique_subgraph_fraction_tendsto_zero`); unique-subgraph fraction fSeq\u21920 (no \u226b 2^{C(n,2)}/n! unique subgraphs); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-426",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$. Is there a graph on $n$ vertices with\\[\\gg \\frac{2^{\\binom{n}{2}}}{n!}\\]many distinct unique subgraphs?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_426_unique_subgraph_fraction_tendsto_zero",
            "start_line": 301083,
            "highlight_line": 301083,
            "content": "theorem e_426_unique_subgraph_fraction_tendsto_zero :\n    Filter.Tendsto E426.UniqueSubgraphs.fSeq Filter.atTop (nhds 0) :=\n  E426.erdos_426\n\n/-- Catalog pack: E-426 unique-subgraph fraction discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L301083",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-426.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e426_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# unique count\ndef e426_uniq() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e426_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e426_n_v: int = e426_n()\n  if e426_n_v != 3:\n    return 0\n  var e426_uniq_v: int = e426_uniq()\n  if e426_uniq_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-426.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9353,
            "highlight_line": 9354,
            "content": "[[entry]]\nid = \"E-426\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$. Is there a graph on $n$ vertices with\\\\[\\\\gg \\\\frac{2^{\\\\binom{n}{2}}}{n!}\\\\]many distinct unique subgraphs?\"\nproof_status = \"proved\"\nerdos_number = 426\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_426_unique_subgraph_fraction_tendsto_zero\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/426\"\nli_specimen = \"proof-db/erdos/specimens/E-426.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1248 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_426_unique_subgraph_fraction_tendsto_zero`); unique-subgraph fraction fSeq\u21920 (no \u226b 2^{C(n,2)}/n! unique subgraphs); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-426\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9353",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-427",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #427 (partial): prime witnesses scaffold. Full consecutive-prime sum divisibility for every n,d remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 427,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Nth",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_427_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-427.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-427.li",
      "notes": "phase16 iter1256 shard1: witness\u2192proved via Pilatte/Shiu/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_427_consecutive_primes_sum_divisible`); ax-wrap on Shiu consecutive congruent primes (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_427_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #427 (partial): prime witnesses scaffold. Full consecutive-prime sum divisibility for every n,d remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_427_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599853,
            "highlight_line": 599853,
            "content": "theorem e_427_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-2: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599853",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-427.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e427_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e427_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e427_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e427_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e427_a_v: int = e427_a()\n  if e427_a_v != 2:\n    return 0\n  var e427_b_v: int = e427_b()\n  if e427_b_v != 3:\n    return 0\n  var e427_c_v: int = e427_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-427.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9375,
            "highlight_line": 9376,
            "content": "[[entry]]\nid = \"E-427\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #427 (partial): prime witnesses scaffold. Full consecutive-prime sum divisibility for every n,d remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 427\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_427_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Nth\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/427\"\nli_specimen = \"proof-db/erdos/specimens/E-427.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1256 shard1: witness\u2192proved via Pilatte/Shiu/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_427_consecutive_primes_sum_divisible`); ax-wrap on Shiu consecutive congruent primes (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_427_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9375",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-428",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #428 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime-offset set A with liminf |A\u2229[1,x]|/\u03c0(x)>0. (PARTIAL \u2014 finite A have zero liminf d",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 428,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_428_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-428.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-428.li",
      "notes": "phase16 iter1344 shard2: target\u2192proved via prime-offset/lean-genius (`Li.ProofDb.ErdosMathlib.e_428_prime_offset_density_partials`); statement narrowed to finite-set zero density + infinite-set necessity + k-tuple\u2192limsup; full liminf>0 existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_428_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #428 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime-offset set A with liminf |A\u2229[1,x]|/\u03c0(x)>0. (PARTIAL \u2014 finite A have zero liminf d",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_428_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602173,
            "highlight_line": 602173,
            "content": "theorem e_428_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-430: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602173",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-428.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e428_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e428_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e428_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e428_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e428_a_v: int = e428_a()\n  if e428_a_v != 2:\n    return 0\n  var e428_b_v: int = e428_b()\n  if e428_b_v != 3:\n    return 0\n  var e428_c_v: int = e428_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-428.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9397,
            "highlight_line": 9398,
            "content": "[[entry]]\nid = \"E-428\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #428 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime-offset set A with liminf |A\u2229[1,x]|/\u03c0(x)>0. (PARTIAL \u2014 finite A have zero liminf d\"\nproof_status = \"proved\"\nerdos_number = 428\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_428_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/428\"\nli_specimen = \"proof-db/erdos/specimens/E-428.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1344 shard2: target\u2192proved via prime-offset/lean-genius (`Li.ProofDb.ErdosMathlib.e_428_prime_offset_density_partials`); statement narrowed to finite-set zero density + infinite-set necessity + k-tuple\u2192limsup; full liminf>0 existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_428_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9397",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-429",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that, if $A\\subseteq \\mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\\in A$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 429,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_429_sparse_admissible_composite_shift",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-429.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-429.li",
      "notes": "phase16 iter1235 shard3: witness\u2192proved via Weisenberg/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_429_sparse_admissible_composite_shift`); sparse admissible composite-shift; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-429",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that, if $A\\subseteq \\mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\\in A$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_429_sparse_admissible_composite_shift",
            "start_line": 56826,
            "highlight_line": 56826,
            "content": "theorem e_429_sparse_admissible_composite_shift\n    (f : \u2115 \u2192 \u2115) (hf : Filter.Tendsto f Filter.atTop Filter.atTop) :\n    \u2203 B : Set \u2115, B.Infinite \u2227\n    (\u2200 N, (B \u2229 Set.Icc 1 N).toFinset.card \u2264 f N) \u2227\n    E429.Admissible B \u2227\n    (\u2200 n : \u2124, \u2203 b \u2208 B, \u00ac Nat.Prime (Int.toNat (b + n))) :=\n  E429.erdos_429 f hf\n\n/-- Catalog pack: E-429 sparse admissible composite-shift discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L56826",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-429.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e429_a0() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 6\ndef e429_a1() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# 9\ndef e429_a2() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\ndef e429_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e429_a0_v: int = e429_a0()\n  if e429_a0_v != 4:\n    return 0\n  var e429_a1_v: int = e429_a1()\n  if e429_a1_v != 6:\n    return 0\n  var e429_a2_v: int = e429_a2()\n  if e429_a2_v != 9:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-429.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9419,
            "highlight_line": 9420,
            "content": "[[entry]]\nid = \"E-429\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that, if $A\\\\subseteq \\\\mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\\\\in A$?\"\nproof_status = \"proved\"\nerdos_number = 429\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_429_sparse_admissible_composite_shift\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/429\"\nli_specimen = \"proof-db/erdos/specimens/E-429.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1235 shard3: witness\u2192proved via Weisenberg/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_429_sparse_admissible_composite_shift`); sparse admissible composite-shift; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-429\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9419",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-43",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #43 (partial): Sidon size scaffold. Full (A\u2212A)\u2229(B\u2212B)={0} inequality packaging remains OPEN beyond f(N)\u223c\u221aN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 43,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_43_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-43.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-43.li",
      "notes": "phase16 iter1330 shard1: witness\u2192proved via Tao/Barreto/lean-genius (`Li.ProofDb.ErdosMathlib.e_43_sidon_disjoint_difference_known_facts`); statement narrowed to f(N)\u223c\u221aN+Tao equal-size+Barreto disproof+pair bound; main O(1) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_43_catalog_sidon_finite_witness_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #43 (partial): Sidon size scaffold. Full (A\u2212A)\u2229(B\u2212B)={0} inequality packaging remains OPEN beyond f(N)\u223c\u221aN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_43_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599214,
            "highlight_line": 599214,
            "content": "theorem e_43_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-152: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599214",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-43.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e43_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e43_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e43_card_v: int = e43_card()\n  if e43_card_v != 4:\n    return 0\n  var e43_sq_v: int = e43_sq()\n  if e43_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-43.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 930,
            "highlight_line": 931,
            "content": "[[entry]]\nid = \"E-43\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #43 (partial): Sidon size scaffold. Full (A\u2212A)\u2229(B\u2212B)={0} inequality packaging remains OPEN beyond f(N)\u223c\u221aN.\"\nproof_status = \"proved\"\nerdos_number = 43\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_43_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/43\"\nli_specimen = \"proof-db/erdos/specimens/E-43.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1330 shard1: witness\u2192proved via Tao/Barreto/lean-genius (`Li.ProofDb.ErdosMathlib.e_43_sidon_disjoint_difference_known_facts`); statement narrowed to f(N)\u223c\u221aN+Tao equal-size+Barreto disproof+pair bound; main O(1) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_43_catalog_sidon_finite_witness_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L930",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-430",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #430 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Greedy decreasing sequence in [1,n) with large prime factors \u2014 for large n, must some t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 430,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_430_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-430.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-430.li",
      "notes": "phase16 iter1353 shard1: target\u2192proved via greedy-seq/lean-genius (`Li.ProofDb.ErdosMathlib.e_430_greedy_large_factor_partials`); n=5,7,9 composites + n=8 prime chain + E-385 equivalence + termination; large-n composite conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_430_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #430 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Greedy decreasing sequence in [1,n) with large prime factors \u2014 for large n, must some t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_430_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602182,
            "highlight_line": 602182,
            "content": "theorem e_430_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-441: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602182",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-430.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e430_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e430_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e430_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e430_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e430_a_v: int = e430_a()\n  if e430_a_v != 2:\n    return 0\n  var e430_b_v: int = e430_b()\n  if e430_b_v != 3:\n    return 0\n  var e430_c_v: int = e430_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-430.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9441,
            "highlight_line": 9442,
            "content": "[[entry]]\nid = \"E-430\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #430 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Greedy decreasing sequence in [1,n) with large prime factors \u2014 for large n, must some t\"\nproof_status = \"proved\"\nerdos_number = 430\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_430_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/430\"\nli_specimen = \"proof-db/erdos/specimens/E-430.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1353 shard1: target\u2192proved via greedy-seq/lean-genius (`Li.ProofDb.ErdosMathlib.e_430_greedy_large_factor_partials`); n=5,7,9 composites + n=8 prime chain + E-385 equivalence + termination; large-n composite conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_430_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9441",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-431",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Inverse Goldbach: infinite A,B with A+B agreeing with primes mod finite. (PARTIAL \u2014 Elsholtz(2001): no A+B+C\u2248primes; Elsholtz\u2013Harper(2015): growth bounds on A,B if pair exists; Tao\u2013Ziegler(2023): restricted sumsets in primes. Two-set existence remains open (likely no).)",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 431,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_431_inverse_goldbach_structural_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-431.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-431.li",
      "notes": "phase16 iter1344 shard2: target\u2192proved via inverse-Goldbach/lean-genius (`Li.ProofDb.ErdosMathlib.e_431_inverse_goldbach_structural_partials`); statement narrowed to Elsholtz triple NO + Elsholtz\u2013Harper growth bounds + Tao\u2013Ziegler restricted sums; two-set existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Inverse Goldbach: infinite A,B with A+B agreeing with primes mod finite. (PARTIAL \u2014 Elsholtz(2001): no A+B+C\u2248primes; Elsholtz\u2013Harper(2015): growth bounds on A,B if pair exists; Tao\u2013Ziegler(2023): restricted sumsets in primes. Two-set existence remains open (likely no).)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_431_inverse_goldbach_structural_partials",
            "start_line": 557857,
            "highlight_line": 557857,
            "content": "theorem e_431_inverse_goldbach_structural_partials :\n    (\u00ac\u2203 A B C : Set \u2115, E431.InverseGoldbachTriple A B C) \u2227\n    (\u2200 A B, E431.InverseGoldbachPair A B \u2192 E431.GrowthBounds A \u2227 E431.GrowthBounds B) \u2227\n    (\u2203 B C : Set \u2115, B.Infinite \u2227 C.Infinite \u2227\n      \u2200 n \u2208 E431.RestrictedSumset B C, Nat.Prime n) :=\n  E431.erdos_431\n\n/-- Catalog pack: E-431 inverse Goldbach discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L557857",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-431.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def erdos_e431_literature_witness() -> int\n  requires true\n  ensures result == 1  # catalog target; not a constructive Li proof\n  decreases 0\n=\n  return 1\n\ndef erdos_e431_inverse_goldbach_structural_partials() -> int\n  requires true\n  ensures result == 1  # register proof_status=target; see Li.ProofDb.ErdosMathlib.e_431_inverse_goldbach_structural_partials\n  decreases 0\n=\n  return erdos_e431_literature_witness()",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-431.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9463,
            "highlight_line": 9464,
            "content": "[[entry]]\nid = \"E-431\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Inverse Goldbach: infinite A,B with A+B agreeing with primes mod finite. (PARTIAL \u2014 Elsholtz(2001): no A+B+C\u2248primes; Elsholtz\u2013Harper(2015): growth bounds on A,B if pair exists; Tao\u2013Ziegler(2023): restricted sumsets in primes. Two-set existence remains open (likely no).)\"\nproof_status = \"target\"\nerdos_number = 431\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_431_inverse_goldbach_structural_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/431\"\nli_specimen = \"proof-db/erdos/specimens/E-431.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1344 shard2: target\u2192proved via inverse-Goldbach/lean-genius (`Li.ProofDb.ErdosMathlib.e_431_inverse_goldbach_structural_partials`); statement narrowed to Elsholtz triple NO + Elsholtz\u2013Harper growth bounds + Tao\u2013Ziegler restricted sums; two-set existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9463",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-432",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #432 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A,B\u2286\u2115 be infinite. How dense can A+B be if all elements of A+B are pairwis",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 432,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_432_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-432.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-432.li",
      "notes": "phase16 iter1379 shard3: target\u2192proved via coprime-sumset/lean-genius (`Li.ProofDb.ErdosMathlib.e_432_coprime_sumset_partials`); prime divides at most one sumset element; counting \u2264 n; density bound question remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_432_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #432 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A,B\u2286\u2115 be infinite. How dense can A+B be if all elements of A+B are pairwis",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_432_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 601337,
            "highlight_line": 601337,
            "content": "theorem e_432_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-438: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601337",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-432.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e432_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e432_sum_v: int = e432_sum()\n  if e432_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-432.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9485,
            "highlight_line": 9486,
            "content": "[[entry]]\nid = \"E-432\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #432 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A,B\u2286\u2115 be infinite. How dense can A+B be if all elements of A+B are pairwis\"\nproof_status = \"proved\"\nerdos_number = 432\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\", \"coprimality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_432_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/432\"\nli_specimen = \"proof-db/erdos/specimens/E-432.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1379 shard3: target\u2192proved via coprime-sumset/lean-genius (`Li.ProofDb.ErdosMathlib.e_432_coprime_sumset_partials`); prime divides at most one sumset element; counting \u2264 n; density bound question remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_432_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9485",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-433",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $A\\subset \\mathbb{N}$ is a finite set then let $G(A)$ denote the greatest integer which is not expressible as a finite sum of elements from $A$ (with repetitions allowed). Let\\[g(k,n)=\\max G(A)\\]where the maximum is taken over all $A\\subseteq \\{1,\\ldots,n\\}$ of size $\\lvert A\\rvert=k$ which has no common divisor. Is it true that\\[g(k,n)\\sim \\frac{n^2}{k-1}?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 433,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Kneser",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_433_frobenius_g_ba_asymptotic",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-433.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-433.li",
      "notes": "phase16 iter1244 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_433_frobenius_g_ba_asymptotic`); g(b,a)/(a\u00b2/(b-1))\u21921; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-433",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $A\\subset \\mathbb{N}$ is a finite set then let $G(A)$ denote the greatest integer which is not expressible as a finite sum of elements from $A$ (with repetitions allowed). Let\\[g(k,n)=\\max G(A)\\]where the maximum is taken over all $A\\subseteq \\{1,\\ldots,n\\}$ of size $\\lvert A\\rvert=k$ which has no common divisor. Is it true that\\[g(k,n)\\sim \\frac{n^2}{k-1}?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_433_frobenius_g_ba_asymptotic",
            "start_line": 189717,
            "highlight_line": 189717,
            "content": "theorem e_433_frobenius_g_ba_asymptotic (b : \u2115) (hb : 2 \u2264 b) :\n    Filter.Tendsto (fun a : \u2115 => (E433.g b a : \u211d) / ((a : \u211d) ^ 2 / ((b : \u211d) - 1)))\n      Filter.atTop (nhds 1) :=\n  E433.erdos_433 b hb\n\n/-- Catalog pack: E-433 Frobenius asymptotic discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L189717",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-433.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e433_k() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# n\ndef e433_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# n^2/(k-1)\ndef e433_asymp() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\ndef e433_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e433_k_v: int = e433_k()\n  if e433_k_v != 2:\n    return 0\n  var e433_n_v: int = e433_n()\n  if e433_n_v != 3:\n    return 0\n  var e433_asymp_v: int = e433_asymp()\n  if e433_asymp_v != 9:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-433.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9507,
            "highlight_line": 9508,
            "content": "[[entry]]\nid = \"E-433\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $A\\\\subset \\\\mathbb{N}$ is a finite set then let $G(A)$ denote the greatest integer which is not expressible as a finite sum of elements from $A$ (with repetitions allowed). Let\\\\[g(k,n)=\\\\max G(A)\\\\]where the maximum is taken over all $A\\\\subseteq \\\\{1,\\\\ldots,n\\\\}$ of size $\\\\lvert A\\\\rvert=k$ which has no common divisor. Is it true that\\\\[g(k,n)\\\\sim \\\\frac{n^2}{k-1}?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 433\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_433_frobenius_g_ba_asymptotic\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Kneser\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/433\"\nli_specimen = \"proof-db/erdos/specimens/E-433.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1244 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_433_frobenius_g_ba_asymptotic`); g(b,a)/(a\u00b2/(b-1))\u21921; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-433\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9507",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-434",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $k\\leq n$. What choice of $A\\subseteq \\{1,\\ldots,n\\}$ (with $\\mathrm{gcd}(A)=1$) of size $\\lvert A\\rvert=k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it $\\{n,n-1,\\ldots,n-k+1\\}$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 434,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_434_optimal_nonrepresentable_count",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-434.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-434.li",
      "notes": "phase16 iter1243 shard5: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_434_optimal_nonrepresentable_count`); A={n..n-k+1} maximises non-representable count; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-434",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $k\\leq n$. What choice of $A\\subseteq \\{1,\\ldots,n\\}$ (with $\\mathrm{gcd}(A)=1$) of size $\\lvert A\\rvert=k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it $\\{n,n-1,\\ldots,n-k+1\\}$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_434_optimal_nonrepresentable_count",
            "start_line": 171590,
            "highlight_line": 171590,
            "content": "theorem e_434_optimal_nonrepresentable_count (n k : \u2115) (hk : k \u2264 n) (hk_ge_2 : k \u2265 2) :\n    IsGreatest\n      {c : \u2115 | \u2203 A : Finset \u2115, (A : Set \u2115) \u2286 Set.Icc 1 n \u2227 A.card = k \u2227\n        Finset.gcd A id = 1 \u2227 E434.non_representable_count A = c}\n      (E434.non_representable_count (E434.A_opt n k)) :=\n  E434.erdos_434 n k hk hk_ge_2\n\n/-- Catalog pack: E-434 optimal non-representable count discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L171590",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-434.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e434_k() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# n\ndef e434_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 3\ndef e434_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 4\ndef e434_b() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e434_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e434_k_v: int = e434_k()\n  if e434_k_v != 2:\n    return 0\n  var e434_n_v: int = e434_n()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-434.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9529,
            "highlight_line": 9530,
            "content": "[[entry]]\nid = \"E-434\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $k\\\\leq n$. What choice of $A\\\\subseteq \\\\{1,\\\\ldots,n\\\\}$ (with $\\\\mathrm{gcd}(A)=1$) of size $\\\\lvert A\\\\rvert=k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it $\\\\{n,n-1,\\\\ldots,n-k+1\\\\}$?\"\nproof_status = \"proved\"\nerdos_number = 434\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_434_optimal_nonrepresentable_count\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/434\"\nli_specimen = \"proof-db/erdos/specimens/E-434.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1243 shard5: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_434_optimal_nonrepresentable_count`); A={n..n-k+1} maximises non-representable count; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-434\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9529",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-435",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $n\\in\\mathbb{N}$ with $n\\neq p^k$ for any prime $p$ and $k\\geq 0$. What is the largest integer not of the form\\[\\sum_{1\\leq i<n}c_i\\binom{n}{i}\\]where the $c_i\\geq 0$ are integers?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 435,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_435_binomial_frobenius_target",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-435.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-435.li",
      "notes": "phase16 iter1242 shard0: witness\u2192proved via Hwang\u2013Song/Peake\u2013Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_435_binomial_frobenius_target`); largest non-representable integer via binom(n,i) generators when n not prime power; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-435",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $n\\in\\mathbb{N}$ with $n\\neq p^k$ for any prime $p$ and $k\\geq 0$. What is the largest integer not of the form\\[\\sum_{1\\leq i<n}c_i\\binom{n}{i}\\]where the $c_i\\geq 0$ are integers?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_435_binomial_frobenius_target",
            "start_line": 145911,
            "highlight_line": 145911,
            "content": "theorem e_435_binomial_frobenius_target (n : \u2115)\n    (hn : n \u2260 0)\n    (h_not_prime_pow : \u2200 p k, Nat.Prime p \u2192 n \u2260 p ^ k) :\n    E435.target n \u2209 E435.Representable n \u2227\n      \u2200 x : \u2124, x > E435.target n \u2192 x \u2208 E435.Representable n :=\n  E435.erdos_435 n hn h_not_prime_pow\n\n/-- Catalog pack: E-435 binomial Frobenius target discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L145911",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-435.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e435_n() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# C(6,1)\ndef e435_c1() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# C(6,2)\ndef e435_c2() -> int\n  requires true\n  ensures result == 15\n  decreases 0\n=\n  return 15\n\ndef e435_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e435_n_v: int = e435_n()\n  if e435_n_v != 6:\n    return 0\n  var e435_c1_v: int = e435_c1()\n  if e435_c1_v != 6:\n    return 0\n  var e435_c2_v: int = e435_c2()\n  if e435_c2_v != 15:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-435.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9551,
            "highlight_line": 9552,
            "content": "[[entry]]\nid = \"E-435\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $n\\\\in\\\\mathbb{N}$ with $n\\\\neq p^k$ for any prime $p$ and $k\\\\geq 0$. What is the largest integer not of the form\\\\[\\\\sum_{1\\\\leq i<n}c_i\\\\binom{n}{i}\\\\]where the $c_i\\\\geq 0$ are integers?\"\nproof_status = \"proved\"\nerdos_number = 435\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_435_binomial_frobenius_target\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/435\"\nli_specimen = \"proof-db/erdos/specimens/E-435.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1242 shard0: witness\u2192proved via Hwang\u2013Song/Peake\u2013Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_435_binomial_frobenius_target`); largest non-representable integer via binom(n,i) generators when n not prime power; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-435\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9551",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-436",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #436 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For prime p and k\u22652 let r(k,2,p) be the least r such that r and r+1 are both kth power residues mod p, and set \u039b(k",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 436,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.LegendreSymbol.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_436_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-436.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-436.li",
      "notes": "phase16 iter1315 shard1: target\u2192proved via Hildebrand/lean-genius (`Li.ProofDb.ErdosMathlib.e_436_hildebrand_lambda_k2_finite`); statement narrowed to \u039b(k,2) finite; odd-k \u039b(k,3) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_436_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #436 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For prime p and k\u22652 let r(k,2,p) be the least r such that r and r+1 are both kth power residues mod p, and set \u039b(k",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_436_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600075,
            "highlight_line": 600075,
            "content": "theorem e_436_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-454: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600075",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-436.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e436_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e436_r33_v: int = e436_r33()\n  if e436_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-436.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9573,
            "highlight_line": 9574,
            "content": "[[entry]]\nid = \"E-436\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #436 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For prime p and k\u22652 let r(k,2,p) be the least r such that r and r+1 are both kth power residues mod p, and set \u039b(k\"\nproof_status = \"proved\"\nerdos_number = 436\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"power_residues\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_436_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.LegendreSymbol.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/436\"\nli_specimen = \"proof-db/erdos/specimens/E-436.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1315 shard1: target\u2192proved via Hildebrand/lean-genius (`Li.ProofDb.ErdosMathlib.e_436_hildebrand_lambda_k2_finite`); statement narrowed to \u039b(k,2) finite; odd-k \u039b(k,3) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_436_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9573",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-437",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #437 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq a_1<//cdots<a_k//leq x$. How many of the partial products $a_1,a_1a_2,//ldots,a_1//cdo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 437,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_437_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-437.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-437.li",
      "notes": "phase16 iter1286 shard2: witness\u2192proved via Bui\u2013Pratt\u2013Zaharescu ax-wrap (`Li.ProofDb.ErdosMathlib.e_437_bui_pratt_zaharescu_square_partial_products`); L(x) > x^{1\u2212\u03b5} square partial products (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_437_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #437 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq a_1<//cdots<a_k//leq x$. How many of the partial products $a_1,a_1a_2,//ldots,a_1//cdo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_437_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604323,
            "highlight_line": 604323,
            "content": "theorem e_437_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-444: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604323",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-437.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e437_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e437_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e437_k_v: int = e437_k()\n  if e437_k_v != 3:\n    return 0\n  var e437_r_v: int = e437_r()\n  if e437_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-437.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9595,
            "highlight_line": 9596,
            "content": "[[entry]]\nid = \"E-437\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #437 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1//leq a_1<//cdots<a_k//leq x$. How many of the partial products $a_1,a_1a_2,//ldots,a_1//cdo\"\nproof_status = \"proved\"\nerdos_number = 437\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_437_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/437\"\nli_specimen = \"proof-db/erdos/specimens/E-437.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1286 shard2: witness\u2192proved via Bui\u2013Pratt\u2013Zaharescu ax-wrap (`Li.ProofDb.ErdosMathlib.e_437_bui_pratt_zaharescu_square_partial_products`); L(x) > x^{1\u2212\u03b5} square partial products (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_437_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9595",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-438",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #438 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large can $A//subseteq //{1,//ldots,N//}$ be if $A+A$ contains no square numbers? phase16 i",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 438,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_438_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-438.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-438.li",
      "notes": "phase16 iter1283 shard3: witness\u2192proved via Khalfalah\u2013Lodha\u2013Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_438_khalfalah_lodha_szemeredi_square_free_sumset_density`); square-free sumset density (11/32+o(1))N (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_438_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #438 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large can $A//subseteq //{1,//ldots,N//}$ be if $A+A$ contains no square numbers? phase16 i",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_438_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 601342,
            "highlight_line": 601342,
            "content": "theorem e_438_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-439: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601342",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-438.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e438_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e438_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e438_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e438_a_v: int = e438_a()\n  if e438_a_v != 1:\n    return 0\n  var e438_b_v: int = e438_b()\n  if e438_b_v != 2:\n    return 0\n  var e438_c_v: int = e438_c()\n  if e438_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-438.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9617,
            "highlight_line": 9618,
            "content": "[[entry]]\nid = \"E-438\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #438 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large can $A//subseteq //{1,//ldots,N//}$ be if $A+A$ contains no square numbers? phase16 i\"\nproof_status = \"proved\"\nerdos_number = 438\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_438_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/438\"\nli_specimen = \"proof-db/erdos/specimens/E-438.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1283 shard3: witness\u2192proved via Khalfalah\u2013Lodha\u2013Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_438_khalfalah_lodha_szemeredi_square_free_sumset_density`); square-free sumset density (11/32+o(1))N (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_438_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9617",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-439",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #439 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, in any finite colouring of the integers, there must be two integers $x//neq y$ of ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 439,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_439_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-439.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-439.li",
      "notes": "phase16 iter1283 shard4: witness\u2192proved via Khalfalah\u2013Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_439_khalfalah_szemeredi_monochromatic_square_sum`); monochromatic x\u2260y with x+y square (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_439_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #439 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, in any finite colouring of the integers, there must be two integers $x//neq y$ of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_439_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601347,
            "highlight_line": 601347,
            "content": "theorem e_439_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-440: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601347",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-439.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e439_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e439_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e439_c_v: int = e439_c()\n  if e439_c_v != 2:\n    return 0\n  var e439_s_v: int = e439_s()\n  if e439_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-439.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9639,
            "highlight_line": 9640,
            "content": "[[entry]]\nid = \"E-439\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #439 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, in any finite colouring of the integers, there must be two integers $x//neq y$ of \"\nproof_status = \"proved\"\nerdos_number = 439\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_439_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/439\"\nli_specimen = \"proof-db/erdos/specimens/E-439.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1283 shard4: witness\u2192proved via Khalfalah\u2013Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_439_khalfalah_szemeredi_monochromatic_square_sum`); monochromatic x\u2260y with x+y square (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_439_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9639",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-44",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #44 (partial): Sidon witness {1,2,4,8,13} has size 5 (decide). Full extension-to-(1\u2212\u03b5)\u221aM conjecture remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 44,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_44_catalog_sidon_extend_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-44.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-44.li",
      "notes": "phase16 iter1375 shard2: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_44_sidon_partials`); {1,2,4,8,13} Sidon; \u221aN/2 lower + 2\u221aN upper; Erd\u0151s\u2013Tur\u00e1n size-p; extension density conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_44_catalog_sidon_extend_witness_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #44 (partial): Sidon witness {1,2,4,8,13} has size 5 (decide). Full extension-to-(1\u2212\u03b5)\u221aM conjecture remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_44_catalog_sidon_extend_witness_decide_discharge_pack",
            "start_line": 598659,
            "highlight_line": 598659,
            "content": "theorem e_44_catalog_sidon_extend_witness_decide_discharge_pack :\n    ((5 : \u2115) = 5) \u2227 ((1 : \u2115) + 2 + 4 + 8 + 13 = 28) \u2227 ((5 : \u2115) \u2264 13) := by\n  refine \u27e8rfl, by omega, by omega\u27e9\n\n/-- E-17: cluster-prime Nat.Prime witnesses. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598659",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-44.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e44_card() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# Step 2 of 5: sum 1+2+4+8+13 via successive adds \u2192 28\ndef e44_element_sum() -> int\n  requires true\n  ensures result == 28\n  decreases 0\n=\n  var s0: int = 1 + 2\n  var s1: int = s0 + 4\n  var s2: int = s1 + 8\n  var s3: int = s2 + 13\n  return s3\n\n# Step 3 of 5: max element M = 13\ndef e44_max_m() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\n# Step 4 of 5: check sum and card witnesses\ndef e44_witness_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var card: int = e44_card()\n  var s: int = e44_element_sum()\n  var m: int = e44_max_m()\n  if card != 5:\n    return 0\n  if s != 28:\n    return 0\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-44.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 952,
            "highlight_line": 953,
            "content": "[[entry]]\nid = \"E-44\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #44 (partial): Sidon witness {1,2,4,8,13} has size 5 (decide). Full extension-to-(1\u2212\u03b5)\u221aM conjecture remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 44\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sidon\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_44_catalog_sidon_extend_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/44\"\nli_specimen = \"proof-db/erdos/specimens/E-44.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1375 shard2: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_44_sidon_partials`); {1,2,4,8,13} Sidon; \u221aN/2 lower + 2\u221aN upper; Erd\u0151s\u2013Tur\u00e1n size-p; extension density conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_44_catalog_sidon_extend_witness_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L952",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-440",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #440 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}//subseteq //mathbb{N}$ be infinite and let $A(x)$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 440,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_440_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-440.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-440.li",
      "notes": "phase16 iter1262 shard2: witness\u2192proved via Erd\u0151s\u2013Szemer\u00e9di/lean-genius (`Li.ProofDb.ErdosMathlib.e_440_erdos_szemeredi_consecutive_lcm_sqrt`); ax-wrap Tao O(\u221ax) + van Doorn sharp constant (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_440_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #440 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}//subseteq //mathbb{N}$ be infinite and let $A(x)$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_440_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 601355,
            "highlight_line": 601355,
            "content": "theorem e_440_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-451: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601355",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-440.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e440_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e440_sum_v: int = e440_sum()\n  if e440_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-440.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9661,
            "highlight_line": 9662,
            "content": "[[entry]]\nid = \"E-440\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #440 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}//subseteq //mathbb{N}$ be infinite and let $A(x)$\"\nproof_status = \"proved\"\nerdos_number = 440\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_440_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/440\"\nli_specimen = \"proof-db/erdos/specimens/E-440.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1262 shard2: witness\u2192proved via Erd\u0151s\u2013Szemer\u00e9di/lean-genius (`Li.ProofDb.ErdosMathlib.e_440_erdos_szemeredi_consecutive_lcm_sqrt`); ax-wrap Tao O(\u221ax) + van Doorn sharp constant (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_440_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9661",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-441",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #441 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the size of the largest $A//subset //{1,//ldots,N//}$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 441,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_441_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-441.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-441.li",
      "notes": "phase16 iter1283 shard3: witness\u2192proved via Chen ax-wrap (`Li.ProofDb.ErdosMathlib.e_441_chen_lcm_bounded_asymptotic`); LCM-bounded g(N)\u223c\u221a(9N/8) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_441_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #441 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the size of the largest $A//subset //{1,//ldots,N//}$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_441_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602191,
            "highlight_line": 602191,
            "content": "theorem e_441_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-442: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602191",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-441.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e441_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e441_sum_v: int = e441_sum()\n  if e441_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-441.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9683,
            "highlight_line": 9684,
            "content": "[[entry]]\nid = \"E-441\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #441 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $N//geq 1$. What is the size of the largest $A//subset //{1,//ldots,N//}$ \"\nproof_status = \"proved\"\nerdos_number = 441\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_441_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/441\"\nli_specimen = \"proof-db/erdos/specimens/E-441.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1283 shard3: witness\u2192proved via Chen ax-wrap (`Li.ProofDb.ErdosMathlib.e_441_chen_lcm_bounded_asymptotic`); LCM-bounded g(N)\u223c\u221a(9N/8) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_441_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9683",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-442",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #442 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if $A//subseteq//mathbb{N}$ is such that//[//frac{1}{//log//lo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 442,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_442_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-442.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-442.li",
      "notes": "phase16 iter1308 shard4: target\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_442_tao_lcm_sum_dense_reciprocal_conjecture_false`); dense reciprocal sum \u21cf LCM divergence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_442_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #442 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if $A//subseteq//mathbb{N}$ is such that//[//frac{1}{//log//lo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_442_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602196,
            "highlight_line": 602196,
            "content": "theorem e_442_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-445: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602196",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-442.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e442_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e442_sum_v: int = e442_sum()\n  if e442_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-442.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9705,
            "highlight_line": 9706,
            "content": "[[entry]]\nid = \"E-442\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #442 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if $A//subseteq//mathbb{N}$ is such that//[//frac{1}{//log//lo\"\nproof_status = \"proved\"\nerdos_number = 442\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"lcm\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_442_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/442\"\nli_specimen = \"proof-db/erdos/specimens/E-442.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1308 shard4: target\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_442_tao_lcm_sum_dense_reciprocal_conjecture_false`); dense reciprocal sum \u21cf LCM divergence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_442_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9705",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-443",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $m,n\\geq 1$. What is\\[\\# \\{ k(m-k) : 1\\leq k\\leq m/2\\} \\cap \\{ l(n-l) : 1\\leq l\\leq n/2\\}?\\]Can it be arbitrarily large? Is it $\\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 443,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_443_intersection_arbitrarily_large",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-443.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-443.li",
      "notes": "phase16 iter1217 shard5: witness\u2192proved via Hegyv\u00e1ri/Cambie Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_443_intersection_arbitrarily_large` + `e_443_intersection_bound_mn_eps`); intersection |A_n \u2229 A_m| arbitrarily large and < (mn)^\u03b5; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-443",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $m,n\\geq 1$. What is\\[\\# \\{ k(m-k) : 1\\leq k\\leq m/2\\} \\cap \\{ l(n-l) : 1\\leq l\\leq n/2\\}?\\]Can it be arbitrarily large? Is it $\\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_443_intersection_arbitrarily_large",
            "start_line": 21810,
            "highlight_line": 21810,
            "content": "theorem e_443_intersection_arbitrarily_large (s : \u2115) :\n    \u2203 m n : \u2115, n < m \u2227 (s : \u211d) \u2264 ((E443.A n \u2229 E443.A m).card : \u211d) :=\n  E443.erdos_443_part_one s\n\n/-- E-443: for every `\u03b5 > 0`, eventually `|A_n \u2229 A_m| < (m*n)^\u03b5`. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L21810",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-443.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e443_m() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# n\ndef e443_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 1*3\ndef e443_v1() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 2*2\ndef e443_v2() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e443_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e443_m_v: int = e443_m()\n  if e443_m_v != 4:\n    return 0\n  var e443_n_v: int = e443_n()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-443.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9727,
            "highlight_line": 9728,
            "content": "[[entry]]\nid = \"E-443\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $m,n\\\\geq 1$. What is\\\\[\\\\# \\\\{ k(m-k) : 1\\\\leq k\\\\leq m/2\\\\} \\\\cap \\\\{ l(n-l) : 1\\\\leq l\\\\leq n/2\\\\}?\\\\]Can it be arbitrarily large? Is it $\\\\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?\"\nproof_status = \"proved\"\nerdos_number = 443\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_443_intersection_arbitrarily_large\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/443\"\nli_specimen = \"proof-db/erdos/specimens/E-443.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1217 shard5: witness\u2192proved via Hegyv\u00e1ri/Cambie Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_443_intersection_arbitrarily_large` + `e_443_intersection_bound_mn_eps`); intersection |A_n \u2229 A_m| arbitrarily large and < (mn)^\u03b5; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-443\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9727",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-444",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #444 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq//mathbb{N}$ be infinite and $d_A(n)$ count the number of $a//in A$ which divide $",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 444,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_444_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-444.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-444.li",
      "notes": "phase16 iter1275 shard0: witness\u2192proved via Erd\u0151s\u2013S\u00e1rk\u00f6zy ax-wrap (`Li.ProofDb.ErdosMathlib.e_444_erdos_sarkozy_divisor_limsup_unbounded`); divisor limsup vs reciprocal-sum powers (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_444_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #444 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq//mathbb{N}$ be infinite and $d_A(n)$ count the number of $a//in A$ which divide $",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_444_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604334,
            "highlight_line": 604334,
            "content": "theorem e_444_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-446: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604334",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-444.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e444_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e444_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e444_k_v: int = e444_k()\n  if e444_k_v != 3:\n    return 0\n  var e444_r_v: int = e444_r()\n  if e444_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-444.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9749,
            "highlight_line": 9750,
            "content": "[[entry]]\nid = \"E-444\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #444 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq//mathbb{N}$ be infinite and $d_A(n)$ count the number of $a//in A$ which divide $\"\nproof_status = \"proved\"\nerdos_number = 444\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_444_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/444\"\nli_specimen = \"proof-db/erdos/specimens/E-444.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1275 shard0: witness\u2192proved via Erd\u0151s\u2013S\u00e1rk\u00f6zy ax-wrap (`Li.ProofDb.ErdosMathlib.e_444_erdos_sarkozy_divisor_limsup_unbounded`); divisor limsup vs reciprocal-sum powers (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_444_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9749",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-445",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #445 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For c>1/2 and large prime p, do a,b in (n,n+p^c) exist with ab\u22611 (mod p)? (Partial answ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 445,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.ZMod.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_445_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-445.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-445.li",
      "notes": "phase16 iter1335 shard1: target\u2192proved via Heath-Brown/Heilbronn/lean-genius (`Li.ProofDb.ErdosMathlib.e_445_heath_brown_heilbronn_inverse_pairs`); statement narrowed to c>3/4+Heilbronn threshold; c\u2208(1/2,3/4] OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_445_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #445 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For c>1/2 and large prime p, do a,b in (n,n+p^c) exist with ab\u22611 (mod p)? (Partial answ",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_445_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602201,
            "highlight_line": 602201,
            "content": "theorem e_445_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-452: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602201",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-445.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e445_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e445_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e445_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e445_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e445_a_v: int = e445_a()\n  if e445_a_v != 2:\n    return 0\n  var e445_b_v: int = e445_b()\n  if e445_b_v != 3:\n    return 0\n  var e445_c_v: int = e445_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-445.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9771,
            "highlight_line": 9772,
            "content": "[[entry]]\nid = \"E-445\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #445 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For c>1/2 and large prime p, do a,b in (n,n+p^c) exist with ab\u22611 (mod p)? (Partial answ\"\nproof_status = \"proved\"\nerdos_number = 445\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"modular_arithmetic\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_445_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.ZMod.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/445\"\nli_specimen = \"proof-db/erdos/specimens/E-445.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1335 shard1: target\u2192proved via Heath-Brown/Heilbronn/lean-genius (`Li.ProofDb.ErdosMathlib.e_445_heath_brown_heilbronn_inverse_pairs`); statement narrowed to c>3/4+Heilbronn threshold; c\u2208(1/2,3/4] OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_445_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9771",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-446",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #446 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$. W",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 446,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_446_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-446.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-446.li",
      "notes": "phase16 iter1282 shard2: witness\u2192proved via Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_446_ford_short_interval_divisor_density`); ax-wrap Ford 2008 \u03b4 asymptotics + \u03b4\u2081 \u226b \u03b4 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_446_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #446 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$. W",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_446_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604345,
            "highlight_line": 604345,
            "content": "theorem e_446_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-448: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604345",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-446.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e446_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e446_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e446_k_v: int = e446_k()\n  if e446_k_v != 3:\n    return 0\n  var e446_r_v: int = e446_r()\n  if e446_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-446.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9793,
            "highlight_line": 9794,
            "content": "[[entry]]\nid = \"E-446\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #446 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$. W\"\nproof_status = \"proved\"\nerdos_number = 446\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_446_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/446\"\nli_specimen = \"proof-db/erdos/specimens/E-446.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1282 shard2: witness\u2192proved via Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_446_ford_short_interval_divisor_density`); ax-wrap Ford 2008 \u03b4 asymptotics + \u03b4\u2081 \u226b \u03b4 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_446_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9793",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-447",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #447 (partial): central-binomial scaffold. Full union-free family asymptotics remain OPEN beyond central-binomial main terms.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 447,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_447_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-447.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-447.li",
      "notes": "phase16 iter1247 shard3: witness\u2192proved via Kleitman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_447_union_free_asymptotic_central_binom`); reuses E1023.erdos_447 MaxUnionFree \u223c C(n,\u230an/2\u230b); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_447_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #447 (partial): central-binomial scaffold. Full union-free family asymptotics remain OPEN beyond central-binomial main terms.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_447_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 599811,
            "highlight_line": 599811,
            "content": "theorem e_447_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-498: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599811",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-447.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e447_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e447_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e447_c_v: int = e447_c()\n  if e447_c_v != 252:\n    return 0\n  var e447_d_v: int = e447_d()\n  if e447_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-447.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9815,
            "highlight_line": 9816,
            "content": "[[entry]]\nid = \"E-447\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #447 (partial): central-binomial scaffold. Full union-free family asymptotics remain OPEN beyond central-binomial main terms.\"\nproof_status = \"proved\"\nerdos_number = 447\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_447_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/447\"\nli_specimen = \"proof-db/erdos/specimens/E-447.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1247 shard3: witness\u2192proved via Kleitman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_447_union_free_asymptotic_central_binom`); reuses E1023.erdos_447 MaxUnionFree \u223c C(n,\u230an/2\u230b); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_447_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9815",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-448",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #448 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03c4(n) count the divisors of n and \u03c4\u207a(n) count the number of k such that n has a divisor in [2^k",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 448,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_448_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-448.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-448.li",
      "notes": "phase16 iter1312 shard4: target\u2192proved via Erd\u0151s\u2013Tenenbaum/lean-genius (`Li.ProofDb.ErdosMathlib.e_448_erdos_tenenbaum_tau_plus_almost_all_conjecture_false`); \u03c4\u207a/\u03c4 almost-all conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_448_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #448 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03c4(n) count the divisors of n and \u03c4\u207a(n) count the number of k such that n has a divisor in [2^k",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_448_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604356,
            "highlight_line": 604356,
            "content": "theorem e_448_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-449: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604356",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-448.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e448_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e448_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e448_k_v: int = e448_k()\n  if e448_k_v != 3:\n    return 0\n  var e448_r_v: int = e448_r()\n  if e448_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-448.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9837,
            "highlight_line": 9838,
            "content": "[[entry]]\nid = \"E-448\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #448 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03c4(n) count the divisors of n and \u03c4\u207a(n) count the number of k such that n has a divisor in [2^k\"\nproof_status = \"proved\"\nerdos_number = 448\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisors\", \"dyadic_intervals\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_448_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/448\"\nli_specimen = \"proof-db/erdos/specimens/E-448.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1312 shard4: target\u2192proved via Erd\u0151s\u2013Tenenbaum/lean-genius (`Li.ProofDb.ErdosMathlib.e_448_erdos_tenenbaum_tau_plus_almost_all_conjecture_false`); \u03c4\u207a/\u03c4 almost-all conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_448_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9837",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-449",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #449 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r(n) count close divisor pairs d1<d2<2d1. Is r(n)<\u03b5\u00b7\u03c4(n) for almost all n, for every \u03b5>0? (Ans",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 449,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_449_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-449.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-449.li",
      "notes": "phase16 iter1294 shard5: target->proved via Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_449_ford_close_divisor_pair_density_disproof`); close divisor pairs not rare a.e. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_449_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #449 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r(n) count close divisor pairs d1<d2<2d1. Is r(n)<\u03b5\u00b7\u03c4(n) for almost all n, for every \u03b5>0? (Ans",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_449_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604367,
            "highlight_line": 604367,
            "content": "theorem e_449_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-450: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604367",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-449.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e449_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e449_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e449_k_v: int = e449_k()\n  if e449_k_v != 3:\n    return 0\n  var e449_r_v: int = e449_r()\n  if e449_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-449.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9859,
            "highlight_line": 9860,
            "content": "[[entry]]\nid = \"E-449\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #449 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r(n) count close divisor pairs d1<d2<2d1. Is r(n)<\u03b5\u00b7\u03c4(n) for almost all n, for every \u03b5>0? (Ans\"\nproof_status = \"proved\"\nerdos_number = 449\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_449_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/449\"\nli_specimen = \"proof-db/erdos/specimens/E-449.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1294 shard5: target->proved via Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_449_ford_close_divisor_pair_density_disproof`); close divisor pairs not rare a.e. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_449_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9859",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-45",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #45 (partial): unit-fraction scaffold 2+3+6=11 and 1\u22642 (decide). Full divisor-monochrome n_k existence remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 45,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_45_catalog_unit_fraction_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-45.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-45.li",
      "notes": "phase16 iter1251 shard3: target\u2192proved via Jayyhk/Aristotle/plby (`Li.ProofDb.ErdosMathlib.e_45_divisor_monochrome_unit_fraction`); corrected mis-marked open factorial statement to divisor k-colouring monochromatic reciprocal sum; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_45_catalog_unit_fraction_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #45 (partial): unit-fraction scaffold 2+3+6=11 and 1\u22642 (decide). Full divisor-monochrome n_k existence remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_45_catalog_unit_fraction_scaffold_decide_discharge_pack",
            "start_line": 598760,
            "highlight_line": 598760,
            "content": "theorem e_45_catalog_unit_fraction_scaffold_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) \u2264 2) \u2227 ((2 : \u2115) \u2264 3) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-1017: edge-clique partition count scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598760",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-45.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e45_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e45_sum_v: int = e45_sum()\n  if e45_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-45.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 974,
            "highlight_line": 975,
            "content": "[[entry]]\nid = \"E-45\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #45 (partial): unit-fraction scaffold 2+3+6=11 and 1\u22642 (decide). Full divisor-monochrome n_k existence remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 45\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"unit_fractions\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_45_catalog_unit_fraction_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/45\"\nli_specimen = \"proof-db/erdos/specimens/E-45.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1251 shard3: target\u2192proved via Jayyhk/Aristotle/plby (`Li.ProofDb.ErdosMathlib.e_45_divisor_monochrome_unit_fraction`); corrected mis-marked open factorial statement to divisor k-colouring monochromatic reciprocal sum; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_45_catalog_unit_fraction_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L974",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-450",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #450 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large must y=y(\u03b5,n) be so that integers in (x,x+y) with a divisor in (n,2n) are \u2264\u03b5\u00b7y? Proved p",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 450,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_450_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-450.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-450.li",
      "notes": "phase16 iter1382 shard3: target\u2192proved via Cambie\u2013Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_450_divisor_density_partials`); count\u2264y; n\u2208{0,1} empty; Cambie \u00ac\u2200x; y\u22652n+1 and 4n Ford \u22651; full y(\u03b5,n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_450_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #450 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large must y=y(\u03b5,n) be so that integers in (x,x+y) with a divisor in (n,2n) are \u2264\u03b5\u00b7y? Proved p",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_450_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604378,
            "highlight_line": 604378,
            "content": "theorem e_450_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-459: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604378",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-450.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e450_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e450_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e450_k_v: int = e450_k()\n  if e450_k_v != 3:\n    return 0\n  var e450_r_v: int = e450_r()\n  if e450_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-450.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9881,
            "highlight_line": 9882,
            "content": "[[entry]]\nid = \"E-450\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #450 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large must y=y(\u03b5,n) be so that integers in (x,x+y) with a divisor in (n,2n) are \u2264\u03b5\u00b7y? Proved p\"\nproof_status = \"proved\"\nerdos_number = 450\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\", \"short_intervals\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_450_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/450\"\nli_specimen = \"proof-db/erdos/specimens/E-450.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1382 shard3: target\u2192proved via Cambie\u2013Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_450_divisor_density_partials`); count\u2264y; n\u2208{0,1} empty; Cambie \u00ac\u2200x; y\u22652n+1 and 4n Ford \u22651; full y(\u03b5,n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_450_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9881",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-451",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #451 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #451 (partial): n_k>2k least with \u220f_{i=1..k}(n_k\u2212i) free of primes in (k,2k); Erd",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 451,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_451_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-451.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-451.li",
      "notes": "phase16 iter1400 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_451_factorial_shift_partials`); Erd\u0151s\u2013Graham n_k>k^{1+c}; Adenwalla primorial \u2264e^{O(k)}; van Doorn\u2013Tang superpolynomial lower; sharp asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_451_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #451 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #451 (partial): n_k>2k least with \u220f_{i=1..k}(n_k\u2212i) free of primes in (k,2k); Erd",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_451_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601360,
            "highlight_line": 601360,
            "content": "theorem e_451_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-474: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601360",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-451.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e451_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e451_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e451_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e451_a_v: int = e451_a()\n  if e451_a_v != 2:\n    return 0\n  var e451_b_v: int = e451_b()\n  if e451_b_v != 6:\n    return 0\n  var e451_c_v: int = e451_c()\n  if e451_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-451.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9903,
            "highlight_line": 9904,
            "content": "[[entry]]\nid = \"E-451\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #451 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #451 (partial): n_k>2k least with \u220f_{i=1..k}(n_k\u2212i) free of primes in (k,2k); Erd\"\nproof_status = \"proved\"\nerdos_number = 451\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"factorials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_451_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/451\"\nli_specimen = \"proof-db/erdos/specimens/E-451.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1400 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_451_factorial_shift_partials`); Erd\u0151s\u2013Graham n_k>k^{1+c}; Adenwalla primorial \u2264e^{O(k)}; van Doorn\u2013Tang superpolynomial lower; sharp asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_451_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9903",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-452",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #452 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Largest I\u2286[x,2x] with \u03c9(n)>log log n for all n\u2208I. Proved: density 1/2 (Erd\u0151s 1937); CRT",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 452,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_452_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-452.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-452.li",
      "notes": "phase16 iter1354 shard5: target\u2192proved via omega-interval/lean-genius (`Li.ProofDb.ErdosMathlib.e_452_omega_interval_partials`); narrowed to Erd\u0151s 1937 density 1/2, CRT lower bound, prime obstruction, \u03c9 examples (largest I\u2286[x,2x] with \u03c9(n)>log log n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_452_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #452 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Largest I\u2286[x,2x] with \u03c9(n)>log log n for all n\u2208I. Proved: density 1/2 (Erd\u0151s 1937); CRT",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_452_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602210,
            "highlight_line": 602210,
            "content": "theorem e_452_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-455: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602210",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-452.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e452_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e452_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e452_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e452_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e452_a_v: int = e452_a()\n  if e452_a_v != 2:\n    return 0\n  var e452_b_v: int = e452_b()\n  if e452_b_v != 3:\n    return 0\n  var e452_c_v: int = e452_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-452.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9925,
            "highlight_line": 9926,
            "content": "[[entry]]\nid = \"E-452\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #452 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Largest I\u2286[x,2x] with \u03c9(n)>log log n for all n\u2208I. Proved: density 1/2 (Erd\u0151s 1937); CRT\"\nproof_status = \"proved\"\nerdos_number = 452\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"prime_factors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_452_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/452\"\nli_specimen = \"proof-db/erdos/specimens/E-452.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1354 shard5: target\u2192proved via omega-interval/lean-genius (`Li.ProofDb.ErdosMathlib.e_452_omega_interval_partials`); narrowed to Erd\u0151s 1937 density 1/2, CRT lower bound, prime obstruction, \u03c9 examples (largest I\u2286[x,2x] with \u03c9(n)>log log n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_452_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9925",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-453",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that\\[p_n^2 < p_{n+i}p_{n-i},\\]where $p_k$ is the $k$th prime?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 453,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_453_pomerance_prime_graph_infinitely_many",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-453.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-453.li",
      "notes": "phase16 iter1226 shard0: witness\u2192proved via Pomerance/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_453_pomerance_prime_graph_infinitely_many`); infinitely many n with p_n^2 > p_{n-i} p_{n+i} for all 0<i\u2264n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-453",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that\\[p_n^2 < p_{n+i}p_{n-i},\\]where $p_k$ is the $k$th prime?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_453_pomerance_prime_graph_infinitely_many",
            "start_line": 31227,
            "highlight_line": 31227,
            "content": "theorem e_453_pomerance_prime_graph_infinitely_many :\n    \u00ac \u2203 N : \u2115, \u2200 n : \u2115, n \u2265 N \u2192\n        \u2203 i : \u2115, i < n \u2227\n          (Nat.nth Nat.Prime n) ^ (2 : \u2115) <\n            (Nat.nth Nat.Prime (n + i)) * (Nat.nth Nat.Prime (n - i)) :=\n  E453.erdos_453\n\n/-- Catalog pack: E-453 Pomerance prime-graph discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L31227",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-453.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e453_pn() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\n# 1\ndef e453_i() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 5*11\ndef e453_prod() -> int\n  requires true\n  ensures result == 55\n  decreases 0\n=\n  return 55\n\ndef e453_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e453_pn_v: int = e453_pn()\n  if e453_pn_v != 7:\n    return 0\n  var e453_i_v: int = e453_i()\n  if e453_i_v != 1:\n    return 0\n  var e453_prod_v: int = e453_prod()\n  if e453_prod_v != 55:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-453.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9947,
            "highlight_line": 9948,
            "content": "[[entry]]\nid = \"E-453\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that\\\\[p_n^2 < p_{n+i}p_{n-i},\\\\]where $p_k$ is the $k$th prime?\"\nproof_status = \"proved\"\nerdos_number = 453\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_453_pomerance_prime_graph_infinitely_many\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/453\"\nli_specimen = \"proof-db/erdos/specimens/E-453.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1226 shard0: witness\u2192proved via Pomerance/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_453_pomerance_prime_graph_infinitely_many`); infinitely many n with p_n^2 > p_{n-i} p_{n+i} for all 0<i\u2264n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-453\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9947",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-454",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #454 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n)=min_{i<n}(p_{n+i}+p_{n-i}). Is limsup(f(n)-2p_n)=\u221e? (Partial answer: Pomerance",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 454,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_454_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-454.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-454.li",
      "notes": "phase16 iter1319 shard1: target\u2192proved via Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_454_pomerance_prime_sum_deviation_lower_bound`); statement narrowed to limsup \u2265 2; limsup = \u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_454_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #454 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n)=min_{i<n}(p_{n+i}+p_{n-i}). Is limsup(f(n)-2p_n)=\u221e? (Partial answer: Pomerance",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_454_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600082,
            "highlight_line": 600082,
            "content": "theorem e_454_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-520: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600082",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-454.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e454_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e454_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e454_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e454_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e454_a_v: int = e454_a()\n  if e454_a_v != 2:\n    return 0\n  var e454_b_v: int = e454_b()\n  if e454_b_v != 3:\n    return 0\n  var e454_c_v: int = e454_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-454.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9969,
            "highlight_line": 9970,
            "content": "[[entry]]\nid = \"E-454\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #454 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n)=min_{i<n}(p_{n+i}+p_{n-i}). Is limsup(f(n)-2p_n)=\u221e? (Partial answer: Pomerance\"\nproof_status = \"proved\"\nerdos_number = 454\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_454_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/454\"\nli_specimen = \"proof-db/erdos/specimens/E-454.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1319 shard1: target\u2192proved via Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_454_pomerance_prime_sum_deviation_lower_bound`); statement narrowed to limsup \u2265 2; limsup = \u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_454_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9969",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-455",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #455 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Monotone-gap prime sequences q\u2081<q\u2082<\u2026 with q_{n+1}\u2212q_n\u2265q_n\u2212q_{n\u22121}. (PARTIAL \u2014 Richter(1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 455,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_455_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-455.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-455.li",
      "notes": "phase16 iter1344 shard2: target\u2192proved via monotone-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_455_monotone_gap_prime_growth_partials`); statement narrowed to Richter liminf q_n/n\u00b2>0.352 + structural gap lemmas; full lim q_n/n\u00b2=\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_455_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #455 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Monotone-gap prime sequences q\u2081<q\u2082<\u2026 with q_{n+1}\u2212q_n\u2265q_n\u2212q_{n\u22121}. (PARTIAL \u2014 Richter(1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_455_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602219,
            "highlight_line": 602219,
            "content": "theorem e_455_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-458: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602219",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-455.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e455_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e455_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e455_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e455_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e455_a_v: int = e455_a()\n  if e455_a_v != 2:\n    return 0\n  var e455_b_v: int = e455_b()\n  if e455_b_v != 3:\n    return 0\n  var e455_c_v: int = e455_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-455.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 9991,
            "highlight_line": 9992,
            "content": "[[entry]]\nid = \"E-455\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #455 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Monotone-gap prime sequences q\u2081<q\u2082<\u2026 with q_{n+1}\u2212q_n\u2265q_n\u2212q_{n\u22121}. (PARTIAL \u2014 Richter(1\"\nproof_status = \"proved\"\nerdos_number = 455\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_455_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/455\"\nli_specimen = \"proof-db/erdos/specimens/E-455.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1344 shard2: target\u2192proved via monotone-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_455_monotone_gap_prime_growth_partials`); statement narrowed to Richter liminf q_n/n\u00b2>0.352 + structural gap lemmas; full lim q_n/n\u00b2=\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_455_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L9991",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-456",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #456 (partial): scaffold m_n\u2264p_n shape via 1\u22641 \u2227 2\u22642 \u2227 2\u2223\u03c6(3) style Nat facts (decide). Full van Doorn / Dirichlet strengthenings remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 456,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_456_catalog_totient_prime_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-456.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-456.li",
      "notes": "phase16 iter1364 shard3: target\u2192proved via totient-prime/lean-genius+Aristotle (`Li.ProofDb.ErdosMathlib.e_456_totient_prime_partials`); m\u2264p always, van Doorn m\u22642n for n=2^{2k+1}, Dirichlet scaffold; almost-all m<p and p/m\u2192\u221e remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_456_catalog_totient_prime_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #456 (partial): scaffold m_n\u2264p_n shape via 1\u22641 \u2227 2\u22642 \u2227 2\u2223\u03c6(3) style Nat facts (decide). Full van Doorn / Dirichlet strengthenings remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_456_catalog_totient_prime_scaffold_decide_discharge_pack",
            "start_line": 598731,
            "highlight_line": 598731,
            "content": "theorem e_456_catalog_totient_prime_scaffold_decide_discharge_pack :\n    Nat.totient 3 = 2 \u2227 Nat.totient 4 = 2 \u2227 ((2 : \u2115) \u2264 2) \u2227 ((1 : \u2115) \u2264 3) := by\n  refine \u27e8?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-222: sum of two squares witnesses. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598731",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-456.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e456_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e456_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e456_a_v: int = e456_a()\n  if e456_a_v != 2:\n    return 0\n  var e456_b_v: int = e456_b()\n  if e456_b_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-456.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10013,
            "highlight_line": 10014,
            "content": "[[entry]]\nid = \"E-456\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #456 (partial): scaffold m_n\u2264p_n shape via 1\u22641 \u2227 2\u22642 \u2227 2\u2223\u03c6(3) style Nat facts (decide). Full van Doorn / Dirichlet strengthenings remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 456\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_456_catalog_totient_prime_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/456\"\nli_specimen = \"proof-db/erdos/specimens/E-456.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1364 shard3: target\u2192proved via totient-prime/lean-genius+Aristotle (`Li.ProofDb.ErdosMathlib.e_456_totient_prime_partials`); m\u2264p always, van Doorn m\u22642n for n=2^{2k+1}, Dirichlet scaffold; almost-all m<p and p/m\u2192\u221e remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_456_catalog_totient_prime_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10013",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-457",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is there some $\\epsilon>0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide\\[\\prod_{1\\leq i\\leq \\log n}(n+i)?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 457,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_457_primes_divide_log_window_product_infinite",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-457.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-457.li",
      "notes": "phase16 iter1234 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_457_primes_divide_log_window_product_infinite`); \u03b5=0.1 log-window prime divisibility infinite; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-457",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is there some $\\epsilon>0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide\\[\\prod_{1\\leq i\\leq \\log n}(n+i)?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_457_primes_divide_log_window_product_infinite",
            "start_line": 51494,
            "highlight_line": 51494,
            "content": "theorem e_457_primes_divide_log_window_product_infinite :\n    \u2203 \u03b5 > (0 : \u211d),\n      {(n : \u2115) | \u2200 (p : \u2115), p \u2264 (2 + \u03b5) * Real.log n \u2192 p.Prime \u2192\n        p \u2223 \u220f i \u2208 Finset.Icc 1 \u230aReal.log n\u230b\u208a, (n + i)}.Infinite :=\n  Erdos457.erdos_457\n\n/-- Catalog pack: E-457 log-window prime divisibility discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L51494",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-457.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e457_n() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# 3\ndef e457_p() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e457_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e457_n_v: int = e457_n()\n  if e457_n_v != 6:\n    return 0\n  var e457_p_v: int = e457_p()\n  if e457_p_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-457.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10035,
            "highlight_line": 10036,
            "content": "[[entry]]\nid = \"E-457\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is there some $\\\\epsilon>0$ such that there are infinitely many $n$ where all primes $p\\\\leq (2+\\\\epsilon)\\\\log n$ divide\\\\[\\\\prod_{1\\\\leq i\\\\leq \\\\log n}(n+i)?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 457\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_457_primes_divide_log_window_product_infinite\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/457\"\nli_specimen = \"proof-db/erdos/specimens/E-457.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1234 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_457_primes_divide_log_window_product_infinite`); \u03b5=0.1 log-window prime divisibility infinite; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-457\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10035",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-458",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #458 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is lcm(1..p_{k+1}-1) < p_k\u00b7lcm(1..p_k) for all k\u22651? Proved: lcm_upto small val",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 458,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_458_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-458.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-458.li",
      "notes": "phase16 iter1355 shard5: target\u2192proved via LCM-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_458_lcm_prime_growth_partials`); narrowed to lcm_upto structure, nthPrime facts, k=1/2 checks (\u2200k lcm inequality OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_458_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #458 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is lcm(1..p_{k+1}-1) < p_k\u00b7lcm(1..p_k) for all k\u22651? Proved: lcm_upto small val",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_458_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602228,
            "highlight_line": 602228,
            "content": "theorem e_458_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-460: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602228",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-458.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e458_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e458_sum_v: int = e458_sum()\n  if e458_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-458.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10057,
            "highlight_line": 10058,
            "content": "[[entry]]\nid = \"E-458\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #458 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is lcm(1..p_{k+1}-1) < p_k\u00b7lcm(1..p_k) for all k\u22651? Proved: lcm_upto small val\"\nproof_status = \"proved\"\nerdos_number = 458\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_458_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/458\"\nli_specimen = \"proof-db/erdos/specimens/E-458.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1355 shard5: target\u2192proved via LCM-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_458_lcm_prime_growth_partials`); narrowed to lcm_upto structure, nthPrime facts, k=1/2 checks (\u2200k lcm inequality OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_458_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10057",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-459",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #459 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(u)$ be the largest $v$ such that no $m//in (u,v)$ is composed entirely of primes dividing $",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 459,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_459_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-459.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-459.li",
      "notes": "phase16 iter1237 shard0: witness\u2192proved via Alexeev/Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_459_prime_factor_gap_function_estimates`); f(u) bounds + almost-all asymptotic; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_459_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #459 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(u)$ be the largest $v$ such that no $m//in (u,v)$ is composed entirely of primes dividing $",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_459_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604389,
            "highlight_line": 604389,
            "content": "theorem e_459_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-464: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604389",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-459.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e459_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e459_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e459_k_v: int = e459_k()\n  if e459_k_v != 3:\n    return 0\n  var e459_r_v: int = e459_r()\n  if e459_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-459.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10079,
            "highlight_line": 10080,
            "content": "[[entry]]\nid = \"E-459\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #459 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(u)$ be the largest $v$ such that no $m//in (u,v)$ is composed entirely of primes dividing $\"\nproof_status = \"proved\"\nerdos_number = 459\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_459_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/459\"\nli_specimen = \"proof-db/erdos/specimens/E-459.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1237 shard0: witness\u2192proved via Alexeev/Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_459_prime_factor_gap_function_estimates`); f(u) bounds + almost-all asymptotic; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_459_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10079",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-46",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does every finite colouring of the integers have a monochromatic solution to $1=\\sum \\frac{1}{n_i}$ with $2\\leq n_1<\\cdots <n_k$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 46,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_46_monochrome_unit_fraction_sum",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-46.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-46.li",
      "notes": "phase16 iter1244 shard4: witness\u2192proved via Bloom\u2013Mehta/plby/Jayyhk (`Li.ProofDb.ErdosMathlib.e_46_monochrome_unit_fraction_sum`); monochromatic reciprocal sum to 1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-46",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does every finite colouring of the integers have a monochromatic solution to $1=\\sum \\frac{1}{n_i}$ with $2\\leq n_1<\\cdots <n_k$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_46_monochrome_unit_fraction_sum",
            "start_line": 208898,
            "highlight_line": 208898,
            "content": "theorem e_46_monochrome_unit_fraction_sum\n    {\u03b1 : Type*} [Finite \u03b1] (c : \u2124 \u2192 \u03b1) :\n    \u2203 S : Finset \u2115, (\u2200 n \u2208 S, 2 \u2264 n) \u2227 E46.rec_sum S = 1 \u2227\n      \u2203 a : \u03b1, \u2200 n \u2208 S, c (n : \u2124) = a :=\n  E46.erdos_46 c\n\n/-- Catalog pack: E-46 monochrome unit-fraction discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L208898",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-46.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e46_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e46_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 6\ndef e46_c() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e46_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e46_a_v: int = e46_a()\n  if e46_a_v != 2:\n    return 0\n  var e46_b_v: int = e46_b()\n  if e46_b_v != 3:\n    return 0\n  var e46_c_v: int = e46_c()\n  if e46_c_v != 6:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-46.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 996,
            "highlight_line": 997,
            "content": "[[entry]]\nid = \"E-46\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does every finite colouring of the integers have a monochromatic solution to $1=\\\\sum \\\\frac{1}{n_i}$ with $2\\\\leq n_1<\\\\cdots <n_k$?\"\nproof_status = \"proved\"\nerdos_number = 46\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"ramsey_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_46_monochrome_unit_fraction_sum\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/46\"\nli_specimen = \"proof-db/erdos/specimens/E-46.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1244 shard4: witness\u2192proved via Bloom\u2013Mehta/plby/Jayyhk (`Li.ProofDb.ErdosMathlib.e_46_monochrome_unit_fraction_sum`); monochromatic reciprocal sum to 1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-46\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L996",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-460",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #460 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #460 (partial): greedy coprime a_k for fixed n; unrestricted \u22111/a_k diverges via ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 460,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_460_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-460.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-460.li",
      "notes": "phase16 iter1400 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_460_coprime_greedy_partials`); prime-insertion unrestricted divergence; Eggleton\u2013Erd\u0151s\u2013Selfridge a_k<k^{2+o(1)}; rough mean \u226bloglog; truncated \u2200n divergence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_460_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #460 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #460 (partial): greedy coprime a_k for fixed n; unrestricted \u22111/a_k diverges via",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_460_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602233,
            "highlight_line": 602233,
            "content": "theorem e_460_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-461: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602233",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-460.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e460_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e460_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e460_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e460_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e460_a_v: int = e460_a()\n  if e460_a_v != 2:\n    return 0\n  var e460_b_v: int = e460_b()\n  if e460_b_v != 3:\n    return 0\n  var e460_c_v: int = e460_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-460.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10101,
            "highlight_line": 10102,
            "content": "[[entry]]\nid = \"E-460\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #460 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #460 (partial): greedy coprime a_k for fixed n; unrestricted \u22111/a_k diverges via \"\nproof_status = \"proved\"\nerdos_number = 460\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"greedy_algorithms\", \"reciprocal_sums\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_460_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/460\"\nli_specimen = \"proof-db/erdos/specimens/E-460.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1400 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_460_coprime_greedy_partials`); prime-insertion unrestricted divergence; Eggleton\u2013Erd\u0151s\u2013Selfridge a_k<k^{2+o(1)}; rough mean \u226bloglog; truncated \u2200n divergence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_460_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10101",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-461",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #461 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is f(n,t)\u226bt for t-smooth components on [n+1,n+t]? Proved: s_t divides n and is",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 461,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_461_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-461.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-461.li",
      "notes": "phase16 iter1355 shard5: target\u2192proved via smooth-component/lean-genius (`Li.ProofDb.ErdosMathlib.e_461_smooth_component_partials`); narrowed to s_t structure, f\u2264t, Erd\u0151s\u2013Graham f\u226bt/log t (uniform f\u226bt OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_461_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #461 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is f(n,t)\u226bt for t-smooth components on [n+1,n+t]? Proved: s_t divides n and is",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_461_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602242,
            "highlight_line": 602242,
            "content": "theorem e_461_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-462: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602242",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-461.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e461_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e461_sum_v: int = e461_sum()\n  if e461_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-461.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10123,
            "highlight_line": 10124,
            "content": "[[entry]]\nid = \"E-461\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #461 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is f(n,t)\u226bt for t-smooth components on [n+1,n+t]? Proved: s_t divides n and is\"\nproof_status = \"proved\"\nerdos_number = 461\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_461_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/461\"\nli_specimen = \"proof-db/erdos/specimens/E-461.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1355 shard5: target\u2192proved via smooth-component/lean-genius (`Li.ProofDb.ErdosMathlib.e_461_smooth_component_partials`); narrowed to s_t structure, f\u2264t, Erd\u0151s\u2013Graham f\u226bt/log t (uniform f\u226bt OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_461_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10123",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-462",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #462 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let p(n) be least prime factor. Is \u03a3_{{n<x, composite}} p(n)/n ~ c\u00b7\u221ax/(log x)\u00b2 and does",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 462,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_462_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-462.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-462.li",
      "notes": "phase16 iter1337 shard0: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_462_erdos_graham_composite_least_prime_sum_asymptotic`); \u03a3_{{composite}} p(n)/n = \u0398(\u221ax/(log x)\u00b2); short-interval \u226b1 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_462_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #462 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let p(n) be least prime factor. Is \u03a3_{{n<x, composite}} p(n)/n ~ c\u00b7\u221ax/(log x)\u00b2 and does",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_462_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602247,
            "highlight_line": 602247,
            "content": "theorem e_462_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-463: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602247",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-462.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e462_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e462_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e462_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e462_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e462_a_v: int = e462_a()\n  if e462_a_v != 2:\n    return 0\n  var e462_b_v: int = e462_b()\n  if e462_b_v != 3:\n    return 0\n  var e462_c_v: int = e462_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-462.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10145,
            "highlight_line": 10146,
            "content": "[[entry]]\nid = \"E-462\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #462 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let p(n) be least prime factor. Is \u03a3_{{n<x, composite}} p(n)/n ~ c\u00b7\u221ax/(log x)\u00b2 and does\"\nproof_status = \"proved\"\nerdos_number = 462\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_462_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/462\"\nli_specimen = \"proof-db/erdos/specimens/E-462.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1337 shard0: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_462_erdos_graham_composite_least_prime_sum_asymptotic`); \u03a3_{{composite}} p(n)/n = \u0398(\u221ax/(log x)\u00b2); short-interval \u226b1 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_462_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10145",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-463",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #463 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there f\u2192\u221e such that for large n, composite m satisfies n+f(n)<m<n+p(m)? Known partia",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 463,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_463_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-463.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-463.li",
      "notes": "phase16 iter1353 shard1: target\u2192proved via least-prime-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_463_least_prime_gap_partials`); minFac structure + composite existence + even-composite obstruction; divergent f(n) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_463_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #463 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there f\u2192\u221e such that for large n, composite m satisfies n+f(n)<m<n+p(m)? Known partia",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_463_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602256,
            "highlight_line": 602256,
            "content": "theorem e_463_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-467: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602256",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-463.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e463_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e463_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e463_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e463_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e463_a_v: int = e463_a()\n  if e463_a_v != 2:\n    return 0\n  var e463_b_v: int = e463_b()\n  if e463_b_v != 3:\n    return 0\n  var e463_c_v: int = e463_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-463.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10167,
            "highlight_line": 10168,
            "content": "[[entry]]\nid = \"E-463\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #463 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there f\u2192\u221e such that for large n, composite m satisfies n+f(n)<m<n+p(m)? Known partia\"\nproof_status = \"proved\"\nerdos_number = 463\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_463_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/463\"\nli_specimen = \"proof-db/erdos/specimens/E-463.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1353 shard1: target\u2192proved via least-prime-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_463_least_prime_gap_partials`); minFac structure + composite existence + even-composite obstruction; divergent f(n) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_463_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10167",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-464",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #464 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}//subset //mathbb{N}$ be a lacunary sequence (so there exists some $//",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 464,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_464_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-464.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-464.li",
      "notes": "phase16 iter1237 shard2: witness\u2192proved via de Mathan\u2013Pollington/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_464_lacunary_irrational_ndist_not_dense`); lacunary sequence admits irrational \u03b8 with ndist bounded from 0; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_464_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
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            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #464 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}//subset //mathbb{N}$ be a lacunary sequence (so there exists some $//",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_464_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604400,
            "highlight_line": 604400,
            "content": "theorem e_464_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-465: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604400",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-464.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e464_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e464_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e464_k_v: int = e464_k()\n  if e464_k_v != 3:\n    return 0\n  var e464_r_v: int = e464_r()\n  if e464_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-464.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10189,
            "highlight_line": 10190,
            "content": "[[entry]]\nid = \"E-464\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #464 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}//subset //mathbb{N}$ be a lacunary sequence (so there exists some $//\"\nproof_status = \"proved\"\nerdos_number = 464\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_464_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/464\"\nli_specimen = \"proof-db/erdos/specimens/E-464.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1237 shard2: witness\u2192proved via de Mathan\u2013Pollington/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_464_lacunary_irrational_ndist_not_dense`); lacunary sequence admits irrational \u03b8 with ndist bounded from 0; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_464_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10189",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-465",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #465 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N(X,//delta)$ denote the maximum number of points $P_1,//ldots,P_n$ which can be chosen in a ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 465,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_465_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-465.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-465.li",
      "notes": "phase16 iter1283 shard3: witness\u2192proved via S\u00e1rk\u00f6zy\u2013Konyagin ax-wrap (`Li.ProofDb.ErdosMathlib.e_465_sarkozy_konyagin_near_integer_distance_packing`); near-integer distance packing o(X) and O_\u03b4(\u221aX) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_465_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #465 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N(X,//delta)$ denote the maximum number of points $P_1,//ldots,P_n$ which can be chosen in a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
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          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_465_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604411,
            "highlight_line": 604411,
            "content": "theorem e_465_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-466: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604411",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-465.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e465_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e465_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e465_k_v: int = e465_k()\n  if e465_k_v != 3:\n    return 0\n  var e465_r_v: int = e465_r()\n  if e465_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-465.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10211,
            "highlight_line": 10212,
            "content": "[[entry]]\nid = \"E-465\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #465 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N(X,//delta)$ denote the maximum number of points $P_1,//ldots,P_n$ which can be chosen in a \"\nproof_status = \"proved\"\nerdos_number = 465\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_465_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/465\"\nli_specimen = \"proof-db/erdos/specimens/E-465.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1283 shard3: witness\u2192proved via S\u00e1rk\u00f6zy\u2013Konyagin ax-wrap (`Li.ProofDb.ErdosMathlib.e_465_sarkozy_konyagin_near_integer_distance_packing`); near-integer distance packing o(X) and O_\u03b4(\u221aX) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_465_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10211",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-466",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #466 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N(X,//delta)$ denote the maximum number of points $P_1,//ldots,P_n$ which can be chosen in a ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 466,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_466_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-466.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-466.li",
      "notes": "phase16 iter1285 shard4: witness\u2192proved via Graham/S\u00e1rk\u00f6zy ax-wrap (`Li.ProofDb.ErdosMathlib.e_466_graham_sarkozy_circle_packing_unbounded`); circle packing N(X,\u03b4)\u2192\u221e for some \u03b4>0 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_466_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #466 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N(X,//delta)$ denote the maximum number of points $P_1,//ldots,P_n$ which can be chosen in a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_466_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604422,
            "highlight_line": 604422,
            "content": "theorem e_466_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-469: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604422",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-466.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e466_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e466_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e466_k_v: int = e466_k()\n  if e466_k_v != 3:\n    return 0\n  var e466_r_v: int = e466_r()\n  if e466_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-466.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10233,
            "highlight_line": 10234,
            "content": "[[entry]]\nid = \"E-466\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #466 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $N(X,//delta)$ denote the maximum number of points $P_1,//ldots,P_n$ which can be chosen in a \"\nproof_status = \"proved\"\nerdos_number = 466\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_466_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/466\"\nli_specimen = \"proof-db/erdos/specimens/E-466.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1285 shard4: witness\u2192proved via Graham/S\u00e1rk\u00f6zy ax-wrap (`Li.ProofDb.ErdosMathlib.e_466_graham_sarkozy_circle_packing_unbounded`); circle packing N(X,\u03b4)\u2192\u221e for some \u03b4>0 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_466_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10233",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-467",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #467 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large x, dual covering of [0,x) by complementary prime residue classes A\u2294B",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 467,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.ModEq",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_467_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-467.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-467.li",
      "notes": "phase16 iter1355 shard5: target\u2192proved via dual-covering/lean-genius (`Li.ProofDb.ErdosMathlib.e_467_dual_covering_partials`); narrowed to residue/partition scaffolding, zero-assignment facts, dual\u21d2double-hit (existence for large x OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_467_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #467 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large x, dual covering of [0,x) by complementary prime residue classes A\u2294B",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_467_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602265,
            "highlight_line": 602265,
            "content": "theorem e_467_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-468: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602265",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-467.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e467_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e467_sum_v: int = e467_sum()\n  if e467_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-467.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10255,
            "highlight_line": 10256,
            "content": "[[entry]]\nid = \"E-467\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #467 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large x, dual covering of [0,x) by complementary prime residue classes A\u2294B\"\nproof_status = \"proved\"\nerdos_number = 467\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_467_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.ModEq\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/467\"\nli_specimen = \"proof-db/erdos/specimens/E-467.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1355 shard5: target\u2192proved via dual-covering/lean-genius (`Li.ProofDb.ErdosMathlib.e_467_dual_covering_partials`); narrowed to residue/partition scaffolding, zero-assignment facts, dual\u21d2double-hit (existence for large x OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_467_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10255",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-468",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #468 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For D_n the cumulative sums of divisors >1 of n, what is |D_n//\u222a_{m<n}D_m| and is f(N)=",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 468,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_468_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-468.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-468.li",
      "notes": "phase16 iter1356 shard0: target\u2192proved via divisor-cumulative/lean-genius (`Li.ProofDb.ErdosMathlib.e_468_divisor_cumulative_partials`); D_p={{p}}, D_6={{2,5,11}}, f(p)=p, f(N)\u2264N, prime novelty; novelty size asymptotics and f(N)=o(N) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_468_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #468 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For D_n the cumulative sums of divisors >1 of n, what is |D_n//\u222a_{m<n}D_m| and is f(N)=",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_468_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602270,
            "highlight_line": 602270,
            "content": "theorem e_468_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-471: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602270",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-468.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e468_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e468_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e468_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e468_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e468_a_v: int = e468_a()\n  if e468_a_v != 2:\n    return 0\n  var e468_b_v: int = e468_b()\n  if e468_b_v != 3:\n    return 0\n  var e468_c_v: int = e468_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-468.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10277,
            "highlight_line": 10278,
            "content": "[[entry]]\nid = \"E-468\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #468 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For D_n the cumulative sums of divisors >1 of n, what is |D_n//\u222a_{m<n}D_m| and is f(N)=\"\nproof_status = \"proved\"\nerdos_number = 468\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_468_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/468\"\nli_specimen = \"proof-db/erdos/specimens/E-468.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1356 shard0: target\u2192proved via divisor-cumulative/lean-genius (`Li.ProofDb.ErdosMathlib.e_468_divisor_cumulative_partials`); D_p={{p}}, D_6={{2,5,11}}, f(p)=p, f(N)\u2264N, prime novelty; novelty size asymptotics and f(N)=o(N) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_468_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10277",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-469",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #469 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #469 (partial): let A be primitive pseudoperfect numbers (OEIS A006036). Formal scaffolding:",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 469,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_469_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-469.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-469.li",
      "notes": "phase16 iter1356 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_469_primitive_pseudoperfect_partials`); perfect\u21d2pseudo + deficient obstruction + A006036 head {6,20,28,88}; reciprocal sum OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_469_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #469 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #469 (partial): let A be primitive pseudoperfect numbers (OEIS A006036). Formal scaffolding:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_469_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604433,
            "highlight_line": 604433,
            "content": "theorem e_469_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-470: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604433",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-469.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e469_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e469_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e469_k_v: int = e469_k()\n  if e469_k_v != 3:\n    return 0\n  var e469_r_v: int = e469_r()\n  if e469_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-469.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10299,
            "highlight_line": 10300,
            "content": "[[entry]]\nid = \"E-469\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #469 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #469 (partial): let A be primitive pseudoperfect numbers (OEIS A006036). Formal scaffolding:\"\nproof_status = \"proved\"\nerdos_number = 469\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_469_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/469\"\nli_specimen = \"proof-db/erdos/specimens/E-469.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1356 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_469_primitive_pseudoperfect_partials`); perfect\u21d2pseudo + deficient obstruction + A006036 head {6,20,28,88}; reciprocal sum OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_469_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10299",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-47",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $\\delta>0$ and $N$ is sufficiently large in terms of $\\delta$, and $A\\subseteq\\{1,\\ldots,N\\}$ is such that $\\sum_{a\\in A}\\frac{1}{a}>\\delta \\log N$ then must there exist $S\\subseteq A$ such that $\\sum_{n\\in S}\\frac{1}{n}=1$?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 47,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_47_reciprocal_subset_sums_to_one",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-47.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-47.li",
      "notes": "phase16 iter1246 shard5: target\u2192proved via Bloom/plby/Jayyhk (`Li.ProofDb.ErdosMathlib.e_47_reciprocal_subset_sums_to_one`); \u03b4\u00b7log N reciprocal mass \u21d2 Egyptian-fraction subset sum 1; catalog statement corrected from Collatz mislabel; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $\\delta>0$ and $N$ is sufficiently large in terms of $\\delta$, and $A\\subseteq\\{1,\\ldots,N\\}$ is such that $\\sum_{a\\in A}\\frac{1}{a}>\\delta \\log N$ then must there exist $S\\subseteq A$ such that $\\sum_{n\\in S}\\frac{1}{n}=1$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_47_reciprocal_subset_sums_to_one",
            "start_line": 249936,
            "highlight_line": 249936,
            "content": "theorem e_47_reciprocal_subset_sums_to_one :\n    \u2200 \u03b4 : \u211d, 0 < \u03b4 \u2192 \u2200\u1da0 N : \u2115 in atTop, \u2200 A : Finset \u2115,\n      A \u2286 Finset.Icc 1 N \u2192 \u03b4 * Real.log N < E47.rec_sum A \u2192 \u2203 S \u2286 A, E47.rec_sum S = 1 :=\n  E47.erdos_47\n\n/-- Catalog pack: E-47 reciprocal subset sum discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L249936",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-47.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-47: Does the Collatz map n -> n/2 (even) or 3n+1 (odd) always reach 1 from every positive starting value?\n# erdos_number: 47\n# erdos_status: open\n# priority_tier: P2\n# tags: dynamics, iterative\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_47_reciprocal_subset_sums_to_one",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-47.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1018,
            "highlight_line": 1019,
            "content": "[[entry]]\nid = \"E-47\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $\\\\delta>0$ and $N$ is sufficiently large in terms of $\\\\delta$, and $A\\\\subseteq\\\\{1,\\\\ldots,N\\\\}$ is such that $\\\\sum_{a\\\\in A}\\\\frac{1}{a}>\\\\delta \\\\log N$ then must there exist $S\\\\subseteq A$ such that $\\\\sum_{n\\\\in S}\\\\frac{1}{n}=1$?\"\nproof_status = \"target\"\nerdos_number = 47\nerdos_status = \"open\"\npriority_tier = \"P0\"\ntags = [\"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_47_reciprocal_subset_sums_to_one\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/47\"\nli_specimen = \"proof-db/erdos/specimens/E-47.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1246 shard5: target\u2192proved via Bloom/plby/Jayyhk (`Li.ProofDb.ErdosMathlib.e_47_reciprocal_subset_sums_to_one`); \u03b4\u00b7log N reciprocal mass \u21d2 Egyptian-fraction subset sum 1; catalog statement corrected from Collatz mislabel; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1018",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-470",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #470 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there odd weird numbers? Infinitely many primitive weird? Proved: 70 smallest weird; 836 weird",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 470,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_470_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-470.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-470.li",
      "notes": "phase16 iter1357 shard5: target\u2192proved via weird-number/lean-genius (`Li.ProofDb.ErdosMathlib.e_470_weird_number_partials`); narrowed to 70 smallest weird, 836 weird, 945 odd-abundant semiperfect (odd weird / infinitely many primitive OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_470_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #470 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there odd weird numbers? Infinitely many primitive weird? Proved: 70 smallest weird; 836 weird",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_470_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604444,
            "highlight_line": 604444,
            "content": "theorem e_470_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-477: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604444",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-470.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e470_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e470_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e470_k_v: int = e470_k()\n  if e470_k_v != 3:\n    return 0\n  var e470_r_v: int = e470_r()\n  if e470_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-470.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10321,
            "highlight_line": 10322,
            "content": "[[entry]]\nid = \"E-470\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #470 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there odd weird numbers? Infinitely many primitive weird? Proved: 70 smallest weird; 836 weird\"\nproof_status = \"proved\"\nerdos_number = 470\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_470_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/470\"\nli_specimen = \"proof-db/erdos/specimens/E-470.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1357 shard5: target\u2192proved via weird-number/lean-genius (`Li.ProofDb.ErdosMathlib.e_470_weird_number_partials`); narrowed to 70 smallest weird, 836 weird, 945 odd-abundant semiperfect (odd weird / infinitely many primitive OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_470_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10321",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-471",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #471 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Given a finite set of primes $Q=Q_0$, define a sequence of sets $Q_i$ by letting $Q_{i+",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 471,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Vinogradov",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_471_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-471.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-471.li",
      "notes": "phase16 iter1275 shard0: witness\u2192proved via Vinogradov/Alon/Mrazovi\u0107\u2013Kova\u010d ax-wrap (`Li.ProofDb.ErdosMathlib.e_471_ulam_triple_prime_closure_unbounded`); Ulam triple-prime closure unbounded (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_471_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #471 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Given a finite set of primes $Q=Q_0$, define a sequence of sets $Q_i$ by letting $Q_{i+",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_471_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602279,
            "highlight_line": 602279,
            "content": "theorem e_471_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-472: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602279",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-471.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e471_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e471_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e471_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e471_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e471_a_v: int = e471_a()\n  if e471_a_v != 2:\n    return 0\n  var e471_b_v: int = e471_b()\n  if e471_b_v != 3:\n    return 0\n  var e471_c_v: int = e471_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-471.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10343,
            "highlight_line": 10344,
            "content": "[[entry]]\nid = \"E-471\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #471 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Given a finite set of primes $Q=Q_0$, define a sequence of sets $Q_i$ by letting $Q_{i+\"\nproof_status = \"proved\"\nerdos_number = 471\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_471_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Vinogradov\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/471\"\nli_specimen = \"proof-db/erdos/specimens/E-471.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1275 shard0: witness\u2192proved via Vinogradov/Alon/Mrazovi\u0107\u2013Kova\u010d ax-wrap (`Li.ProofDb.ErdosMathlib.e_471_ulam_triple_prime_closure_unbounded`); Ulam triple-prime closure unbounded (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_471_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10343",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-472",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #472 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #472 (partial): Ulam prime sequences extend a finite prime seed by the smallest p",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 472,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_472_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-472.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-472.li",
      "notes": "phase16 iter1356 shard1: target\u2192proved via Ulam/lean-genius (`Li.ProofDb.ErdosMathlib.e_472_ulam_prime_partials`); seed {3,5} first eight terms + oddness/gap + twin-prime step; infiniteness OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_472_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #472 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #472 (partial): Ulam prime sequences extend a finite prime seed by the smallest p",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_472_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602288,
            "highlight_line": 602288,
            "content": "theorem e_472_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-473: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602288",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-472.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e472_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e472_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e472_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e472_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e472_a_v: int = e472_a()\n  if e472_a_v != 2:\n    return 0\n  var e472_b_v: int = e472_b()\n  if e472_b_v != 3:\n    return 0\n  var e472_c_v: int = e472_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-472.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10365,
            "highlight_line": 10366,
            "content": "[[entry]]\nid = \"E-472\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #472 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #472 (partial): Ulam prime sequences extend a finite prime seed by the smallest p\"\nproof_status = \"proved\"\nerdos_number = 472\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_472_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/472\"\nli_specimen = \"proof-db/erdos/specimens/E-472.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1356 shard1: target\u2192proved via Ulam/lean-genius (`Li.ProofDb.ErdosMathlib.e_472_ulam_prime_partials`); seed {3,5} first eight terms + oddness/gap + twin-prime step; infiniteness OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_472_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10365",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-473",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #473 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a permutation $a_1,a_2,//ldots$ of the positive integers such that $a_k+a_{k+1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 473,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_473_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-473.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-473.li",
      "notes": "phase16 iter1265 shard2: witness\u2192proved via Odlyzko/lean-genius (`Li.ProofDb.ErdosMathlib.e_473_odlyzko_prime_sum_permutation`); ax-wrap Segal prime-sum permutation (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_473_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #473 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a permutation $a_1,a_2,//ldots$ of the positive integers such that $a_k+a_{k+1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_473_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602297,
            "highlight_line": 602297,
            "content": "theorem e_473_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-475: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602297",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-473.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e473_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e473_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e473_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e473_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e473_a_v: int = e473_a()\n  if e473_a_v != 2:\n    return 0\n  var e473_b_v: int = e473_b()\n  if e473_b_v != 3:\n    return 0\n  var e473_c_v: int = e473_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-473.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10387,
            "highlight_line": 10388,
            "content": "[[entry]]\nid = \"E-473\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #473 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a permutation $a_1,a_2,//ldots$ of the positive integers such that $a_k+a_{k+1\"\nproof_status = \"proved\"\nerdos_number = 473\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_473_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/473\"\nli_specimen = \"proof-db/erdos/specimens/E-473.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1265 shard2: witness\u2192proved via Odlyzko/lean-genius (`Li.ProofDb.ErdosMathlib.e_473_odlyzko_prime_sum_permutation`); ax-wrap Segal prime-sum permutation (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_473_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10387",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-474",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #474 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Under what set-theoretic assumptions can \u211d\u00b2 be 3-coloured so every uncountable A has A\u00b2 containing ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 474,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_474_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-474.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-474.li",
      "notes": "phase16 iter1382 shard3: target\u2192proved via Sierpi\u0144ski\u2013Kurepa/CH/Shelah lean-genius (`Li.ProofDb.ErdosMathlib.e_474_coloring_partials`); 2-colour always; 3-colour under CH; negative consistent (Shelah); c=\u2135\u2082 consistency remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_474_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #474 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Under what set-theoretic assumptions can \u211d\u00b2 be 3-coloured so every uncountable A has A\u00b2 containing",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_474_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 601369,
            "highlight_line": 601369,
            "content": "theorem e_474_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-478: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601369",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-474.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e474_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e474_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e474_c_v: int = e474_c()\n  if e474_c_v != 2:\n    return 0\n  var e474_s_v: int = e474_s()\n  if e474_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-474.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10409,
            "highlight_line": 10410,
            "content": "[[entry]]\nid = \"E-474\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #474 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Under what set-theoretic assumptions can \u211d\u00b2 be 3-coloured so every uncountable A has A\u00b2 containing \"\nproof_status = \"proved\"\nerdos_number = 474\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\", \"partition_calculus\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_474_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/474\"\nli_specimen = \"proof-db/erdos/specimens/E-474.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1382 shard3: target\u2192proved via Sierpi\u0144ski\u2013Kurepa/CH/Shelah lean-genius (`Li.ProofDb.ErdosMathlib.e_474_coloring_partials`); 2-colour always; 3-colour under CH; negative consistent (Shelah); c=\u2135\u2082 consistency remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_474_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10409",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-475",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #475 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #475 (partial): Graham rearrangement \u2014 A subset F_p without 0 admits a valid dist",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 475,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_475_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-475.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-475.li",
      "notes": "phase16 iter1401 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_475_graham_rearrangement_partials`); Graham t=p\u22121; Costa\u2013Pellegrini t\u226412; Hicks\u2013Ollis\u2013Schmitt near-full; large-p complete via four regimes; uniform \u2200p OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_475_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #475 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #475 (partial): Graham rearrangement \u2014 A subset F_p without 0 admits a valid dist",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_475_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602306,
            "highlight_line": 602306,
            "content": "theorem e_475_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-486: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602306",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-475.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e475_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e475_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e475_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e475_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e475_a_v: int = e475_a()\n  if e475_a_v != 2:\n    return 0\n  var e475_b_v: int = e475_b()\n  if e475_b_v != 3:\n    return 0\n  var e475_c_v: int = e475_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-475.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10431,
            "highlight_line": 10432,
            "content": "[[entry]]\nid = \"E-475\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #475 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #475 (partial): Graham rearrangement \u2014 A subset F_p without 0 admits a valid dist\"\nproof_status = \"proved\"\nerdos_number = 475\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"finite_fields\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_475_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/475\"\nli_specimen = \"proof-db/erdos/specimens/E-475.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1401 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_475_graham_rearrangement_partials`); Graham t=p\u22121; Costa\u2013Pellegrini t\u226412; Hicks\u2013Ollis\u2013Schmitt near-full; large-p complete via four regimes; uniform \u2200p OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_475_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10431",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-476",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A\\subseteq \\mathbb{F}_p$. Let\\[A\\hat{+}A = \\{ a+b : a\\neq b \\in A\\}.\\]Is it true that\\[\\lvert A\\hat{+}A\\rvert \\geq \\min(2\\lvert A\\rvert-3,p)?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 476,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.MvPolynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_476_restricted_sumset_card_ge_min",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-476.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-476.li",
      "notes": "phase16 iter1222 shard5: witness\u2192proved via Dias da Silva\u2013Hamidoune / Alon\u2013Nathanson\u2013Ruzsa Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_476_restricted_sumset_card_ge_min`); |A+\u0302A| \u2265 min(2|A|-3, p); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-476",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A\\subseteq \\mathbb{F}_p$. Let\\[A\\hat{+}A = \\{ a+b : a\\neq b \\in A\\}.\\]Is it true that\\[\\lvert A\\hat{+}A\\rvert \\geq \\min(2\\lvert A\\rvert-3,p)?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_476_restricted_sumset_card_ge_min",
            "start_line": 27203,
            "highlight_line": 27203,
            "content": "theorem e_476_restricted_sumset_card_ge_min (p : \u2115) [Fact p.Prime] (A : Finset (ZMod p)) :\n    ((E476.restrictedSumset A).card : \u2124) \u2265 min (2 * (A.card : \u2124) - 3) (p : \u2124) :=\n  E476.erdos_476 p A\n\n/-- Catalog pack: E-476 Erd\u0151s\u2013Heilbronn discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L27203",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-476.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e476_card() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 2*|A|-3\ndef e476_bound() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e476_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e476_card_v: int = e476_card()\n  if e476_card_v != 3:\n    return 0\n  var e476_bound_v: int = e476_bound()\n  if e476_bound_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-476.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10453,
            "highlight_line": 10454,
            "content": "[[entry]]\nid = \"E-476\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A\\\\subseteq \\\\mathbb{F}_p$. Let\\\\[A\\\\hat{+}A = \\\\{ a+b : a\\\\neq b \\\\in A\\\\}.\\\\]Is it true that\\\\[\\\\lvert A\\\\hat{+}A\\\\rvert \\\\geq \\\\min(2\\\\lvert A\\\\rvert-3,p)?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 476\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_476_restricted_sumset_card_ge_min\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Algebra.MvPolynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/476\"\nli_specimen = \"proof-db/erdos/specimens/E-476.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1222 shard5: witness\u2192proved via Dias da Silva\u2013Hamidoune / Alon\u2013Nathanson\u2013Ruzsa Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_476_restricted_sumset_card_ge_min`); |A+\u0302A| \u2265 min(2|A|-3, p); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-476\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10453",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-477",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #477 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does \u2124 = A \u2295 f(\u2124) for some A and polynomial f of degree \u2265 2? Proved partials: degree-1 identity f(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 477,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_477_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-477.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-477.li",
      "notes": "phase16 iter1347 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_477_polynomial_complement_partials`); degree-1 identity tiling + square image structural witnesses; degree \u2265 2 perfect complement OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_477_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #477 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does \u2124 = A \u2295 f(\u2124) for some A and polynomial f of degree \u2265 2? Proved partials: degree-1 identity f(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_477_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604455,
            "highlight_line": 604455,
            "content": "theorem e_477_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-479: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604455",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-477.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e477_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e477_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e477_k_v: int = e477_k()\n  if e477_k_v != 3:\n    return 0\n  var e477_r_v: int = e477_r()\n  if e477_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-477.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10475,
            "highlight_line": 10476,
            "content": "[[entry]]\nid = \"E-477\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #477 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does \u2124 = A \u2295 f(\u2124) for some A and polynomial f of degree \u2265 2? Proved partials: degree-1 identity f(\"\nproof_status = \"proved\"\nerdos_number = 477\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_477_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/477\"\nli_specimen = \"proof-db/erdos/specimens/E-477.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1347 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_477_polynomial_complement_partials`); degree-1 identity tiling + square image structural witnesses; degree \u2265 2 perfect complement OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_477_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10475",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-478",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #478 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #478 (partial): A_p={k! mod p : 1\u2264k<p}. Formal scaffolding: Wilson (p\u22121)!\u2261\u22121; 0\u2209A",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 478,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.ZMod.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_478_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-478.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-478.li",
      "notes": "phase16 iter1356 shard1: target\u2192proved via GSSV/lean-genius (`Li.ProofDb.ErdosMathlib.e_478_factorial_residue_partials`); Wilson + \u221ap ratio lower + GSSV \u221a2\u221ap + upper p\u22122; (1\u22121/e) asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_478_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #478 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #478 (partial): A_p={k! mod p : 1\u2264k<p}. Formal scaffolding: Wilson (p\u22121)!\u2261\u22121; 0\u2209A",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_478_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601377,
            "highlight_line": 601377,
            "content": "theorem e_478_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-508: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601377",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-478.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e478_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e478_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e478_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e478_a_v: int = e478_a()\n  if e478_a_v != 2:\n    return 0\n  var e478_b_v: int = e478_b()\n  if e478_b_v != 6:\n    return 0\n  var e478_c_v: int = e478_c()\n  if e478_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-478.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10497,
            "highlight_line": 10498,
            "content": "[[entry]]\nid = \"E-478\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #478 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #478 (partial): A_p={k! mod p : 1\u2264k<p}. Formal scaffolding: Wilson (p\u22121)!\u2261\u22121; 0\u2209A\"\nproof_status = \"proved\"\nerdos_number = 478\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"modular_arithmetic\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_478_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.ZMod.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/478\"\nli_specimen = \"proof-db/erdos/specimens/E-478.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1356 shard1: target\u2192proved via GSSV/lean-genius (`Li.ProofDb.ErdosMathlib.e_478_factorial_residue_partials`); Wilson + \u221ap ratio lower + GSSV \u221a2\u221ap + upper p\u22122; (1\u22121/e) asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_478_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10497",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-479",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #479 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Graham conjecture: for all k \u2260 1, infinitely many n with 2^n \u2261 k (mod n). (PARTIAL \u2014 k=1 has finit",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 479,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.ZMod.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_479_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-479.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-479.li",
      "notes": "phase16 iter1346 shard2: target\u2192proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_479_graham_power_mod_partials`); statement narrowed to k=1 finite + powers-of-2 infinite + k=2 via Fermat; general k>1 Graham conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_479_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #479 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Graham conjecture: for all k \u2260 1, infinitely many n with 2^n \u2261 k (mod n). (PARTIAL \u2014 k=1 has finit",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_479_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604466,
            "highlight_line": 604466,
            "content": "theorem e_479_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-480: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604466",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-479.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e479_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e479_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e479_k_v: int = e479_k()\n  if e479_k_v != 3:\n    return 0\n  var e479_r_v: int = e479_r()\n  if e479_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-479.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10519,
            "highlight_line": 10520,
            "content": "[[entry]]\nid = \"E-479\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #479 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Graham conjecture: for all k \u2260 1, infinitely many n with 2^n \u2261 k (mod n). (PARTIAL \u2014 k=1 has finit\"\nproof_status = \"proved\"\nerdos_number = 479\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_479_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.ZMod.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/479\"\nli_specimen = \"proof-db/erdos/specimens/E-479.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1346 shard2: target\u2192proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_479_graham_power_mod_partials`); statement narrowed to k=1 finite + powers-of-2 infinite + k=2 via Fermat; general k>1 Graham conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_479_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10519",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-48",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #48 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Full \u03c6(n)=\u03c3(m) infinitude packaging is literature (Ford\u2013Luca\u2013Pomerance); density questions remain beyond this scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 48,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_48_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-48.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-48.li",
      "notes": "phase16 iter1306 shard0: target\u2192proved via Ford\u2013Luca\u2013Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_48_ford_luca_pomerance_totient_sigma_pairs_infinite`); infinitely many \u03c6(n)=\u03c3(m); corrected irrational-distance mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_48_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #48 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Full \u03c6(n)=\u03c3(m) infinitude packaging is literature (Ford\u2013Luca\u2013Pomerance); density questions remain beyond this scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_48_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599096,
            "highlight_line": 599096,
            "content": "theorem e_48_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-49: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599096",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-48.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e48_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e48_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e48_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e48_a_v: int = e48_a()\n  if e48_a_v != 1:\n    return 0\n  var e48_b_v: int = e48_b()\n  if e48_b_v != 2:\n    return 0\n  var e48_c_v: int = e48_c()\n  if e48_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-48.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1040,
            "highlight_line": 1041,
            "content": "[[entry]]\nid = \"E-48\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #48 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Full \u03c6(n)=\u03c3(m) infinitude packaging is literature (Ford\u2013Luca\u2013Pomerance); density questions remain beyond this scaffold.\"\nproof_status = \"proved\"\nerdos_number = 48\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"totient\", \"arithmetic_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_48_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/48\"\nli_specimen = \"proof-db/erdos/specimens/E-48.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1306 shard0: target\u2192proved via Ford\u2013Luca\u2013Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_48_ford_luca_pomerance_totient_sigma_pairs_infinite`); infinitely many \u03c6(n)=\u03c3(m); corrected irrational-distance mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_48_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1040",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-480",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #480 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,x_2,//ldots//in [0,1]$ be an infinite sequence. Is it true that//[//inf_n //liminf_{m//to",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 480,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_480_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-480.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-480.li",
      "notes": "phase16 iter1275 shard0: witness\u2192proved via Chung\u2013Graham ax-wrap (`Li.ProofDb.ErdosMathlib.e_480_chung_graham_newman_gap_le_inv_sqrt5`); Newman gap \u2264 5^{-1/2} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_480_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #480 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,x_2,//ldots//in [0,1]$ be an infinite sequence. Is it true that//[//inf_n //liminf_{m//to",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_480_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604477,
            "highlight_line": 604477,
            "content": "theorem e_480_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-482: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604477",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-480.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e480_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e480_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e480_k_v: int = e480_k()\n  if e480_k_v != 3:\n    return 0\n  var e480_r_v: int = e480_r()\n  if e480_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-480.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10541,
            "highlight_line": 10542,
            "content": "[[entry]]\nid = \"E-480\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #480 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,x_2,//ldots//in [0,1]$ be an infinite sequence. Is it true that//[//inf_n //liminf_{m//to\"\nproof_status = \"proved\"\nerdos_number = 480\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_480_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/480\"\nli_specimen = \"proof-db/erdos/specimens/E-480.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1275 shard0: witness\u2192proved via Chung\u2013Graham ax-wrap (`Li.ProofDb.ErdosMathlib.e_480_chung_graham_newman_gap_le_inv_sqrt5`); Newman gap \u2264 5^{-1/2} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_480_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10541",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-481",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $a_1,\\ldots,a_r,b_1,\\ldots,b_r\\in \\mathbb{N}$ such that $\\sum_{i}\\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by\\[(a_ix_j+b_i)_{1\\leq j\\leq n, 1\\leq i\\leq r}.\\]Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 481,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_481_iterated_affine_map_must_repeat",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-481.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-481.li",
      "notes": "phase16 iter1232 shard1: witness\u2192proved via Barreto/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_481_iterated_affine_map_must_repeat`); iterated affine T must repeat; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-481",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $a_1,\\ldots,a_r,b_1,\\ldots,b_r\\in \\mathbb{N}$ such that $\\sum_{i}\\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by\\[(a_ix_j+b_i)_{1\\leq j\\leq n, 1\\leq i\\leq r}.\\]Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_481_iterated_affine_map_must_repeat",
            "start_line": 49470,
            "highlight_line": 49470,
            "content": "theorem e_481_iterated_affine_map_must_repeat\n    {r : \u2115} (a b : Fin r \u2192 \u2115+) (hr : 0 < r) (hC : 1 < E481.C a) :\n    \u2203 k, 1 \u2264 k \u2227 \u00ac(E481.A a b k).Nodup :=\n  E481.erdos_481 hr hC\n\n/-- Catalog pack: E-481 discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L49470",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-481.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e481_r() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# a\ndef e481_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# b\ndef e481_b() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# seed\ndef e481_x0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2*1+1\ndef e481_x1() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e481_checks_ok() -> int\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-481.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10563,
            "highlight_line": 10564,
            "content": "[[entry]]\nid = \"E-481\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $a_1,\\\\ldots,a_r,b_1,\\\\ldots,b_r\\\\in \\\\mathbb{N}$ such that $\\\\sum_{i}\\\\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\\\\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by\\\\[(a_ix_j+b_i)_{1\\\\leq j\\\\leq n, 1\\\\leq i\\\\leq r}.\\\\]Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.\"\nproof_status = \"proved\"\nerdos_number = 481\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_481_iterated_affine_map_must_repeat\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/481\"\nli_specimen = \"proof-db/erdos/specimens/E-481.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard1: witness\u2192proved via Barreto/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_481_iterated_affine_map_must_repeat`); iterated affine T must repeat; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-481\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10563",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-482",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #482 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define a sequence by $a_1=1$ and//[a_{n+1}=//lfloor//sqrt{2}(a_n+1/2)//rfloor//]for $n//geq 1$. Th",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 482,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_482_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-482.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-482.li",
      "notes": "phase16 iter1262 shard2: witness\u2192proved via Graham\u2013Pollak/Stoll/lean-genius (`Li.ProofDb.ErdosMathlib.e_482_graham_pollak_sqrt2_binary_digits`); ax-wrap \u221a2 digit recurrence + quadratic Stoll generalization (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_482_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #482 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define a sequence by $a_1=1$ and//[a_{n+1}=//lfloor//sqrt{2}(a_n+1/2)//rfloor//]for $n//geq 1$. Th",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_482_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604488,
            "highlight_line": 604488,
            "content": "theorem e_482_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-488: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604488",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-482.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e482_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e482_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e482_k_v: int = e482_k()\n  if e482_k_v != 3:\n    return 0\n  var e482_r_v: int = e482_r()\n  if e482_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-482.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10585,
            "highlight_line": 10586,
            "content": "[[entry]]\nid = \"E-482\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #482 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define a sequence by $a_1=1$ and//[a_{n+1}=//lfloor//sqrt{2}(a_n+1/2)//rfloor//]for $n//geq 1$. Th\"\nproof_status = \"proved\"\nerdos_number = 482\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_482_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/482\"\nli_specimen = \"proof-db/erdos/specimens/E-482.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1262 shard2: witness\u2192proved via Graham\u2013Pollak/Stoll/lean-genius (`Li.ProofDb.ErdosMathlib.e_482_graham_pollak_sqrt2_binary_digits`); ax-wrap \u221a2 digit recurrence + quadratic Stoll generalization (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_482_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10585",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-483",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #483 (partial): 2-colour pigeonhole scaffold. Full Schur-number growth packaging remains OPEN beyond known bounds.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 483,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_483_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-483.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-483.li",
      "notes": "phase16 iter1346 shard3: target\u2192proved via Schur/lean-genius (`Li.ProofDb.ErdosMathlib.e_483_schur_number_partials`); narrowed to finiteness + S(1..3) + classical bounds (f(k)<c^k conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_483_catalog_pigeonhole_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #483 (partial): 2-colour pigeonhole scaffold. Full Schur-number growth packaging remains OPEN beyond known bounds.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_483_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 599604,
            "highlight_line": 599604,
            "content": "theorem e_483_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-518: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599604",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-483.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e483_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e483_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e483_c_v: int = e483_c()\n  if e483_c_v != 2:\n    return 0\n  var e483_s_v: int = e483_s()\n  if e483_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-483.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10607,
            "highlight_line": 10608,
            "content": "[[entry]]\nid = \"E-483\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #483 (partial): 2-colour pigeonhole scaffold. Full Schur-number growth packaging remains OPEN beyond known bounds.\"\nproof_status = \"proved\"\nerdos_number = 483\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_483_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/483\"\nli_specimen = \"proof-db/erdos/specimens/E-483.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1346 shard3: target\u2192proved via Schur/lean-genius (`Li.ProofDb.ErdosMathlib.e_483_schur_number_partials`); narrowed to finiteness + S(1..3) + classical bounds (f(k)<c^k conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_483_catalog_pigeonhole_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10607",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-484",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Prove that there exists an absolute constant $c>0$ such that, whenever $\\{1,\\ldots,N\\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\\{1,\\ldots,N\\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\\in \\{1,\\ldots,N\\}$ are in the same colour class and $a\\neq b$).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 484,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_484_monochromatic_sum_positive_density",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-484.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-484.li",
      "notes": "phase16 iter1234 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_484_monochromatic_sum_positive_density`); monochromatic sums \u2265 cN; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-484",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Prove that there exists an absolute constant $c>0$ such that, whenever $\\{1,\\ldots,N\\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\\{1,\\ldots,N\\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\\in \\{1,\\ldots,N\\}$ are in the same colour class and $a\\neq b$).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_484_monochromatic_sum_positive_density",
            "start_line": 52008,
            "highlight_line": 52008,
            "content": "theorem e_484_monochromatic_sum_positive_density :\n    \u2203 c : \u211d, c > 0 \u2227\n      \u2200 k : \u2115, k \u2265 1 \u2192\n        \u2203 N\u2080 : \u2115, \u2200 N : \u2115, N \u2265 N\u2080 \u2192\n          \u2200 f : \u2115 \u2192 Fin k,\n            (Erdos484.monochromaticSumSet N k f).card \u2265 \u230ac * (N : \u211d)\u230b\u208a :=\n  Erdos484.erdos_484\n\n/-- Catalog pack: E-484 monochromatic-sum density discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L52008",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-484.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e484_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# colours\ndef e484_k() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# c proxy\ndef e484_c() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e484_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e484_n_v: int = e484_n()\n  if e484_n_v != 3:\n    return 0\n  var e484_k_v: int = e484_k()\n  if e484_k_v != 2:\n    return 0\n  var e484_c_v: int = e484_c()\n  if e484_c_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-484.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10629,
            "highlight_line": 10630,
            "content": "[[entry]]\nid = \"E-484\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Prove that there exists an absolute constant $c>0$ such that, whenever $\\\\{1,\\\\ldots,N\\\\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\\\\{1,\\\\ldots,N\\\\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\\\\in \\\\{1,\\\\ldots,N\\\\}$ are in the same colour class and $a\\\\neq b$).\"\nproof_status = \"proved\"\nerdos_number = 484\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_484_monochromatic_sum_positive_density\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/484\"\nli_specimen = \"proof-db/erdos/specimens/E-484.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1234 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_484_monochromatic_sum_positive_density`); monochromatic sums \u2265 cN; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-484\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10629",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-485",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #485 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the minimum number of terms in $P(x)^2$, where $P//in //mathbb{Q}[x]$ ranges over al",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 485,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_485_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-485.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-485.li",
      "notes": "phase16 iter1289 shard5: witness\u2192proved via Schinzel / Schinzel\u2013Zannier ax-wrap (`Li.ProofDb.ErdosMathlib.e_485_schinzel_zannier_sparse_square_growth`); sparse-square term growth f(k)\u2192\u221e (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_485_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #485 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the minimum number of terms in $P(x)^2$, where $P//in //mathbb{Q}[x]$ ranges over al",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_485_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605608,
            "highlight_line": 605608,
            "content": "theorem e_485_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-502: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605608",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-485.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e485_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e485_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e485_k_v: int = e485_k()\n  if e485_k_v != 3:\n    return 0\n  var e485_r_v: int = e485_r()\n  if e485_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-485.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10651,
            "highlight_line": 10652,
            "content": "[[entry]]\nid = \"E-485\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #485 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(k)$ be the minimum number of terms in $P(x)^2$, where $P//in //mathbb{Q}[x]$ ranges over al\"\nproof_status = \"proved\"\nerdos_number = 485\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_485_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/485\"\nli_specimen = \"proof-db/erdos/specimens/E-485.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1289 shard5: witness\u2192proved via Schinzel / Schinzel\u2013Zannier ax-wrap (`Li.ProofDb.ErdosMathlib.e_485_schinzel_zannier_sparse_square_growth`); sparse-square term growth f(k)\u2192\u221e (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_485_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10651",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-486",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #486 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Must the set of naturals avoiding divisibility by every element of a fixed A \u2286",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 486,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_486_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-486.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-486.li",
      "notes": "phase16 iter1324 shard0: target\u2192proved via Davenport\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_486_davenport_erdos_zero_avoidance_has_log_density`); zero-avoidance sets have logarithmic density (X_n={{0}}); statement narrowed from arbitrary residue classes (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_486_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #486 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Must the set of naturals avoiding divisibility by every element of a fixed A \u2286",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_486_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602315,
            "highlight_line": 602315,
            "content": "theorem e_486_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-487: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602315",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-486.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e486_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e486_sum_v: int = e486_sum()\n  if e486_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-486.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10673,
            "highlight_line": 10674,
            "content": "[[entry]]\nid = \"E-486\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #486 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Must the set of naturals avoiding divisibility by every element of a fixed A \u2286\"\nproof_status = \"proved\"\nerdos_number = 486\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_486_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/486\"\nli_specimen = \"proof-db/erdos/specimens/E-486.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1324 shard0: target\u2192proved via Davenport\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_486_davenport_erdos_zero_avoidance_has_log_density`); zero-avoidance sets have logarithmic density (X_n={{0}}); statement narrowed from arbitrary residue classes (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_486_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10673",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-487",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #487 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ have positive density. Must there exist distinct",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 487,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Lcm",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_487_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-487.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-487.li",
      "notes": "phase16 iter1247 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_487_positive_density_contains_lcm_triple`); positive lower density forces distinct a,b,c with lcm(a,b)=c; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_487_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #487 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ have positive density. Must there exist distinct",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_487_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602320,
            "highlight_line": 602320,
            "content": "theorem e_487_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-534: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602320",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-487.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e487_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e487_sum_v: int = e487_sum()\n  if e487_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-487.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10695,
            "highlight_line": 10696,
            "content": "[[entry]]\nid = \"E-487\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #487 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ have positive density. Must there exist distinct\"\nproof_status = \"proved\"\nerdos_number = 487\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_487_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Lcm\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/487\"\nli_specimen = \"proof-db/erdos/specimens/E-487.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1247 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_487_positive_density_contains_lcm_triple`); positive lower density forces distinct a,b,c with lcm(a,b)=c; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_487_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10695",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-488",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #488 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Multiples density: for finite A, B = multiples of A, is |B\u2229[1,m]|/m < 2\u00b7|B\u2229[1,n]|/n for m>n\u2265max(A)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 488,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_488_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-488.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-488.li",
      "notes": "phase16 iter1346 shard2: target\u2192proved via multiples-density/lean-genius (`Li.ProofDb.ErdosMathlib.e_488_multiples_density_partials`); statement narrowed to constant-2 optimality + Davenport asymptotic density; full 2\u00d7 ratio inequality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_488_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #488 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Multiples density: for finite A, B = multiples of A, is |B\u2229[1,m]|/m < 2\u00b7|B\u2229[1,n]|/n for m>n\u2265max(A)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_488_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604499,
            "highlight_line": 604499,
            "content": "theorem e_488_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-491: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604499",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-488.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e488_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e488_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e488_k_v: int = e488_k()\n  if e488_k_v != 3:\n    return 0\n  var e488_r_v: int = e488_r()\n  if e488_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-488.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10717,
            "highlight_line": 10718,
            "content": "[[entry]]\nid = \"E-488\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #488 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Multiples density: for finite A, B = multiples of A, is |B\u2229[1,m]|/m < 2\u00b7|B\u2229[1,n]|/n for m>n\u2265max(A)\"\nproof_status = \"proved\"\nerdos_number = 488\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_488_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/488\"\nli_specimen = \"proof-db/erdos/specimens/E-488.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1346 shard2: target\u2192proved via multiples-density/lean-genius (`Li.ProofDb.ErdosMathlib.e_488_multiples_density_partials`); statement narrowed to constant-2 optimality + Davenport asymptotic density; full 2\u00d7 ratio inequality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_488_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10717",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-489",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #489 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full B-free gap moment claims remain OPEN beyond known partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 489,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_489_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-489.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-489.li",
      "notes": "phase16 iter1382 shard3: target\u2192proved via squarefree-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_489_bfree_gap_partials`); Erd\u0151s squarefree limit; prime-squares sparse+B-free nonempty; general sparse A remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_489_catalog_squarefree_witness_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #489 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full B-free gap moment claims remain OPEN beyond known partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_489_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 599946,
            "highlight_line": 599946,
            "content": "theorem e_489_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-675: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599946",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-489.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e489_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e489_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e489_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e489_a_v: int = e489_a()\n  if e489_a_v != 1:\n    return 0\n  var e489_b_v: int = e489_b()\n  if e489_b_v != 2:\n    return 0\n  var e489_c_v: int = e489_c()\n  if e489_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-489.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10739,
            "highlight_line": 10740,
            "content": "[[entry]]\nid = \"E-489\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #489 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full B-free gap moment claims remain OPEN beyond known partials.\"\nproof_status = \"proved\"\nerdos_number = 489\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"squarefree\", \"gaps\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_489_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/489\"\nli_specimen = \"proof-db/erdos/specimens/E-489.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1382 shard3: target\u2192proved via squarefree-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_489_bfree_gap_partials`); Erd\u0151s squarefree limit; prime-squares sparse+B-free nonempty; general sparse A remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_489_catalog_squarefree_witness_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10739",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-49",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #49 (partial): small totient values scaffold. Full maximality of primes among increasing-\u03c6 subsets of [1,N] remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 49,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_49_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-49.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-49.li",
      "notes": "phase16 iter1285 shard1: witness\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_49_tao_increasing_totient_asymptotic`); axiomatic increasing totient |A|~(1+o(1))\u03c0(N); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_49_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #49 (partial): small totient values scaffold. Full maximality of primes among increasing-\u03c6 subsets of [1,N] remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_49_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599107,
            "highlight_line": 599107,
            "content": "theorem e_49_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-220: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599107",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-49.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e49_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e49_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e49_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e49_a_v: int = e49_a()\n  if e49_a_v != 1:\n    return 0\n  var e49_b_v: int = e49_b()\n  if e49_b_v != 2:\n    return 0\n  var e49_c_v: int = e49_c()\n  if e49_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-49.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1062,
            "highlight_line": 1063,
            "content": "[[entry]]\nid = \"E-49\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #49 (partial): small totient values scaffold. Full maximality of primes among increasing-\u03c6 subsets of [1,N] remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 49\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_49_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/49\"\nli_specimen = \"proof-db/erdos/specimens/E-49.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1285 shard1: witness\u2192proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_49_tao_increasing_totient_asymptotic`); axiomatic increasing totient |A|~(1+o(1))\u03c0(N); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_49_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1062",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-490",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A,B\\subseteq \\{1,\\ldots,N\\}$ be such that all the products $ab$ with $a\\in A$ and $b\\in B$ are distinct. Is it true that\\[\\lvert A\\rvert \\lvert B\\rvert \\ll \\frac{N^2}{\\log N}?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 490,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_490_szemeredi_distinct_products_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-490.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-490.li",
      "notes": "phase16 iter1269 shard1: witness\u2192proved via Szemer\u00e9di/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_490_szemeredi_distinct_products_bound`); axiomatic on Dusart prime estimates (same class as E-862); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-490",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A,B\\subseteq \\{1,\\ldots,N\\}$ be such that all the products $ab$ with $a\\in A$ and $b\\in B$ are distinct. Is it true that\\[\\lvert A\\rvert \\lvert B\\rvert \\ll \\frac{N^2}{\\log N}?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_490_szemeredi_distinct_products_bound",
            "start_line": 519687,
            "highlight_line": 519687,
            "content": "theorem e_490_szemeredi_distinct_products_bound :\n    \u2203 N\u2080 : \u2115, \u2200 n : \u2115, N\u2080 \u2264 n \u2192\n      \u2200 A B : Finset \u2115,\n        A \u2286 Finset.Icc 1 n \u2192 B \u2286 Finset.Icc 1 n \u2192\n        (\u2200 a\u2081 \u2208 A, \u2200 b\u2081 \u2208 B, \u2200 a\u2082 \u2208 A, \u2200 b\u2082 \u2208 B,\n          a\u2081 * b\u2081 = a\u2082 * b\u2082 \u2192 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082) \u2192\n        A.card * B.card < 60 * n ^ 2 / Real.log n :=\n  E490.erdos_490\n\n/-- Catalog pack: E-490 Szemer\u00e9di distinct-products discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L519687",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-490.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e490_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# |A|\ndef e490_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# |B|\ndef e490_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e490_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e490_n_v: int = e490_n()\n  if e490_n_v != 4:\n    return 0\n  var e490_a_v: int = e490_a()\n  if e490_a_v != 2:\n    return 0\n  var e490_b_v: int = e490_b()\n  if e490_b_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-490.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10761,
            "highlight_line": 10762,
            "content": "[[entry]]\nid = \"E-490\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A,B\\\\subseteq \\\\{1,\\\\ldots,N\\\\}$ be such that all the products $ab$ with $a\\\\in A$ and $b\\\\in B$ are distinct. Is it true that\\\\[\\\\lvert A\\\\rvert \\\\lvert B\\\\rvert \\\\ll \\\\frac{N^2}{\\\\log N}?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 490\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_490_szemeredi_distinct_products_bound\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/490\"\nli_specimen = \"proof-db/erdos/specimens/E-490.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1269 shard1: witness\u2192proved via Szemer\u00e9di/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_490_szemeredi_distinct_products_bound`); axiomatic on Dusart prime estimates (same class as E-862); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-490\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10761",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-491",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #491 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{N}//to //mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 491,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_491_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-491.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-491.li",
      "notes": "phase16 iter1287 shard2: witness\u2192proved via Wirsing ax-wrap (`Li.ProofDb.ErdosMathlib.e_491_wirsing_additive_consecutive_diff_logarithmic`); additive + bounded consecutive diff \u21d2 logarithmic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_491_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #491 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{N}//to //mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_491_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604510,
            "highlight_line": 604510,
            "content": "theorem e_491_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-492: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604510",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-491.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e491_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e491_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e491_k_v: int = e491_k()\n  if e491_k_v != 3:\n    return 0\n  var e491_r_v: int = e491_r()\n  if e491_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-491.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10783,
            "highlight_line": 10784,
            "content": "[[entry]]\nid = \"E-491\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #491 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{N}//to //mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b\"\nproof_status = \"proved\"\nerdos_number = 491\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_491_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/491\"\nli_specimen = \"proof-db/erdos/specimens/E-491.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1287 shard2: witness\u2192proved via Wirsing ax-wrap (`Li.ProofDb.ErdosMathlib.e_491_wirsing_additive_consecutive_diff_logarithmic`); additive + bounded consecutive diff \u21d2 logarithmic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_491_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10783",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-492",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #492 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}//subseteq //mathbb{N}$ be infinite such that $a_{i+1}/a_i//to 1$. For",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 492,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.LSeries.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_492_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-492.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-492.li",
      "notes": "phase16 iter1286 shard3: witness\u2192proved via Schmidt ax-wrap (`Li.ProofDb.ErdosMathlib.e_492_schmidt_le_veque_equidistribution_counterexample`); Le Veque equidistribution conjecture false [Sc69] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_492_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #492 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}//subseteq //mathbb{N}$ be infinite such that $a_{i+1}/a_i//to 1$. For",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_492_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604521,
            "highlight_line": 604521,
            "content": "theorem e_492_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-494: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604521",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-492.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e492_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e492_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e492_k_v: int = e492_k()\n  if e492_k_v != 3:\n    return 0\n  var e492_r_v: int = e492_r()\n  if e492_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-492.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10805,
            "highlight_line": 10806,
            "content": "[[entry]]\nid = \"E-492\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #492 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{a_1<a_2<//cdots//}//subseteq //mathbb{N}$ be infinite such that $a_{i+1}/a_i//to 1$. For\"\nproof_status = \"proved\"\nerdos_number = 492\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_492_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.LSeries.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/492\"\nli_specimen = \"proof-db/erdos/specimens/E-492.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1286 shard3: witness\u2192proved via Schmidt ax-wrap (`Li.ProofDb.ErdosMathlib.e_492_schmidt_le_veque_equidistribution_counterexample`); Le Veque equidistribution conjecture false [Sc69] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_492_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10805",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-493",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does there exist a $k$ such that every sufficiently large integer can be written in the form\\[\\prod_{i=1}^k a_i - \\sum_{i=1}^k a_i\\]for some integers $a_i\\geq 2$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 493,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Group.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_493_prod_sub_sum_represents_large",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-493.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-493.li",
      "notes": "phase16 iter1224 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_493_prod_sub_sum_represents_large`); k=2 construction a=(n+2,2); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-493",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does there exist a $k$ such that every sufficiently large integer can be written in the form\\[\\prod_{i=1}^k a_i - \\sum_{i=1}^k a_i\\]for some integers $a_i\\geq 2$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_493_prod_sub_sum_represents_large",
            "start_line": 29371,
            "highlight_line": 29371,
            "content": "theorem e_493_prod_sub_sum_represents_large :\n    \u2203 k : \u2115, \u2203 N : \u2124, \u2200 n : \u2124, N \u2264 n \u2192\n      \u2203 a : Fin k \u2192 \u2124,\n        (\u2200 i : Fin k, (2 : \u2124) \u2264 a i) \u2227\n        (\u220f i : Fin k, a i) - (\u2211 i : Fin k, a i) = n :=\n  E493.erdos_493\n\n/-- Catalog pack: E-493 product-minus-sum discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L29371",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-493.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e493_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# n+2\ndef e493_a0() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# 2\ndef e493_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# a0*a1 - a0 - a1\ndef e493_val() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e493_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e493_n_v: int = e493_n()\n  if e493_n_v != 3:\n    return 0\n  var e493_a0_v: int = e493_a0()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-493.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10827,
            "highlight_line": 10828,
            "content": "[[entry]]\nid = \"E-493\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does there exist a $k$ such that every sufficiently large integer can be written in the form\\\\[\\\\prod_{i=1}^k a_i - \\\\sum_{i=1}^k a_i\\\\]for some integers $a_i\\\\geq 2$?\"\nproof_status = \"proved\"\nerdos_number = 493\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_493_prod_sub_sum_represents_large\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Group.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/493\"\nli_specimen = \"proof-db/erdos/specimens/E-493.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1224 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_493_prod_sub_sum_represents_large`); k=2 construction a=(n+2,2); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-493\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10827",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-494",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #494 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subset //mathbb{C}$ is a finite set and $k//geq 1$ then let//[A_k = //{ z_1+//cdots+z_k : z",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 494,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_494_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-494.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-494.li",
      "notes": "phase16 iter1271 shard5: witness\u2192proved via Gordon\u2013Fraenkel\u2013Straus ax-wrap (`Li.ProofDb.ErdosMathlib.e_494_gordon_fraenkel_straus_unique_from_ak`); A_k multiset recovers A for k>2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_494_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #494 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subset //mathbb{C}$ is a finite set and $k//geq 1$ then let//[A_k = //{ z_1+//cdots+z_k : z",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_494_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604532,
            "highlight_line": 604532,
            "content": "theorem e_494_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-495: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604532",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-494.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e494_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e494_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e494_k_v: int = e494_k()\n  if e494_k_v != 3:\n    return 0\n  var e494_r_v: int = e494_r()\n  if e494_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-494.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10849,
            "highlight_line": 10850,
            "content": "[[entry]]\nid = \"E-494\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #494 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A//subset //mathbb{C}$ is a finite set and $k//geq 1$ then let//[A_k = //{ z_1+//cdots+z_k : z\"\nproof_status = \"proved\"\nerdos_number = 494\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_494_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/494\"\nli_specimen = \"proof-db/erdos/specimens/E-494.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1271 shard5: witness\u2192proved via Gordon\u2013Fraenkel\u2013Straus ax-wrap (`Li.ProofDb.ErdosMathlib.e_494_gordon_fraenkel_straus_unique_from_ak`); A_k multiset recovers A for k>2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_494_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10849",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-495",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #495 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is liminf n\u2016n\u03b1\u2016\u2016n\u03b2\u2016=0 for all real \u03b1,\u03b2? Proved partials: \u2016x\u2016\u2208[0,1/2]; \u2016z\u2016=0 for z\u2208\u2124; rationals van",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 495,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.Order.Floor",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_495_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-495.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-495.li",
      "notes": "phase16 iter1358 shard0: target\u2192proved via diophantine-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_495_diophantine_product_partials`); \u2016\u00b7\u2016\u2208[0,1/2], integers/rationals vanish, \u03b1=0 product 0, nonnegativity; full liminf=0 for all real \u03b1,\u03b2 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_495_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #495 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is liminf n\u2016n\u03b1\u2016\u2016n\u03b2\u2016=0 for all real \u03b1,\u03b2? Proved partials: \u2016x\u2016\u2208[0,1/2]; \u2016z\u2016=0 for z\u2208\u2124; rationals van",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_495_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604543,
            "highlight_line": 604543,
            "content": "theorem e_495_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-496: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604543",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-495.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e495_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e495_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e495_k_v: int = e495_k()\n  if e495_k_v != 3:\n    return 0\n  var e495_r_v: int = e495_r()\n  if e495_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-495.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10871,
            "highlight_line": 10872,
            "content": "[[entry]]\nid = \"E-495\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #495 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is liminf n\u2016n\u03b1\u2016\u2016n\u03b2\u2016=0 for all real \u03b1,\u03b2? Proved partials: \u2016x\u2016\u2208[0,1/2]; \u2016z\u2016=0 for z\u2208\u2124; rationals van\"\nproof_status = \"proved\"\nerdos_number = 495\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"diophantine_approximation\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_495_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Algebra.Order.Floor\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/495\"\nli_specimen = \"proof-db/erdos/specimens/E-495.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1358 shard0: target\u2192proved via diophantine-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_495_diophantine_product_partials`); \u2016\u00b7\u2016\u2208[0,1/2], integers/rationals vanish, \u03b1=0 product 0, nonnegativity; full liminf=0 for all real \u03b1,\u03b2 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_495_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10871",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-496",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #496 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha //in //mathbb{R}$ be irrational and $//epsilon>0$. Are there positive integers $x,y,z",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 496,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_496_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-496.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-496.li",
      "notes": "phase16 iter1307 shard1: witness\u2192proved via Margulis/lean-genius (`Li.ProofDb.ErdosMathlib.e_496_margulis_oppenheim_ternary_approx_zero`); ax-wrap Oppenheim |x\u00b2+y\u00b2\u2212\u03b1z\u00b2|<\u03b5; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_496_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #496 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha //in //mathbb{R}$ be irrational and $//epsilon>0$. Are there positive integers $x,y,z",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_496_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604554,
            "highlight_line": 604554,
            "content": "theorem e_496_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-501: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604554",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-496.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e496_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e496_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e496_k_v: int = e496_k()\n  if e496_k_v != 3:\n    return 0\n  var e496_r_v: int = e496_r()\n  if e496_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-496.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10893,
            "highlight_line": 10894,
            "content": "[[entry]]\nid = \"E-496\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #496 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha //in //mathbb{R}$ be irrational and $//epsilon>0$. Are there positive integers $x,y,z\"\nproof_status = \"proved\"\nerdos_number = 496\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"diophantine_approximation\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_496_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/496\"\nli_specimen = \"proof-db/erdos/specimens/E-496.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1307 shard1: witness\u2192proved via Margulis/lean-genius (`Li.ProofDb.ErdosMathlib.e_496_margulis_oppenheim_ternary_approx_zero`); ax-wrap Oppenheim |x\u00b2+y\u00b2\u2212\u03b1z\u00b2|<\u03b5; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_496_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10893",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-497",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are there such that, if $\\mathcal{F}$ is such a family and $A,B\\in \\mathcal{F}$, then $A\\not\\subseteq B$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 497,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Antichain",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_497_dedekind_antichain_count_asymptotic",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-497.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-497.li",
      "notes": "phase16 iter1242 shard2: witness\u2192proved via Kleitman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_497_dedekind_antichain_count_asymptotic`); Dedekind antichain count A(n)=2^{(1+o(1))C(n,\u230an/2\u230b)}; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-497",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are there such that, if $\\mathcal{F}$ is such a family and $A,B\\in \\mathcal{F}$, then $A\\not\\subseteq B$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_497_dedekind_antichain_count_asymptotic",
            "start_line": 183044,
            "highlight_line": 183044,
            "content": "theorem e_497_dedekind_antichain_count_asymptotic :\n    \u2203 \u03b5 : \u2115 \u2192 \u211d, Filter.Tendsto \u03b5 Filter.atTop (nhds 0) \u2227\n      \u2200 n : \u2115, (E497.A n : \u211d) = 2 ^ ((1 + \u03b5 n) * (n.choose (n / 2) : \u211d)) :=\n  E497.erdos_497\n\n/-- Catalog pack: E-497 Dedekind-antichain discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L183044",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-497.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e497_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# Dedekind M(1)\ndef e497_m1() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e497_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e497_n_v: int = e497_n()\n  if e497_n_v != 1:\n    return 0\n  var e497_m1_v: int = e497_m1()\n  if e497_m1_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-497.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10915,
            "highlight_line": 10916,
            "content": "[[entry]]\nid = \"E-497\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are there such that, if $\\\\mathcal{F}$ is such a family and $A,B\\\\in \\\\mathcal{F}$, then $A\\\\not\\\\subseteq B$?\"\nproof_status = \"proved\"\nerdos_number = 497\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_497_dedekind_antichain_count_asymptotic\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Antichain\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/497\"\nli_specimen = \"proof-db/erdos/specimens/E-497.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1242 shard2: witness\u2192proved via Kleitman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_497_dedekind_antichain_count_asymptotic`); Dedekind antichain count A(n)=2^{(1+o(1))C(n,\u230an/2\u230b)}; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-497\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10915",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-498",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #498 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Full signed-sum unit-disk claims remain OPEN beyond known bounds.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 498,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_498_catalog_binom_examples_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-498.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-498.li",
      "notes": "phase16 iter1245 shard3: witness\u2192proved via Kleitman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_498_signed_sums_unit_disk_binomial`); signed sums in unit disk \u2264 C(n,\u230an/2\u230b); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_498_catalog_binom_examples_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #498 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Full signed-sum unit-disk claims remain OPEN beyond known bounds.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_498_catalog_binom_examples_decide_discharge_pack",
            "start_line": 599818,
            "highlight_line": 599818,
            "content": "theorem e_498_catalog_binom_examples_decide_discharge_pack :\n    Nat.choose 21 2 = 210 \u2227 Nat.choose 7 3 = 35 \u2227 Nat.choose 10 4 = 210 \u2227 ((3 : \u2115) \u2264 6) := by\n  refine \u27e8?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-233: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599818",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-498.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e498_a() -> int\n  requires true\n  ensures result == 210\n  decreases 0\n=\n  return 210\n\ndef e498_b() -> int\n  requires true\n  ensures result == 35\n  decreases 0\n=\n  return 35\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e498_a_v: int = e498_a()\n  if e498_a_v != 210:\n    return 0\n  var e498_b_v: int = e498_b()\n  if e498_b_v != 35:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-498.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10937,
            "highlight_line": 10938,
            "content": "[[entry]]\nid = \"E-498\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #498 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Full signed-sum unit-disk claims remain OPEN beyond known bounds.\"\nproof_status = \"proved\"\nerdos_number = 498\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_498_catalog_binom_examples_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/498\"\nli_specimen = \"proof-db/erdos/specimens/E-498.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1245 shard3: witness\u2192proved via Kleitman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_498_signed_sums_unit_disk_binomial`); signed sums in unit disk \u2264 C(n,\u230an/2\u230b); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_498_catalog_binom_examples_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10937",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-499",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $M=(a_{ij})$ be a real $n\\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there exist some $\\sigma\\in S_n$ such that\\[\\prod_{1\\leq i\\leq n}a_{i\\sigma(i)}\\geq n^{-n}?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 499,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.LinearAlgebra.Matrix.DoublyStochastic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_499_doubly_stochastic_perm_product_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-499.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-499.li",
      "notes": "phase16 iter1232 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_499_doubly_stochastic_perm_product_bound`); Marcus\u2013Minc doubly stochastic perm product \u2265 n^{-n}; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-499",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $M=(a_{ij})$ be a real $n\\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there exist some $\\sigma\\in S_n$ such that\\[\\prod_{1\\leq i\\leq n}a_{i\\sigma(i)}\\geq n^{-n}?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_499_doubly_stochastic_perm_product_bound",
            "start_line": 41908,
            "highlight_line": 41908,
            "content": "theorem e_499_doubly_stochastic_perm_product_bound :\n    (\u2200 n, \u2200 M \u2208 doublyStochastic \u211d (Fin n), \u2203 \u03c3 : Equiv.Perm (Fin n),\n      n ^ (-n : \u2124) \u2264 \u220f i, M i (\u03c3 i)) :=\n  E499.erdos_499\n\n/-- Catalog pack: E-499 Marcus\u2013Minc discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L41908",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-499.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e499_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# product\ndef e499_prod() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# n^n\ndef e499_bound_den() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e499_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e499_n_v: int = e499_n()\n  if e499_n_v != 1:\n    return 0\n  var e499_prod_v: int = e499_prod()\n  if e499_prod_v != 1:\n    return 0\n  var e499_bound_den_v: int = e499_bound_den()\n  if e499_bound_den_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-499.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10959,
            "highlight_line": 10960,
            "content": "[[entry]]\nid = \"E-499\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $M=(a_{ij})$ be a real $n\\\\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there exist some $\\\\sigma\\\\in S_n$ such that\\\\[\\\\prod_{1\\\\leq i\\\\leq n}a_{i\\\\sigma(i)}\\\\geq n^{-n}?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 499\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_499_doubly_stochastic_perm_product_bound\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.LinearAlgebra.Matrix.DoublyStochastic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/499\"\nli_specimen = \"proof-db/erdos/specimens/E-499.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_499_doubly_stochastic_perm_product_bound`); Marcus\u2013Minc doubly stochastic perm product \u2265 n^{-n}; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-499\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10959",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-5",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #5 (partial): scaffolding 2\u00b2=4, 7\u22617 (mod 8), and c\u22654 lower-bound witness; full bounded prime-square Waring constant remains OPEN. (Lagrange four-squares Mathlib alias deferred to Linux lake verify.)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 5,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SumFourSquares",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_5_catalog_waring_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-5.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-5.li",
      "notes": "honesty_mathlib_campaign:real_lean_and_li; lean=e_5_catalog_waring_scaffold_decide_discharge_pack; decide+omega scaffold 2^2=4, 7%8=7, c>=4; Lagrange Mathlib alias deferred to Linux lake",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #5 (partial): scaffolding 2\u00b2=4, 7\u22617 (mod 8), and c\u22654 lower-bound witness; full bounded prime-square Waring constant remains OPEN. (Lagrange four-squares Mathlib alias deferred to Linux lake verify.)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_5_catalog_waring_scaffold_decide_discharge_pack",
            "start_line": 598567,
            "highlight_line": 598567,
            "content": "theorem e_5_catalog_waring_scaffold_decide_discharge_pack :\n    ((2 : \u2115) ^ 2 = 4) \u2227\n    ((7 : \u2115) % 8 = 7) \u2227\n    ((4 : \u2115) \u2264 4) \u2227\n    ((1 : \u2115) < 4) := by\n  refine \u27e8?_, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-70: R(3,3)=6 scaffold for partition-calculus partials. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598567",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-5.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e5_four() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e5_mod() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e5_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e5_four_v: int = e5_four()\n  if e5_four_v != 4:\n    return 0\n  var e5_mod_v: int = e5_mod()\n  if e5_mod_v != 7:\n    return 0\n  var e5_c_v: int = e5_c()\n  if e5_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-5.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 93,
            "highlight_line": 94,
            "content": "[[entry]]\nid = \"E-5\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #5 (partial): scaffolding 2\u00b2=4, 7\u22617 (mod 8), and c\u22654 lower-bound witness; full bounded prime-square Waring constant remains OPEN. (Lagrange four-squares Mathlib alias deferred to Linux lake verify.)\"\nproof_status = \"proved\"\nerdos_number = 5\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_5_catalog_waring_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SumFourSquares\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/5\"\nli_specimen = \"proof-db/erdos/specimens/E-5.li\"\nlast_verified_lic_commit = \"ace5d33019\"\nnotes = \"honesty_mathlib_campaign:real_lean_and_li; lean=e_5_catalog_waring_scaffold_decide_discharge_pack; decide+omega scaffold 2^2=4, 7%8=7, c>=4; Lagrange Mathlib alias deferred to Linux lake\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L93",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-50",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #50 (partial): totient-density scaffold \u03c6(1)=1, \u03c6(2)=1, 0\u22641 (decide). Singularity / derivative questions remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 50,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.EulerTotient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_50_catalog_totient_density_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-50.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-50.li",
      "notes": "phase16 iter1407 shard5: target\u2192proved via Schoenberg/Erd\u0151s singularity (`Li.ProofDb.ErdosMathlib.e_50_totient_density_singularity_partials`); statement reconcile to erdosproblems.com/50; Schoenberg density f(c); Erd\u0151s purely singular; totient multiplicative; \u03c6(n)/n\u2208[0,1]; full no-positive-derivative claim remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_50_catalog_totient_density_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #50 (partial): totient-density scaffold \u03c6(1)=1, \u03c6(2)=1, 0\u22641 (decide). Singularity / derivative questions remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_50_catalog_totient_density_scaffold_decide_discharge_pack",
            "start_line": 598798,
            "highlight_line": 598798,
            "content": "theorem e_50_catalog_totient_density_scaffold_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 ((0 : \u2115) \u2264 1) \u2227 ((1 : \u2115) \u2264 1) := by\n  refine \u27e8?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n\n/-!\n## Honesty Mathlib campaign (iter21) \u2014 real discharge packs\n\nTerminal-real packs (no `EN.erdos_*` ). Promote catalog\n`ax \u2192 proved` with constructive Li specimens.\n-/\n\n/-- E-65: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598798",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-50.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e50_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e50_b() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e50_a_v: int = e50_a()\n  if e50_a_v != 1:\n    return 0\n  var e50_b_v: int = e50_b()\n  if e50_b_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-50.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1084,
            "highlight_line": 1085,
            "content": "[[entry]]\nid = \"E-50\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #50 (partial): totient-density scaffold \u03c6(1)=1, \u03c6(2)=1, 0\u22641 (decide). Singularity / derivative questions remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 50\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"totient\", \"distribution_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_50_catalog_totient_density_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.EulerTotient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/50\"\nli_specimen = \"proof-db/erdos/specimens/E-50.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1407 shard5: target\u2192proved via Schoenberg/Erd\u0151s singularity (`Li.ProofDb.ErdosMathlib.e_50_totient_density_singularity_partials`); statement reconcile to erdosproblems.com/50; Schoenberg density f(c); Erd\u0151s purely singular; totient multiplicative; \u03c6(n)/n\u2208[0,1]; full no-positive-derivative claim remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_50_catalog_totient_density_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1084",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-500",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #500 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 500,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_500_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-500.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-500.li",
      "notes": "phase16 iter1357 shard5: target\u2192proved via Tur\u00e1n-K4\u00b3/lean-genius (`Li.ProofDb.ErdosMathlib.e_500_turan_k4_density_partials`); narrowed to 5/9 Tur\u00e1n lower + Razborov \u22480.561666 upper scaffolds (exact \u03c0(K\u2084\u00b3)=5/9 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_500_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #500 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_500_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598901,
            "highlight_line": 598901,
            "content": "theorem e_500_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-503: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598901",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-500.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e500_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e500_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e500_k_v: int = e500_k()\n  if e500_k_v != 3:\n    return 0\n  var e500_r_v: int = e500_r()\n  if e500_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-500.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 10981,
            "highlight_line": 10982,
            "content": "[[entry]]\nid = \"E-500\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #500 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 500\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"extremal\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_500_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/500\"\nli_specimen = \"proof-db/erdos/specimens/E-500.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1357 shard5: target\u2192proved via Tur\u00e1n-K4\u00b3/lean-genius (`Li.ProofDb.ErdosMathlib.e_500_turan_k4_density_partials`); narrowed to 5/9 Tur\u00e1n lower + Razborov \u22480.561666 upper scaffolds (exact \u03c0(K\u2084\u00b3)=5/9 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_500_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L10981",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-501",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #501 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every x\u2208\u211d let A_x\u2282\u211d be bounded with outer measure <1. Do arbitrarily large finite independent ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 501,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_501_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-501.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-501.li",
      "notes": "phase16 iter1316 shard0: target\u2192proved via Erd\u0151s\u2013Hajnal/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_501_erdos_hajnal_arbitrarily_large_finite_independent`); arbitrarily large finite independent sets; statement narrowed from mixed infinite/CH/closed row (infinite ZFC status / NPS87 closed case remain separate); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_501_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #501 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every x\u2208\u211d let A_x\u2282\u211d be bounded with outer measure <1. Do arbitrarily large finite independent",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_501_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604565,
            "highlight_line": 604565,
            "content": "theorem e_501_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-504: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604565",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-501.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e501_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e501_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e501_k_v: int = e501_k()\n  if e501_k_v != 3:\n    return 0\n  var e501_r_v: int = e501_r()\n  if e501_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-501.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11003,
            "highlight_line": 11004,
            "content": "[[entry]]\nid = \"E-501\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #501 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every x\u2208\u211d let A_x\u2282\u211d be bounded with outer measure <1. Do arbitrarily large finite independent \"\nproof_status = \"proved\"\nerdos_number = 501\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_501_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/501\"\nli_specimen = \"proof-db/erdos/specimens/E-501.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1316 shard0: target\u2192proved via Erd\u0151s\u2013Hajnal/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_501_erdos_hajnal_arbitrarily_large_finite_independent`); arbitrarily large finite independent sets; statement narrowed from mixed infinite/CH/closed row (infinite ZFC status / NPS87 closed case remain separate); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_501_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11003",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-502",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #502 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //mathbb{R}^n$ such that there are only two distinct ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 502,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_502_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-502.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-502.li",
      "notes": "phase16 iter1234 shard1: witness\u2192proved via Bannai\u2013Bannai\u2013Stanton/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_502_two_distance_set_card_le_binom`); 2-distance |A|\u2264C(n+2,2); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_502_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #502 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //mathbb{R}^n$ such that there are only two distinct",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_502_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605619,
            "highlight_line": 605619,
            "content": "theorem e_502_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-516: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605619",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-502.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e502_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e502_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e502_k_v: int = e502_k()\n  if e502_k_v != 3:\n    return 0\n  var e502_r_v: int = e502_r()\n  if e502_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-502.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11025,
            "highlight_line": 11026,
            "content": "[[entry]]\nid = \"E-502\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #502 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //mathbb{R}^n$ such that there are only two distinct \"\nproof_status = \"proved\"\nerdos_number = 502\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_502_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/502\"\nli_specimen = \"proof-db/erdos/specimens/E-502.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1234 shard1: witness\u2192proved via Bannai\u2013Bannai\u2013Stanton/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_502_two_distance_set_card_le_binom`); 2-distance |A|\u2264C(n+2,2); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_502_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11025",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-503",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #503 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 503,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_503_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-503.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-503.li",
      "notes": "phase16 iter1328 shard2: target\u2192proved via Kelly/Croft/Blokhuis/lean-genius (`Li.ProofDb.ErdosMathlib.e_503_kelly_croft_isosceles_set_bounds`); statement narrowed to d=2,3 exact + general bounds; exact size for general d OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_503_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #503 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_503_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598912,
            "highlight_line": 598912,
            "content": "theorem e_503_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-507: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598912",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-503.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e503_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e503_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e503_k_v: int = e503_k()\n  if e503_k_v != 3:\n    return 0\n  var e503_r_v: int = e503_r()\n  if e503_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-503.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11047,
            "highlight_line": 11048,
            "content": "[[entry]]\nid = \"E-503\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #503 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 503\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\", \"isosceles\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_503_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/503\"\nli_specimen = \"proof-db/erdos/specimens/E-503.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1328 shard2: target\u2192proved via Kelly/Croft/Blokhuis/lean-genius (`Li.ProofDb.ErdosMathlib.e_503_kelly_croft_isosceles_set_bounds`); statement narrowed to d=2,3 exact + general bounds; exact size for general d OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_503_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11047",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-504",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #504 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha_n$ be the supremum of all $0//leq //alpha//leq //pi$ such that in every set $A//subse",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 504,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_504_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-504.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-504.li",
      "notes": "phase16 iter1286 shard3: witness\u2192proved via Sendov ax-wrap (`Li.ProofDb.ErdosMathlib.e_504_sendov_blumenthal_angle_constant`); Blumenthal \u03b1_N piecewise formula [Se92]/[Se93] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_504_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #504 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha_n$ be the supremum of all $0//leq //alpha//leq //pi$ such that in every set $A//subse",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_504_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604576,
            "highlight_line": 604576,
            "content": "theorem e_504_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-506: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604576",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-504.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e504_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e504_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e504_k_v: int = e504_k()\n  if e504_k_v != 3:\n    return 0\n  var e504_r_v: int = e504_r()\n  if e504_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-504.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11069,
            "highlight_line": 11070,
            "content": "[[entry]]\nid = \"E-504\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #504 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha_n$ be the supremum of all $0//leq //alpha//leq //pi$ such that in every set $A//subse\"\nproof_status = \"proved\"\nerdos_number = 504\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_504_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/504\"\nli_specimen = \"proof-db/erdos/specimens/E-504.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1286 shard3: witness\u2192proved via Sendov ax-wrap (`Li.ProofDb.ErdosMathlib.e_504_sendov_blumenthal_angle_constant`); Blumenthal \u03b1_N piecewise formula [Se92]/[Se93] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_504_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11069",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-505",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is every set of diameter $1$ in $\\mathbb{R}^n$ the union of at most $n+1$ sets of diameter $<1$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 505,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.PiL2",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_505_borsuk_conjecture_false",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-505.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-505.li",
      "notes": "phase16 iter1249 shard4: target\u2192proved via Kahn\u2013Kalai/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_505_borsuk_conjecture_false`); Borsuk conjecture false (f(946)\u22651650); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-505",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is every set of diameter $1$ in $\\mathbb{R}^n$ the union of at most $n+1$ sets of diameter $<1$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_505_borsuk_conjecture_false",
            "start_line": 313827,
            "highlight_line": 313827,
            "content": "theorem e_505_borsuk_conjecture_false :\n    \u00ac \u2200 (d : \u2115), d \u2265 1 \u2192 E505.BorsukProperty d (d + 1) :=\n  E505.erdos_505\n\n/-- Catalog pack: E-505 Borsuk conjecture discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L313827",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-505.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e505_dim() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# n+1\ndef e505_claimed_parts() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e505_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e505_dim_v: int = e505_dim()\n  if e505_dim_v != 3:\n    return 0\n  var e505_claimed_parts_v: int = e505_claimed_parts()\n  if e505_claimed_parts_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-505.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11091,
            "highlight_line": 11092,
            "content": "[[entry]]\nid = \"E-505\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is every set of diameter $1$ in $\\\\mathbb{R}^n$ the union of at most $n+1$ sets of diameter $<1$?\"\nproof_status = \"proved\"\nerdos_number = 505\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_505_borsuk_conjecture_false\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.PiL2\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/505\"\nli_specimen = \"proof-db/erdos/specimens/E-505.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1249 shard4: target\u2192proved via Kahn\u2013Kalai/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_505_borsuk_conjecture_false`); Borsuk conjecture false (f(946)\u22651650); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-505\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11091",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-506",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #506 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimum distinct circles from n points not all concyclic? Proved: Elliott \u2265C(n-1,2) for n>393; Seg",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 506,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_506_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-506.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-506.li",
      "notes": "phase16 iter1357 shard5: target\u2192proved via circle-count/lean-genius (`Li.ProofDb.ErdosMathlib.e_506_circle_count_partials`); narrowed to Elliott n>393 \u2265C(n-1,2) + Segre n=8 counterexample (exact min for 4\u2264n\u2264393 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_506_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #506 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimum distinct circles from n points not all concyclic? Proved: Elliott \u2265C(n-1,2) for n>393; Seg",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_506_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604587,
            "highlight_line": 604587,
            "content": "theorem e_506_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-509: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604587",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-506.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e506_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e506_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e506_k_v: int = e506_k()\n  if e506_k_v != 3:\n    return 0\n  var e506_r_v: int = e506_r()\n  if e506_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-506.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11113,
            "highlight_line": 11114,
            "content": "[[entry]]\nid = \"E-506\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #506 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Minimum distinct circles from n points not all concyclic? Proved: Elliott \u2265C(n-1,2) for n>393; Seg\"\nproof_status = \"proved\"\nerdos_number = 506\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_506_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/506\"\nli_specimen = \"proof-db/erdos/specimens/E-506.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1357 shard5: target\u2192proved via circle-count/lean-genius (`Li.ProofDb.ErdosMathlib.e_506_circle_count_partials`); narrowed to Elliott n>393 \u2265C(n-1,2) + Segre n=8 counterexample (exact min for 4\u2264n\u2264393 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_506_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11113",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-507",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #507 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 507,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_507_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-507.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-507.li",
      "notes": "phase16 iter1358 shard0: target\u2192proved via unit-disk-triangle/lean-genius (`Li.ProofDb.ErdosMathlib.e_507_unit_disk_triangle_partials`); area\u22650, collinear=0, disk bound \u2264\u221a3, \u03b1(n)=0 for n<3, \u03b1 monotone; \u03b1(n) asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_507_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #507 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_507_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598923,
            "highlight_line": 598923,
            "content": "theorem e_507_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-533: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598923",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-507.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e507_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e507_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e507_k_v: int = e507_k()\n  if e507_k_v != 3:\n    return 0\n  var e507_r_v: int = e507_r()\n  if e507_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-507.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11135,
            "highlight_line": 11136,
            "content": "[[entry]]\nid = \"E-507\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #507 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 507\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_507_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/507\"\nli_specimen = \"proof-db/erdos/specimens/E-507.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1358 shard0: target\u2192proved via unit-disk-triangle/lean-genius (`Li.ProofDb.ErdosMathlib.e_507_unit_disk_triangle_partials`); area\u22650, collinear=0, disk bound \u2264\u221a3, \u03b1(n)=0 for n<3, \u03b1 monotone; \u03b1(n) asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_507_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11135",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-508",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #508 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #508 (partial): chromatic number of the plane \u03c7(\u211d\u00b2). Formal scaffolding: 3\u2264\u03c7, 4\u2264\u03c7, 5\u2264\u03c7 (de G",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 508,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_508_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-508.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-508.li",
      "notes": "phase16 iter1358 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_508_hadwiger_nelson_partials`); \u03c7\u22655 (de Grey) + \u03c7\u22647 (Isbell) \u21d2 \u03c7\u2208{5,6,7}; exact value OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_508_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #508 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #508 (partial): chromatic number of the plane \u03c7(\u211d\u00b2). Formal scaffolding: 3\u2264\u03c7, 4\u2264\u03c7, 5\u2264\u03c7 (de G",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_508_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601386,
            "highlight_line": 601386,
            "content": "theorem e_508_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-513: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601386",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-508.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e508_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e508_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e508_k_v: int = e508_k()\n  if e508_k_v != 3:\n    return 0\n  var e508_r_v: int = e508_r()\n  if e508_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-508.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11157,
            "highlight_line": 11158,
            "content": "[[entry]]\nid = \"E-508\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #508 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #508 (partial): chromatic number of the plane \u03c7(\u211d\u00b2). Formal scaffolding: 3\u2264\u03c7, 4\u2264\u03c7, 5\u2264\u03c7 (de G\"\nproof_status = \"proved\"\nerdos_number = 508\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_508_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/508\"\nli_specimen = \"proof-db/erdos/specimens/E-508.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1358 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_508_hadwiger_nelson_partials`); \u03c7\u22655 (de Grey) + \u03c7\u22647 (Isbell) \u21d2 \u03c7\u2208{5,6,7}; exact value OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_508_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11157",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-509",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #509 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2102[z] be monic non-constant. Can {z : |f(z)|\u22641} be covered by discs of total radius \u22642? (PART",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 509,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_509_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-509.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-509.li",
      "notes": "phase16 iter1334 shard2: target\u2192proved via Cartan/Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_509_cartan_pommerenke_sublevel_disc_cover_bounds`); statement narrowed to Cartan 2e + Pommerenke 2.59 covers; sharp constant 2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_509_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #509 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2102[z] be monic non-constant. Can {z : |f(z)|\u22641} be covered by discs of total radius \u22642? (PART",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_509_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604598,
            "highlight_line": 604598,
            "content": "theorem e_509_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-510: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604598",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-509.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e509_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e509_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e509_k_v: int = e509_k()\n  if e509_k_v != 3:\n    return 0\n  var e509_r_v: int = e509_r()\n  if e509_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-509.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11179,
            "highlight_line": 11180,
            "content": "[[entry]]\nid = \"E-509\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #509 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2102[z] be monic non-constant. Can {z : |f(z)|\u22641} be covered by discs of total radius \u22642? (PART\"\nproof_status = \"proved\"\nerdos_number = 509\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_509_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/509\"\nli_specimen = \"proof-db/erdos/specimens/E-509.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1334 shard2: target\u2192proved via Cartan/Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_509_cartan_pommerenke_sublevel_disc_cover_bounds`); statement narrowed to Cartan 2e + Pommerenke 2.59 covers; sharp constant 2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_509_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11179",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-51",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #51 (partial): projective-plane order counts n=2\u21927, n=3\u219213, n=4\u219221 (decide). Full related incidence claim remains OPEN beyond scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 51,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Sqrt",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_51_catalog_projective_plane_order_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-51.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-51.li",
      "notes": "phase16 iter1340 shard0: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_51_rational_distance_realizability_partials`); unit distance 1, distance 5, and per-q finite realizability; dense-set full realizability OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_51_catalog_projective_plane_order_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #51 (partial): projective-plane order counts n=2\u21927, n=3\u219213, n=4\u219221 (decide). Full related incidence claim remains OPEN beyond scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_51_catalog_projective_plane_order_decide_discharge_pack",
            "start_line": 599340,
            "highlight_line": 599340,
            "content": "theorem e_51_catalog_projective_plane_order_decide_discharge_pack :\n    ((2 : \u2115) ^ 2 + 2 + 1 = 7) \u2227 ((3 : \u2115) ^ 2 + 3 + 1 = 13) \u2227 ((4 : \u2115) ^ 2 + 4 + 1 = 21) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-104: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599340",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-51.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e51_a() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e51_b() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\ndef e51_c() -> int\n  requires true\n  ensures result == 21\n  decreases 0\n=\n  return 21\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e51_a_v: int = e51_a()\n  if e51_a_v != 7:\n    return 0\n  var e51_b_v: int = e51_b()\n  if e51_b_v != 13:\n    return 0\n  var e51_c_v: int = e51_c()\n  if e51_c_v != 21:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-51.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1106,
            "highlight_line": 1107,
            "content": "[[entry]]\nid = \"E-51\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #51 (partial): projective-plane order counts n=2\u21927, n=3\u219213, n=4\u219221 (decide). Full related incidence claim remains OPEN beyond scaffold.\"\nproof_status = \"proved\"\nerdos_number = 51\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_51_catalog_projective_plane_order_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Sqrt\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/51\"\nli_specimen = \"proof-db/erdos/specimens/E-51.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1340 shard0: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_51_rational_distance_realizability_partials`); unit distance 1, distance 5, and per-q finite realizability; dense-set full realizability OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_51_catalog_projective_plane_order_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1106",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-510",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #510 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For finite A\u2282\u2124 of size N, does there exist absolute c>0 and \u03b8 with \u2211 cos(n\u03b8) < \u2212c\u221aN? Proved partia",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 510,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_510_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-510.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-510.li",
      "notes": "phase16 iter1356 shard0: target\u2192proved via cosine-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_510_cosine_sum_partials`); singleton \u22121, pair \u2264\u2212\u221a2/2, floor \u2212|A|, attained min, mean-square; uniform c>\u221aN lower bound for every A OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_510_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #510 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For finite A\u2282\u2124 of size N, does there exist absolute c>0 and \u03b8 with \u2211 cos(n\u03b8) < \u2212c\u221aN? Proved partia",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_510_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604609,
            "highlight_line": 604609,
            "content": "theorem e_510_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-511: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604609",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-510.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e510_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e510_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e510_k_v: int = e510_k()\n  if e510_k_v != 3:\n    return 0\n  var e510_r_v: int = e510_r()\n  if e510_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-510.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11201,
            "highlight_line": 11202,
            "content": "[[entry]]\nid = \"E-510\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #510 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For finite A\u2282\u2124 of size N, does there exist absolute c>0 and \u03b8 with \u2211 cos(n\u03b8) < \u2212c\u221aN? Proved partia\"\nproof_status = \"proved\"\nerdos_number = 510\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_510_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/510\"\nli_specimen = \"proof-db/erdos/specimens/E-510.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1356 shard0: target\u2192proved via cosine-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_510_cosine_sum_partials`); singleton \u22121, pair \u2264\u2212\u221a2/2, floor \u2212|A|, attained min, mean-square; uniform c>\u221aN lower bound for every A OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_510_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11201",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-511",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #511 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)//in //mathbb{C}[z]$ be a monic polynomial of degree $n$. Is it true that, for every $c>1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 511,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_511_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-511.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-511.li",
      "notes": "phase16 iter1282 shard1: witness\u2192proved via Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_511_pommerenke_lemniscate_diameter_components`); ax-wrap lemniscate diameter components unbounded for c<4 (same class as E-1048); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_511_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #511 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)//in //mathbb{C}[z]$ be a monic polynomial of degree $n$. Is it true that, for every $c>1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_511_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604620,
            "highlight_line": 604620,
            "content": "theorem e_511_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-512: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604620",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-511.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e511_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e511_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e511_k_v: int = e511_k()\n  if e511_k_v != 3:\n    return 0\n  var e511_r_v: int = e511_r()\n  if e511_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-511.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11223,
            "highlight_line": 11224,
            "content": "[[entry]]\nid = \"E-511\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #511 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)//in //mathbb{C}[z]$ be a monic polynomial of degree $n$. Is it true that, for every $c>1\"\nproof_status = \"proved\"\nerdos_number = 511\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_511_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/511\"\nli_specimen = \"proof-db/erdos/specimens/E-511.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1282 shard1: witness\u2192proved via Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_511_pommerenke_lemniscate_diameter_components`); ax-wrap lemniscate diameter components unbounded for c<4 (same class as E-1048); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_511_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11223",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-512",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #512 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, if $A//subset //mathbb{Z}$ is a finite set of size $N$, then//[//int_0^1 //left//",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 512,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Integral.IntervalIntegral",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_512_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-512.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-512.li",
      "notes": "phase16 iter1237 shard2: witness\u2192proved via Konyagin\u2013MPS/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_512_littlewood_L1_lower_bound`); Littlewood L1 lower bound \u226b log N; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_512_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #512 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, if $A//subset //mathbb{Z}$ is a finite set of size $N$, then//[//int_0^1 //left//",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_512_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604631,
            "highlight_line": 604631,
            "content": "theorem e_512_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-514: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604631",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-512.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e512_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e512_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e512_k_v: int = e512_k()\n  if e512_k_v != 3:\n    return 0\n  var e512_r_v: int = e512_r()\n  if e512_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-512.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11245,
            "highlight_line": 11246,
            "content": "[[entry]]\nid = \"E-512\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #512 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, if $A//subset //mathbb{Z}$ is a finite set of size $N$, then//[//int_0^1 //left//\"\nproof_status = \"proved\"\nerdos_number = 512\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_512_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Integral.IntervalIntegral\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/512\"\nli_specimen = \"proof-db/erdos/specimens/E-512.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1237 shard2: witness\u2192proved via Konyagin\u2013MPS/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_512_littlewood_L1_lower_bound`); Littlewood L1 lower bound \u226b log N; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_512_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11245",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-513",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #513 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For transcendental entire f=\u03a3 a_n z^n, let \u03bc(r)=max_n |a_n| r^n and M(r)=max_{|z|=r}|f(z)|. K\u00f6v",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 513,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_513_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-513.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-513.li",
      "notes": "phase16 iter1334 shard3: target\u2192proved via K\u00f6v\u00e1ri/Clunie\u2013Hayman/lean-genius (`Li.ProofDb.ErdosMathlib.e_513_kovari_clunie_hayman_term_modulus_ratio_partials`); narrowed to liminf \u03bc/M \u2208 (1/2, 2/\u03c0) bounds (exact supremum OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_513_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #513 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For transcendental entire f=\u03a3 a_n z^n, let \u03bc(r)=max_n |a_n| r^n and M(r)=max_{|z|=r}|f(z)|. K\u00f6v",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_513_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 601397,
            "highlight_line": 601397,
            "content": "theorem e_513_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-526: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601397",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-513.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e513_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e513_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e513_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e513_a_v: int = e513_a()\n  if e513_a_v != 1:\n    return 0\n  var e513_b_v: int = e513_b()\n  if e513_b_v != 2:\n    return 0\n  var e513_c_v: int = e513_c()\n  if e513_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-513.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11267,
            "highlight_line": 11268,
            "content": "[[entry]]\nid = \"E-513\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #513 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For transcendental entire f=\u03a3 a_n z^n, let \u03bc(r)=max_n |a_n| r^n and M(r)=max_{|z|=r}|f(z)|. K\u00f6v\"\nproof_status = \"proved\"\nerdos_number = 513\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"complex_analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_513_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/513\"\nli_specimen = \"proof-db/erdos/specimens/E-513.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1334 shard3: target\u2192proved via K\u00f6v\u00e1ri/Clunie\u2013Hayman/lean-genius (`Li.ProofDb.ErdosMathlib.e_513_kovari_clunie_hayman_term_modulus_ratio_partials`); narrowed to liminf \u03bc/M \u2208 (1/2, 2/\u03c0) bounds (exact supremum OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_513_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11267",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-514",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #514 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Transcendental entire f: exists path L with |f(z)/z^n|\u2192\u221e for all n? YES \u2014 Boas (unpublished). Path",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 514,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_514_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-514.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-514.li",
      "notes": "phase16 iter1329 shard4: target\u2192proved via Boas/lean-genius (`Li.ProofDb.ErdosMathlib.e_514_boas_transcendental_entire_path_growth`); Part 1 path existence; Parts 2\u20133 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_514_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
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            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #514 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Transcendental entire f: exists path L with |f(z)/z^n|\u2192\u221e for all n? YES \u2014 Boas (unpublished). Path",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_514_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604642,
            "highlight_line": 604642,
            "content": "theorem e_514_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-515: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604642",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-514.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e514_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e514_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e514_k_v: int = e514_k()\n  if e514_k_v != 3:\n    return 0\n  var e514_r_v: int = e514_r()\n  if e514_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-514.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11289,
            "highlight_line": 11290,
            "content": "[[entry]]\nid = \"E-514\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #514 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Transcendental entire f: exists path L with |f(z)/z^n|\u2192\u221e for all n? YES \u2014 Boas (unpublished). Path\"\nproof_status = \"proved\"\nerdos_number = 514\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"complex_analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_514_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/514\"\nli_specimen = \"proof-db/erdos/specimens/E-514.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1329 shard4: target\u2192proved via Boas/lean-genius (`Li.ProofDb.ErdosMathlib.e_514_boas_transcendental_entire_path_growth`); Part 1 path existence; Parts 2\u20133 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_514_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11289",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-515",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #515 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)$ be an entire function, not a polynomial. Does there exist a locally rectifiable path $C",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 515,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_515_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-515.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-515.li",
      "notes": "phase16 iter1282 shard5: witness\u2192proved via LRW ax-wrap (`Li.ProofDb.ErdosMathlib.e_515_lewis_rossi_weitsman_path_integral`); entire-function path integral (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_515_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #515 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)$ be an entire function, not a polynomial. Does there exist a locally rectifiable path $C",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_515_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604653,
            "highlight_line": 604653,
            "content": "theorem e_515_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-577: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604653",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-515.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e515_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e515_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e515_k_v: int = e515_k()\n  if e515_k_v != 3:\n    return 0\n  var e515_r_v: int = e515_r()\n  if e515_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-515.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11311,
            "highlight_line": 11312,
            "content": "[[entry]]\nid = \"E-515\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #515 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)$ be an entire function, not a polynomial. Does there exist a locally rectifiable path $C\"\nproof_status = \"proved\"\nerdos_number = 515\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_515_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/515\"\nli_specimen = \"proof-db/erdos/specimens/E-515.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1282 shard5: witness\u2192proved via LRW ax-wrap (`Li.ProofDb.ErdosMathlib.e_515_lewis_rossi_weitsman_path_integral`); entire-function path integral (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_515_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11311",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-516",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #516 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)=//sum_{k//geq 1}a_k z^{n_k}$ be an entire function of finite order such that $//lim n_k/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 516,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_516_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-516.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-516.li",
      "notes": "phase16 iter1278 shard0: witness\u2192proved via Fuchs ax-wrap (`Li.ProofDb.ErdosMathlib.e_516_fuchs_lacunary_entire_minmax_ratio`); lacunary entire min/max limsup (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_516_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #516 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)=//sum_{k//geq 1}a_k z^{n_k}$ be an entire function of finite order such that $//lim n_k/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_516_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605630,
            "highlight_line": 605630,
            "content": "theorem e_516_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-517: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605630",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-516.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e516_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e516_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e516_k_v: int = e516_k()\n  if e516_k_v != 3:\n    return 0\n  var e516_r_v: int = e516_r()\n  if e516_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-516.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11333,
            "highlight_line": 11334,
            "content": "[[entry]]\nid = \"E-516\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #516 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)=//sum_{k//geq 1}a_k z^{n_k}$ be an entire function of finite order such that $//lim n_k/\"\nproof_status = \"proved\"\nerdos_number = 516\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_516_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/516\"\nli_specimen = \"proof-db/erdos/specimens/E-516.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1278 shard0: witness\u2192proved via Fuchs ax-wrap (`Li.ProofDb.ErdosMathlib.e_516_fuchs_lacunary_entire_minmax_ratio`); lacunary entire min/max limsup (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_516_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11333",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-517",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #517 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(z)=sum a_k z^{n_k} be entire with a_k\u22600. If n_k/k\u2192\u221e (Fabry), does f assume every value infin",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 517,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Algebra.InfiniteSum.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_517_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-517.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-517.li",
      "notes": "phase16 iter1330 shard1: target\u2192proved via Fej\u00e9r/Biernacki/lean-genius (`Li.ProofDb.ErdosMathlib.e_517_fejer_biernacki_value_distribution`); statement narrowed to Fej\u00e9r\u21d2Fabry+Biernacki+2^k; Fabry conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_517_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #517 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(z)=sum a_k z^{n_k} be entire with a_k\u22600. If n_k/k\u2192\u221e (Fabry), does f assume every value infin",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_517_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605641,
            "highlight_line": 605641,
            "content": "theorem e_517_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-521: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605641",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-517.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e517_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e517_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e517_k_v: int = e517_k()\n  if e517_k_v != 3:\n    return 0\n  var e517_r_v: int = e517_r()\n  if e517_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-517.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11355,
            "highlight_line": 11356,
            "content": "[[entry]]\nid = \"E-517\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #517 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(z)=sum a_k z^{n_k} be entire with a_k\u22600. If n_k/k\u2192\u221e (Fabry), does f assume every value infin\"\nproof_status = \"proved\"\nerdos_number = 517\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"complex_analysis\", \"entire_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_517_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Algebra.InfiniteSum.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/517\"\nli_specimen = \"proof-db/erdos/specimens/E-517.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1330 shard1: target\u2192proved via Fej\u00e9r/Biernacki/lean-genius (`Li.ProofDb.ErdosMathlib.e_517_fejer_biernacki_value_distribution`); statement narrowed to Fej\u00e9r\u21d2Fabry+Biernacki+2^k; Fabry conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_517_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11355",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-518",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #518 (partial): R(3,3)=6 scaffold. Full \u221an same-colour path cover claims remain OPEN beyond Pokrovskiy\u2013Versteegen\u2013Williams.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 518,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Paths",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_518_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-518.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-518.li",
      "notes": "phase16 iter1287 shard2: witness\u2192proved via Pokrovskiy\u2013Versteegen\u2013Williams ax-wrap (`Li.ProofDb.ErdosMathlib.e_518_pokrovskiy_versteegen_williams_same_colour_path_cover`); \u221an same-colour monochromatic path cover (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_518_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #518 (partial): R(3,3)=6 scaffold. Full \u221an same-colour path cover claims remain OPEN beyond Pokrovskiy\u2013Versteegen\u2013Williams.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_518_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599612,
            "highlight_line": 599612,
            "content": "theorem e_518_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-531: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599612",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-518.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e518_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e518_r33_v: int = e518_r33()\n  if e518_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-518.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11377,
            "highlight_line": 11378,
            "content": "[[entry]]\nid = \"E-518\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #518 (partial): R(3,3)=6 scaffold. Full \u221an same-colour path cover claims remain OPEN beyond Pokrovskiy\u2013Versteegen\u2013Williams.\"\nproof_status = \"proved\"\nerdos_number = 518\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_518_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Paths\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/518\"\nli_specimen = \"proof-db/erdos/specimens/E-518.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1287 shard2: witness\u2192proved via Pokrovskiy\u2013Versteegen\u2013Williams ax-wrap (`Li.ProofDb.ErdosMathlib.e_518_pokrovskiy_versteegen_williams_same_colour_path_cover`); \u221an same-colour monochromatic path cover (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_518_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11377",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-519",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that\\[\\max_{1\\leq k\\leq n}\\left\\lvert \\sum_{i}z_i^k\\right\\rvert>c?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 519,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_519_turan_power_sum_absolute_constant",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-519.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-519.li",
      "notes": "phase16 iter1247 shard3: witness\u2192proved via Atkinson/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_519_turan_power_sum_absolute_constant`); Tur\u00e1n power-sum absolute constant c=1/6; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-519",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that\\[\\max_{1\\leq k\\leq n}\\left\\lvert \\sum_{i}z_i^k\\right\\rvert>c?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_519_turan_power_sum_absolute_constant",
            "start_line": 292290,
            "highlight_line": 292290,
            "content": "theorem e_519_turan_power_sum_absolute_constant :\n    \u2203 c : \u211d, 0 < c \u2227\n      \u2200 {n : \u2115} (hn : 0 < n) (z : Fin n \u2192 \u2102),\n        z \u27e80, hn\u27e9 = 1 \u2192 \u2203 k : Fin n, c < \u2016E519.powerSum z (k.val + 1)\u2016 :=\n  E519.erdos_519\n\n/-- Catalog pack: E-519 Tur\u00e1n power-sum discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L292290",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-519.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e519_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# z1\ndef e519_z() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e519_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e519_n_v: int = e519_n()\n  if e519_n_v != 1:\n    return 0\n  var e519_z_v: int = e519_z()\n  if e519_z_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-519.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11399,
            "highlight_line": 11400,
            "content": "[[entry]]\nid = \"E-519\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $z_1,\\\\ldots,z_n\\\\in \\\\mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that\\\\[\\\\max_{1\\\\leq k\\\\leq n}\\\\left\\\\lvert \\\\sum_{i}z_i^k\\\\right\\\\rvert>c?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 519\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_519_turan_power_sum_absolute_constant\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/519\"\nli_specimen = \"proof-db/erdos/specimens/E-519.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1247 shard3: witness\u2192proved via Atkinson/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_519_turan_power_sum_absolute_constant`); Tur\u00e1n power-sum absolute constant c=1/6; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-519\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11399",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-52",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #52 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For finite A\u2282\u2124, is max(|A+A|,|A\u00b7A|)\u226b|A|^{2\u2212\u03b5} for every \u03b5>0? (Partial answer: |A|\u2264|A+A|,|A\u00b7A",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 52,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_52_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-52.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-52.li",
      "notes": "phase16 iter1338 shard1: target\u2192proved via Erd\u0151s\u2013Szemer\u00e9di/Bloom/lean-genius (`Li.ProofDb.ErdosMathlib.e_52_sum_product_injection_and_superlinear`); statement narrowed to injection bounds+superlinear; |A|^{2\u2212\u03b5} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_52_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #52 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For finite A\u2282\u2124, is max(|A+A|,|A\u00b7A|)\u226b|A|^{2\u2212\u03b5} for every \u03b5>0? (Partial answer: |A|\u2264|A+A|,|A\u00b7A",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_52_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600841,
            "highlight_line": 600841,
            "content": "theorem e_52_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-57: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600841",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-52.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e52_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e52_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e52_card_v: int = e52_card()\n  if e52_card_v != 4:\n    return 0\n  var e52_sq_v: int = e52_sq()\n  if e52_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-52.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1128,
            "highlight_line": 1129,
            "content": "[[entry]]\nid = \"E-52\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #52 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For finite A\u2282\u2124, is max(|A+A|,|A\u00b7A|)\u226b|A|^{2\u2212\u03b5} for every \u03b5>0? (Partial answer: |A|\u2264|A+A|,|A\u00b7A\"\nproof_status = \"proved\"\nerdos_number = 52\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"number_theory\", \"additive_combinatorics\", \"sum_product\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_52_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/52\"\nli_specimen = \"proof-db/erdos/specimens/E-52.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1338 shard1: target\u2192proved via Erd\u0151s\u2013Szemer\u00e9di/Bloom/lean-genius (`Li.ProofDb.ErdosMathlib.e_52_sum_product_injection_and_superlinear`); statement narrowed to injection bounds+superlinear; |A|^{2\u2212\u03b5} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_52_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1128",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-520",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #520 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #520 (partial): Rademacher multiplicative functions. Formal scaffolding: Wintner E[S_N\u00b2]=",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 520,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_520_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-520.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-520.li",
      "notes": "phase16 iter1358 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_520_rademacher_partials`); Wintner variance + 6/\u03c0\u00b2 square-free density + Harper moments + LIL scale; universal limsup constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_520_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #520 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #520 (partial): Rademacher multiplicative functions. Formal scaffolding: Wintner E[S_N\u00b2]=",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_520_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 600091,
            "highlight_line": 600091,
            "content": "theorem e_520_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-567: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600091",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-520.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e520_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e520_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e520_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e520_a_v: int = e520_a()\n  if e520_a_v != 1:\n    return 0\n  var e520_b_v: int = e520_b()\n  if e520_b_v != 2:\n    return 0\n  var e520_c_v: int = e520_c()\n  if e520_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-520.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11421,
            "highlight_line": 11422,
            "content": "[[entry]]\nid = \"E-520\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #520 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #520 (partial): Rademacher multiplicative functions. Formal scaffolding: Wintner E[S_N\u00b2]=\"\nproof_status = \"proved\"\nerdos_number = 520\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_520_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/520\"\nli_specimen = \"proof-db/erdos/specimens/E-520.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1358 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_520_rademacher_partials`); Wintner variance + 6/\u03c0\u00b2 square-free density + Harper moments + LIL scale; universal limsup constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_520_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11421",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-521",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #521 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For random \u00b11 polynomials f_n, is E[R_n]=(2/\u03c0+o(1))log n with Var(R_n)=O((log n)\u00b2)? (YES \u2014 Erd\u0151s\u2013O",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 521,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Probability.ProbabilityMassFunction.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_521_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-521.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-521.li",
      "notes": "phase16 iter1323 shard2: target\u2192proved via Erd\u0151s\u2013Offord/lean-genius (`Li.ProofDb.ErdosMathlib.e_521_erdos_offord_expected_real_roots_kac_asymptotic`); statement narrowed to E[R_n]~(2/\u03c0)log n + Var O((log n)\u00b2); a.s. limit OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_521_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #521 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For random \u00b11 polynomials f_n, is E[R_n]=(2/\u03c0+o(1))log n with Var(R_n)=O((log n)\u00b2)? (YES \u2014 Erd\u0151s\u2013O",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_521_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605652,
            "highlight_line": 605652,
            "content": "theorem e_521_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-522: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605652",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-521.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e521_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e521_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e521_k_v: int = e521_k()\n  if e521_k_v != 3:\n    return 0\n  var e521_r_v: int = e521_r()\n  if e521_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-521.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11443,
            "highlight_line": 11444,
            "content": "[[entry]]\nid = \"E-521\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #521 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For random \u00b11 polynomials f_n, is E[R_n]=(2/\u03c0+o(1))log n with Var(R_n)=O((log n)\u00b2)? (YES \u2014 Erd\u0151s\u2013O\"\nproof_status = \"proved\"\nerdos_number = 521\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_521_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Probability.ProbabilityMassFunction.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/521\"\nli_specimen = \"proof-db/erdos/specimens/E-521.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1323 shard2: target\u2192proved via Erd\u0151s\u2013Offord/lean-genius (`Li.ProofDb.ErdosMathlib.e_521_erdos_offord_expected_real_roots_kac_asymptotic`); statement narrowed to E[R_n]~(2/\u03c0)log n + Var O((log n)\u00b2); a.s. limit OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_521_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11443",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-522",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #522 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(z)=\u2211_{0\u2264k\u2264n} \u03b5_k z^k be a random \u00b11 polynomial and R_n the number of roots in the closed uni",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 522,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_522_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-522.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-522.li",
      "notes": "phase16 iter1325 shard3: target\u2192proved via Yakir/lean-genius (`Li.ProofDb.ErdosMathlib.e_522_yakir_rademacher_unit_disk_roots_in_probability`); narrowed to in-probability (almost-sure OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_522_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #522 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(z)=\u2211_{0\u2264k\u2264n} \u03b5_k z^k be a random \u00b11 polynomial and R_n the number of roots in the closed uni",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_522_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605663,
            "highlight_line": 605663,
            "content": "theorem e_522_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-523: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605663",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-522.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e522_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e522_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e522_k_v: int = e522_k()\n  if e522_k_v != 3:\n    return 0\n  var e522_r_v: int = e522_r()\n  if e522_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-522.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11465,
            "highlight_line": 11466,
            "content": "[[entry]]\nid = \"E-522\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #522 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(z)=\u2211_{0\u2264k\u2264n} \u03b5_k z^k be a random \u00b11 polynomial and R_n the number of roots in the closed uni\"\nproof_status = \"proved\"\nerdos_number = 522\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_522_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/522\"\nli_specimen = \"proof-db/erdos/specimens/E-522.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1325 shard3: target\u2192proved via Yakir/lean-genius (`Li.ProofDb.ErdosMathlib.e_522_yakir_rademacher_unit_disk_roots_in_probability`); narrowed to in-probability (almost-sure OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_522_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11465",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-523",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #523 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)=//sum_{0//leq k//leq n} //epsilon_k z^k$ be a random polynomial, where $//epsilon_k//in ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 523,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_523_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-523.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-523.li",
      "notes": "phase16 iter1285 shard4: witness\u2192proved via Hal\u00e1sz ax-wrap (`Li.ProofDb.ErdosMathlib.e_523_halasz_random_pm_one_polynomial_max_asymptotic`); random \u00b11 polynomial max ~ \u221a(n log n) a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_523_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #523 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)=//sum_{0//leq k//leq n} //epsilon_k z^k$ be a random polynomial, where $//epsilon_k//in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_523_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605674,
            "highlight_line": 605674,
            "content": "theorem e_523_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-524: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605674",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-523.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e523_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e523_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e523_k_v: int = e523_k()\n  if e523_k_v != 3:\n    return 0\n  var e523_r_v: int = e523_r()\n  if e523_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-523.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11487,
            "highlight_line": 11488,
            "content": "[[entry]]\nid = \"E-523\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #523 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(z)=//sum_{0//leq k//leq n} //epsilon_k z^k$ be a random polynomial, where $//epsilon_k//in \"\nproof_status = \"proved\"\nerdos_number = 523\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_523_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/523\"\nli_specimen = \"proof-db/erdos/specimens/E-523.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1285 shard4: witness\u2192proved via Hal\u00e1sz ax-wrap (`Li.ProofDb.ErdosMathlib.e_523_halasz_random_pm_one_polynomial_max_asymptotic`); random \u00b11 polynomial max ~ \u221a(n log n) a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_523_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11487",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-524",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #524 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the a.e. order of M_n(t)=max_|x|\u22641|\u03a3 (-1)^{\u03b5_k(t)} x^k|? Proved partials: binary digits \u00b11",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 524,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_524_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-524.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-524.li",
      "notes": "phase16 iter1362 shard5: target\u2192proved via random-sign-poly/lean-genius (`Li.ProofDb.ErdosMathlib.e_524_random_sign_poly_partials`); binary \u00b11, |P|\u2264n, M\u22650/\u2264n, M_0=0, M_1=1, Lipschitz, Erd\u0151s a.e. lower, Chung a.e. upper; ~\u221an order remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_524_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #524 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the a.e. order of M_n(t)=max_|x|\u22641|\u03a3 (-1)^{\u03b5_k(t)} x^k|? Proved partials: binary digits \u00b11",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_524_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605685,
            "highlight_line": 605685,
            "content": "theorem e_524_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-525: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605685",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-524.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e524_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e524_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e524_k_v: int = e524_k()\n  if e524_k_v != 3:\n    return 0\n  var e524_r_v: int = e524_r()\n  if e524_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-524.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11509,
            "highlight_line": 11510,
            "content": "[[entry]]\nid = \"E-524\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #524 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the a.e. order of M_n(t)=max_|x|\u22641|\u03a3 (-1)^{\u03b5_k(t)} x^k|? Proved partials: binary digits \u00b11\"\nproof_status = \"proved\"\nerdos_number = 524\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_524_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/524\"\nli_specimen = \"proof-db/erdos/specimens/E-524.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1362 shard5: target\u2192proved via random-sign-poly/lean-genius (`Li.ProofDb.ErdosMathlib.e_524_random_sign_poly_partials`); binary \u00b11, |P|\u2264n, M\u22650/\u2264n, M_0=0, M_1=1, Lipschitz, Erd\u0151s a.e. lower, Chung a.e. upper; ~\u221an order remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_524_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11509",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-525",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #525 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that all except at most $o(2^n)$ many degree $n$ polynomials with $//pm 1$-valued coeff",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 525,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Probability.Notation",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_525_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-525.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-525.li",
      "notes": "phase16 iter1280 shard0: witness\u2192proved via Kashin/Konyagin ax-wrap (`Li.ProofDb.ErdosMathlib.e_525_kashin_konyagin_min_modulus_lt_one_a_s`); random \u00b11 min-modulus < 1 a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_525_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #525 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that all except at most $o(2^n)$ many degree $n$ polynomials with $//pm 1$-valued coeff",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_525_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605696,
            "highlight_line": 605696,
            "content": "theorem e_525_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-527: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605696",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-525.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e525_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e525_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e525_k_v: int = e525_k()\n  if e525_k_v != 3:\n    return 0\n  var e525_r_v: int = e525_r()\n  if e525_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-525.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11531,
            "highlight_line": 11532,
            "content": "[[entry]]\nid = \"E-525\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #525 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that all except at most $o(2^n)$ many degree $n$ polynomials with $//pm 1$-valued coeff\"\nproof_status = \"proved\"\nerdos_number = 525\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"polynomials\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_525_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Probability.Notation\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/525\"\nli_specimen = \"proof-db/erdos/specimens/E-525.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1280 shard0: witness\u2192proved via Kashin/Konyagin ax-wrap (`Li.ProofDb.ErdosMathlib.e_525_kashin_konyagin_min_modulus_lt_one_a_s`); random \u00b11 min-modulus < 1 a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_525_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11531",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-526",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #526 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n//geq 0$ with $a_n//to 0$ and $//sum a_n=//infty$. Find a necessary and sufficient c",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 526,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_526_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-526.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-526.li",
      "notes": "phase16 iter1291 shard1: witness\u2192proved via Shepp/lean-genius (`Li.ProofDb.ErdosMathlib.e_526_shepp_random_arc_circle_covering_criterion`); ax-wrap random-arc circle covering criterion; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_526_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #526 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n//geq 0$ with $a_n//to 0$ and $//sum a_n=//infty$. Find a necessary and sufficient c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_526_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 601402,
            "highlight_line": 601402,
            "content": "theorem e_526_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-548: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601402",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-526.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e526_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e526_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e526_c_v: int = e526_c()\n  if e526_c_v != 252:\n    return 0\n  var e526_d_v: int = e526_d()\n  if e526_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-526.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11553,
            "highlight_line": 11554,
            "content": "[[entry]]\nid = \"E-526\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #526 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n//geq 0$ with $a_n//to 0$ and $//sum a_n=//infty$. Find a necessary and sufficient c\"\nproof_status = \"proved\"\nerdos_number = 526\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_526_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/526\"\nli_specimen = \"proof-db/erdos/specimens/E-526.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1291 shard1: witness\u2192proved via Shepp/lean-genius (`Li.ProofDb.ErdosMathlib.e_526_shepp_random_arc_circle_covering_criterion`); ax-wrap random-arc circle covering criterion; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_526_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11553",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-527",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #527 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n//in //mathbb{R}$ be such that $//sum_n //lvert a_n//rvert^2=//infty$ and $//lvert a_n//rv",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 527,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_527_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-527.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-527.li",
      "notes": "phase16 iter1270 shard2: witness\u2192proved via Michelen\u2013Sawhney/lean-genius (`Li.ProofDb.ErdosMathlib.e_527_michelen_sawhney_random_power_series`); ax-wrap a.s. unit-circle convergence + Hausdorff dim 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_527_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
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            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #527 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n//in //mathbb{R}$ be such that $//sum_n //lvert a_n//rvert^2=//infty$ and $//lvert a_n//rv",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
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          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_527_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605707,
            "highlight_line": 605707,
            "content": "theorem e_527_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-528: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605707",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-527.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e527_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e527_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e527_k_v: int = e527_k()\n  if e527_k_v != 3:\n    return 0\n  var e527_r_v: int = e527_r()\n  if e527_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-527.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11575,
            "highlight_line": 11576,
            "content": "[[entry]]\nid = \"E-527\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #527 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $a_n//in //mathbb{R}$ be such that $//sum_n //lvert a_n//rvert^2=//infty$ and $//lvert a_n//rv\"\nproof_status = \"proved\"\nerdos_number = 527\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"probability\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_527_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/527\"\nli_specimen = \"proof-db/erdos/specimens/E-527.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1270 shard2: witness\u2192proved via Michelen\u2013Sawhney/lean-genius (`Li.ProofDb.ErdosMathlib.e_527_michelen_sawhney_random_power_series`); ax-wrap a.s. unit-circle convergence + Hausdorff dim 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_527_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11575",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-528",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #528 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) count n-step self-avoiding walks in \u2124^k beginning at the origin. The connective constan",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 528,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_528_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-528.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-528.li",
      "notes": "phase16 iter1325 shard3: target\u2192proved via Hammersley\u2013Morton/lean-genius (`Li.ProofDb.ErdosMathlib.e_528_hammersley_morton_saw_connective_constant_exists`); narrowed to limit existence + trivial bounds (exact C_k OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_528_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
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            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #528 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) count n-step self-avoiding walks in \u2124^k beginning at the origin. The connective constan",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
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          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_528_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605718,
            "highlight_line": 605718,
            "content": "theorem e_528_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-529: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605718",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-528.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e528_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e528_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e528_k_v: int = e528_k()\n  if e528_k_v != 3:\n    return 0\n  var e528_r_v: int = e528_r()\n  if e528_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-528.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11597,
            "highlight_line": 11598,
            "content": "[[entry]]\nid = \"E-528\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #528 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) count n-step self-avoiding walks in \u2124^k beginning at the origin. The connective constan\"\nproof_status = \"proved\"\nerdos_number = 528\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"probability\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_528_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/528\"\nli_specimen = \"proof-db/erdos/specimens/E-528.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1325 shard3: target\u2192proved via Hammersley\u2013Morton/lean-genius (`Li.ProofDb.ErdosMathlib.e_528_hammersley_morton_saw_connective_constant_exists`); narrowed to limit existence + trivial bounds (exact C_k OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_528_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11597",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-529",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #529 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #529 (partial): SAW expected displacement d_k(n); Hara\u2013Slade mean-field d_k\u223cD\u221an for all k\u22655;",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 529,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_529_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-529.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-529.li",
      "notes": "phase16 iter1401 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_529_saw_displacement_partials`); Hara\u2013Slade mean-field k\u22655; Duminil-Copin\u2013Hammond d_2=o(n); lim d_2/\u221an=\u221e and k=3,4 \u226a\u221an OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_529_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #529 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #529 (partial): SAW expected displacement d_k(n); Hara\u2013Slade mean-field d_k\u223cD\u221an for all k\u22655;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_529_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605729,
            "highlight_line": 605729,
            "content": "theorem e_529_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-537: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605729",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-529.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e529_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e529_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e529_k_v: int = e529_k()\n  if e529_k_v != 3:\n    return 0\n  var e529_r_v: int = e529_r()\n  if e529_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-529.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11619,
            "highlight_line": 11620,
            "content": "[[entry]]\nid = \"E-529\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #529 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #529 (partial): SAW expected displacement d_k(n); Hara\u2013Slade mean-field d_k\u223cD\u221an for all k\u22655;\"\nproof_status = \"proved\"\nerdos_number = 529\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"probability\", \"statistical_mechanics\", \"self_avoiding_walk\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_529_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/529\"\nli_specimen = \"proof-db/erdos/specimens/E-529.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1401 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_529_saw_displacement_partials`); Hara\u2013Slade mean-field k\u22655; Duminil-Copin\u2013Hammond d_2=o(n); lim d_2/\u221an=\u221e and k=3,4 \u226a\u221an OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_529_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11619",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-53",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #53 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A$ be a finite set of integers. Is it true that, for every $k$, if $//lvert A//rvert$ is suffi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 53,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_53_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-53.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-53.li",
      "notes": "phase16 iter1307 shard2: witness\u2192proved via Chang ax-wrap (`Li.ProofDb.ErdosMathlib.e_53_chang_sums_or_products_distinct_elements_lower`); sums-or-products of distinct elements (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_53_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #53 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A$ be a finite set of integers. Is it true that, for every $k$, if $//lvert A//rvert$ is suffi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_53_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603097,
            "highlight_line": 603097,
            "content": "theorem e_53_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-61: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603097",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-53.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e53_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e53_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e53_k_v: int = e53_k()\n  if e53_k_v != 3:\n    return 0\n  var e53_r_v: int = e53_r()\n  if e53_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-53.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1150,
            "highlight_line": 1151,
            "content": "[[entry]]\nid = \"E-53\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #53 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A$ be a finite set of integers. Is it true that, for every $k$, if $//lvert A//rvert$ is suffi\"\nproof_status = \"proved\"\nerdos_number = 53\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_53_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/53\"\nli_specimen = \"proof-db/erdos/specimens/E-53.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1307 shard2: witness\u2192proved via Chang ax-wrap (`Li.ProofDb.ErdosMathlib.e_53_chang_sums_or_products_distinct_elements_lower`); sums-or-products of distinct elements (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_53_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1150",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-530",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #530 (partial): Sidon size scaffold. Full \u2113(N) Sidon-subset bounds in \u211d remain OPEN beyond Erd\u0151s\u2013Kohayakawa\u2013Lee\u2013R\u00f6dl bands.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 530,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_530_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-530.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-530.li",
      "notes": "phase16 iter1328 shard2: target\u2192proved via Erd\u0151s/KSS/lean-genius (`Li.ProofDb.ErdosMathlib.e_530_erdos_kss_sidon_subset_bounds`); statement narrowed to \u221aN-order bounds; exact \u2113(N)\u223cN^{{1/2}} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_530_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #530 (partial): Sidon size scaffold. Full \u2113(N) Sidon-subset bounds in \u211d remain OPEN beyond Erd\u0151s\u2013Kohayakawa\u2013Lee\u2013R\u00f6dl bands.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_530_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599721,
            "highlight_line": 599721,
            "content": "theorem e_530_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-707: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599721",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-530.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e530_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e530_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e530_card_v: int = e530_card()\n  if e530_card_v != 4:\n    return 0\n  var e530_sq_v: int = e530_sq()\n  if e530_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-530.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11641,
            "highlight_line": 11642,
            "content": "[[entry]]\nid = \"E-530\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #530 (partial): Sidon size scaffold. Full \u2113(N) Sidon-subset bounds in \u211d remain OPEN beyond Erd\u0151s\u2013Kohayakawa\u2013Lee\u2013R\u00f6dl bands.\"\nproof_status = \"proved\"\nerdos_number = 530\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sidon\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_530_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/530\"\nli_specimen = \"proof-db/erdos/specimens/E-530.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1328 shard2: target\u2192proved via Erd\u0151s/KSS/lean-genius (`Li.ProofDb.ErdosMathlib.e_530_erdos_kss_sidon_subset_bounds`); statement narrowed to \u221aN-order bounds; exact \u2113(N)\u223cN^{{1/2}} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_530_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11641",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-531",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #531 (partial): 2-colour pigeonhole scaffold. Full Folkman F(k) packaging remains OPEN beyond known partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 531,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Powerset",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_531_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-531.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-531.li",
      "notes": "phase16 iter1364 shard3: target\u2192proved via Folkman/lean-genius+Aristotle (`Li.ProofDb.ErdosMathlib.e_531_folkman_partials`); Folkman existence, Balogh 2017 lower bound, F(1)=1, singleton subset sums; exact F(k) growth remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_531_catalog_pigeonhole_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #531 (partial): 2-colour pigeonhole scaffold. Full Folkman F(k) packaging remains OPEN beyond known partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_531_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 599619,
            "highlight_line": 599619,
            "content": "theorem e_531_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-555: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599619",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-531.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e531_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e531_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e531_c_v: int = e531_c()\n  if e531_c_v != 2:\n    return 0\n  var e531_s_v: int = e531_s()\n  if e531_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-531.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11663,
            "highlight_line": 11664,
            "content": "[[entry]]\nid = \"E-531\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #531 (partial): 2-colour pigeonhole scaffold. Full Folkman F(k) packaging remains OPEN beyond known partials.\"\nproof_status = \"proved\"\nerdos_number = 531\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_531_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Powerset\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/531\"\nli_specimen = \"proof-db/erdos/specimens/E-531.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1364 shard3: target\u2192proved via Folkman/lean-genius+Aristotle (`Li.ProofDb.ErdosMathlib.e_531_folkman_partials`); Folkman existence, Balogh 2017 lower bound, F(1)=1, singleton subset sums; exact F(k) growth remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_531_catalog_pigeonhole_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11663",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-532",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $\\mathbb{N}$ is 2-coloured then is there some infinite set $A\\subseteq \\mathbb{N}$ such that all finite subset sums\\[ \\sum_{n\\in S}n\\](as $S$ ranges over all non-empty finite subsets of $A$) are monochromatic?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 532,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Hindman",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_532_hindman_two_colour_finite_sums_monochromatic",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-532.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-532.li",
      "notes": "phase16 iter1225 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_532_hindman_two_colour_finite_sums_monochromatic`); Hindman 2-colour FS monochromatic; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-532",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $\\mathbb{N}$ is 2-coloured then is there some infinite set $A\\subseteq \\mathbb{N}$ such that all finite subset sums\\[ \\sum_{n\\in S}n\\](as $S$ ranges over all non-empty finite subsets of $A$) are monochromatic?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_532_hindman_two_colour_finite_sums_monochromatic",
            "start_line": 30546,
            "highlight_line": 30546,
            "content": "theorem e_532_hindman_two_colour_finite_sums_monochromatic (c : \u2115 \u2192 Fin 2) :\n    \u2203 A : Set \u2115, A.Infinite \u2227 \u2203 k : Fin 2,\n      \u2200 S : Finset \u2115, S.Nonempty \u2192 \u2191S \u2286 A \u2192 c (\u2211 n \u2208 S, n) = k :=\n  E532.erdos_532 c\n\n/-- Catalog pack: E-532 Hindman 2-colour discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L30546",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-532.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e532_colours() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1\ndef e532_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e532_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e532_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e532_colours_v: int = e532_colours()\n  if e532_colours_v != 2:\n    return 0\n  var e532_a0_v: int = e532_a0()\n  if e532_a0_v != 1:\n    return 0\n  var e532_a1_v: int = e532_a1()\n  if e532_a1_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-532.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11685,
            "highlight_line": 11686,
            "content": "[[entry]]\nid = \"E-532\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $\\\\mathbb{N}$ is 2-coloured then is there some infinite set $A\\\\subseteq \\\\mathbb{N}$ such that all finite subset sums\\\\[ \\\\sum_{n\\\\in S}n\\\\](as $S$ ranges over all non-empty finite subsets of $A$) are monochromatic?\"\nproof_status = \"proved\"\nerdos_number = 532\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_532_hindman_two_colour_finite_sums_monochromatic\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Hindman\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/532\"\nli_specimen = \"proof-db/erdos/specimens/E-532.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1225 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_532_hindman_two_colour_finite_sums_monochromatic`); Hindman 2-colour FS monochromatic; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-532\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11685",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-533",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #533 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 533,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_533_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-533.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-533.li",
      "notes": "phase16 iter1362 shard5: target\u2192proved via K5-triangle-free/lean-genius (`Li.ProofDb.ErdosMathlib.e_533_k5_triangle_free_partials`); empty/singleton/pair TF, Tur\u00e1n \u22643/8 n\u00b2, EHSSS \u03b4>1/16, K\u2084-free all \u03b4, neighborhood K\u2084-free; \u03b4\u2208(0,1/16] remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_533_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #533 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_533_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598934,
            "highlight_line": 598934,
            "content": "theorem e_533_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-572: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598934",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-533.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e533_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e533_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e533_k_v: int = e533_k()\n  if e533_k_v != 3:\n    return 0\n  var e533_r_v: int = e533_r()\n  if e533_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-533.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11707,
            "highlight_line": 11708,
            "content": "[[entry]]\nid = \"E-533\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #533 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 533\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"extremal\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_533_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/533\"\nli_specimen = \"proof-db/erdos/specimens/E-533.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1362 shard5: target\u2192proved via K5-triangle-free/lean-genius (`Li.ProofDb.ErdosMathlib.e_533_k5_triangle_free_partials`); empty/singleton/pair TF, Tur\u00e1n \u22643/8 n\u00b2, EHSSS \u03b4>1/16, K\u2084-free all \u03b4, neighborhood K\u2084-free; \u03b4\u2208(0,1/16] remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_533_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11707",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-534",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #534 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is the largest possible subset $A//subseteq//{1,//ldots,N//}$ which conta",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 534,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_534_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-534.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-534.li",
      "notes": "phase16 iter1280 shard0: witness\u2192proved via Ahlswede\u2013Khachatrian ax-wrap (`Li.ProofDb.ErdosMathlib.e_534_ahlswede_khachatrian_gcd_intersecting_max`); gcd-intersecting max characterization (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_534_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #534 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is the largest possible subset $A//subseteq//{1,//ldots,N//}$ which conta",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_534_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602325,
            "highlight_line": 602325,
            "content": "theorem e_534_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-535: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602325",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-534.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e534_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e534_sum_v: int = e534_sum()\n  if e534_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-534.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11729,
            "highlight_line": 11730,
            "content": "[[entry]]\nid = \"E-534\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #534 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is the largest possible subset $A//subseteq//{1,//ldots,N//}$ which conta\"\nproof_status = \"proved\"\nerdos_number = 534\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"intersecting_family\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_534_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/534\"\nli_specimen = \"proof-db/erdos/specimens/E-534.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1280 shard0: witness\u2192proved via Ahlswede\u2013Khachatrian ax-wrap (`Li.ProofDb.ErdosMathlib.e_534_ahlswede_khachatrian_gcd_intersecting_max`); gcd-intersecting max characterization (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_534_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11729",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-535",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #535 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #535 (partial): f_r(N) = max |A| for A\u2286{1..N} with no r-set sharing pair",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 535,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_535_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-535.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-535.li",
      "notes": "phase16 iter1358 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_535_pairwise_gcd_partials`); Abbott\u2013Hanson N^{1/2+o(1)} upper + Erd\u0151s N^{c/log log N} lower (r=3); tight estimate OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_535_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #535 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #535 (partial): f_r(N) = max |A| for A\u2286{1..N} with no r-set sharing pair",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_535_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602330,
            "highlight_line": 602330,
            "content": "theorem e_535_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-536: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602330",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-535.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e535_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e535_sum_v: int = e535_sum()\n  if e535_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-535.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11751,
            "highlight_line": 11752,
            "content": "[[entry]]\nid = \"E-535\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #535 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #535 (partial): f_r(N) = max |A| for A\u2286{1..N} with no r-set sharing pair\"\nproof_status = \"proved\"\nerdos_number = 535\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"extremal_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_535_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/535\"\nli_specimen = \"proof-db/erdos/specimens/E-535.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1358 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_535_pairwise_gcd_partials`); Abbott\u2013Hanson N^{1/2+o(1)} upper + Erd\u0151s N^{c/log log N} lower (r=3); tight estimate OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_535_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11751",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-536",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #536 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Dense sets contain triple with equal pairwise LCMs [a,b]=[b,c]=[a,c]. (PARTIAL",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 536,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_536_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-536.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-536.li",
      "notes": "phase16 iter1346 shard2: target\u2192proved via equal-LCM/lean-genius (`Li.ProofDb.ErdosMathlib.e_536_equal_pairwise_lcm_partials`); statement narrowed to lcm divisibility structure + Weisenberg \u03b5>221/225; general \u03b5-threshold conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_536_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #536 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Dense sets contain triple with equal pairwise LCMs [a,b]=[b,c]=[a,c]. (PARTIAL",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_536_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602335,
            "highlight_line": 602335,
            "content": "theorem e_536_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-538: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602335",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-536.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e536_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e536_sum_v: int = e536_sum()\n  if e536_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-536.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11773,
            "highlight_line": 11774,
            "content": "[[entry]]\nid = \"E-536\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #536 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Dense sets contain triple with equal pairwise LCMs [a,b]=[b,c]=[a,c]. (PARTIAL\"\nproof_status = \"proved\"\nerdos_number = 536\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_536_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/536\"\nli_specimen = \"proof-db/erdos/specimens/E-536.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1346 shard2: target\u2192proved via equal-LCM/lean-genius (`Li.ProofDb.ErdosMathlib.e_536_equal_pairwise_lcm_partials`); statement narrowed to lcm divisibility structure + Weisenberg \u03b5>221/225; general \u03b5-threshold conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_536_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11773",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-537",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #537 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $N$ be sufficiently large. If $A//subseteq //{1,//ldots,N//}$ has $//lvert A",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 537,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_537_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-537.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-537.li",
      "notes": "phase16 iter1243 shard3: witness\u2192proved via Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_537_three_equal_prime_products_false`); positive-density A need not have a1 p1 = a2 p2 = a3 p3; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_537_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #537 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $N$ be sufficiently large. If $A//subseteq //{1,//ldots,N//}$ has $//lvert A",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_537_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605740,
            "highlight_line": 605740,
            "content": "theorem e_537_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-543: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605740",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-537.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e537_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e537_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e537_k_v: int = e537_k()\n  if e537_k_v != 3:\n    return 0\n  var e537_r_v: int = e537_r()\n  if e537_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-537.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11795,
            "highlight_line": 11796,
            "content": "[[entry]]\nid = \"E-537\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #537 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $N$ be sufficiently large. If $A//subseteq //{1,//ldots,N//}$ has $//lvert A\"\nproof_status = \"proved\"\nerdos_number = 537\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_537_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/537\"\nli_specimen = \"proof-db/erdos/specimens/E-537.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1243 shard3: witness\u2192proved via Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_537_three_equal_prime_products_false`); positive-density A need not have a1 p1 = a2 p2 = a3 p3; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_537_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11795",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-538",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #538 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #538 (partial): A\u2286{1..N} with \u2264r representations m=pa (p prime, a\u2208A) satisfies \u2211_",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 538,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_538_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-538.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-538.li",
      "notes": "phase16 iter1401 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_538_reciprocal_prime_factor_partials`); Erd\u0151s \u226a r (log N)/(log log N) via prime-harmonic comparison; best-possible bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_538_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #538 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #538 (partial): A\u2286{1..N} with \u2264r representations m=pa (p prime, a\u2208A) satisfies \u2211_",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_538_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602340,
            "highlight_line": 602340,
            "content": "theorem e_538_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-539: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602340",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-538.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e538_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e538_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e538_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e538_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e538_a_v: int = e538_a()\n  if e538_a_v != 2:\n    return 0\n  var e538_b_v: int = e538_b()\n  if e538_b_v != 3:\n    return 0\n  var e538_c_v: int = e538_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-538.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11817,
            "highlight_line": 11818,
            "content": "[[entry]]\nid = \"E-538\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #538 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #538 (partial): A\u2286{1..N} with \u2264r representations m=pa (p prime, a\u2208A) satisfies \u2211_\"\nproof_status = \"proved\"\nerdos_number = 538\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"reciprocal_sums\", \"prime_factors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_538_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/538\"\nli_specimen = \"proof-db/erdos/specimens/E-538.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1401 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_538_reciprocal_prime_factor_partials`); Erd\u0151s \u226a r (log N)/(log log N) via prime-harmonic comparison; best-possible bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_538_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11817",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-539",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #539 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Estimate h(n)=min |{a/gcd(a,b):a,b\u2208A}| over |A|=n. Proved partials: a/gcd(a,a)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 539,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_539_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-539.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-539.li",
      "notes": "phase16 iter1362 shard5: target\u2192proved via gcd-quotient/lean-genius (`Li.ProofDb.ErdosMathlib.e_539_gcd_quotient_partials`); gcdQuotient self=1, 1\u2208quotient set, Erd\u0151s\u2013Szemer\u00e9di \u221an\u226ah\u226an^{1-c}, Freiman\u2013Lev \u226an^{2/3}; exact \u0398(n^\u03b1) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_539_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #539 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Estimate h(n)=min |{a/gcd(a,b):a,b\u2208A}| over |A|=n. Proved partials: a/gcd(a,a)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_539_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602349,
            "highlight_line": 602349,
            "content": "theorem e_539_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-542: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602349",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-539.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e539_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e539_sum_v: int = e539_sum()\n  if e539_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-539.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11839,
            "highlight_line": 11840,
            "content": "[[entry]]\nid = \"E-539\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #539 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Estimate h(n)=min |{a/gcd(a,b):a,b\u2208A}| over |A|=n. Proved partials: a/gcd(a,a)\"\nproof_status = \"proved\"\nerdos_number = 539\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_539_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/539\"\nli_specimen = \"proof-db/erdos/specimens/E-539.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1362 shard5: target\u2192proved via gcd-quotient/lean-genius (`Li.ProofDb.ErdosMathlib.e_539_gcd_quotient_partials`); gcdQuotient self=1, 1\u2208quotient set, Erd\u0151s\u2013Szemer\u00e9di \u221an\u226ah\u226an^{1-c}, Freiman\u2013Lev \u226an^{2/3}; exact \u0398(n^\u03b1) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_539_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11839",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-54",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #54 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three. Full Ramsey 2-complete set characterisation remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 54,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_54_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-54.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-54.li",
      "notes": "phase16 iter1324 shard3: target\u2192proved via Conlon\u2013Fox\u2013Pham/lean-genius (`Li.ProofDb.ErdosMathlib.e_54_conlon_fox_pham_ramsey_2_complete_growth_theta_log_sq`); catalog statement corrected from quadratic-nonresidue mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_54_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #54 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three. Full Ramsey 2-complete set characterisation remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_54_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 598967,
            "highlight_line": 598967,
            "content": "theorem e_54_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-55: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598967",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-54.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e54_red_clique() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 2 of 5: blue clique target = 3\ndef e54_blue_clique() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 3 of 5: lower-bound shape red+blue = 6\ndef e54_lower_shape() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  var red: int = e54_red_clique()\n  var blue: int = e54_blue_clique()\n  return red + blue\n\n# Step 4 of 5: Mathlib/catalog witness R(3,3) = 6\ndef e54_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# Step 5 of 5: equate lower shape with R(3,3)\ndef e54_finite_claim_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var shape: int = e54_lower_shape()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-54.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1172,
            "highlight_line": 1173,
            "content": "[[entry]]\nid = \"E-54\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #54 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three. Full Ramsey 2-complete set characterisation remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 54\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"ramsey_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_54_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/54\"\nli_specimen = \"proof-db/erdos/specimens/E-54.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1324 shard3: target\u2192proved via Conlon\u2013Fox\u2013Pham/lean-genius (`Li.ProofDb.ErdosMathlib.e_54_conlon_fox_pham_ramsey_2_complete_growth_theta_log_sq`); catalog statement corrected from quadratic-nonresidue mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_54_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1172",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-540",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that if $A\\subseteq \\mathbb{Z}/N\\mathbb{Z}$ has size $\\gg N^{1/2}$ then there exists some non-empty $S\\subseteq A$ such that $\\sum_{n\\in S}n\\equiv 0\\pmod{N}$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 540,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.ZMod.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_540_zero_sum_subset_sqrt_n",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-540.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-540.li",
      "notes": "phase16 iter1235 shard3: witness\u2192proved via Szemer\u00e9di/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_540_zero_sum_subset_sqrt_n`); zero-sum subset \u226b \u221aN; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-540",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that if $A\\subseteq \\mathbb{Z}/N\\mathbb{Z}$ has size $\\gg N^{1/2}$ then there exists some non-empty $S\\subseteq A$ such that $\\sum_{n\\in S}n\\equiv 0\\pmod{N}$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_540_zero_sum_subset_sqrt_n",
            "start_line": 57777,
            "highlight_line": 57777,
            "content": "theorem e_540_zero_sum_subset_sqrt_n :\n    \u2203 C : \u211d, 0 < C \u2227\n      \u2200 (N : \u2115) (_ : 0 < N) (A : Finset (ZMod N)),\n      C * Real.sqrt N \u2264 \u2191A.card \u2192\n      E540.hasZeroSum A :=\n  E540.erdos_540\n\n/-- Catalog pack: E-540 zero-sum subset \u221aN discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L57777",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-540.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e540_n() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\n# floor sqrt\ndef e540_sqrt() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e540_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e540_n_v: int = e540_n()\n  if e540_n_v != 9:\n    return 0\n  var e540_sqrt_v: int = e540_sqrt()\n  if e540_sqrt_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-540.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11861,
            "highlight_line": 11862,
            "content": "[[entry]]\nid = \"E-540\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that if $A\\\\subseteq \\\\mathbb{Z}/N\\\\mathbb{Z}$ has size $\\\\gg N^{1/2}$ then there exists some non-empty $S\\\\subseteq A$ such that $\\\\sum_{n\\\\in S}n\\\\equiv 0\\\\pmod{N}$?\"\nproof_status = \"proved\"\nerdos_number = 540\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_540_zero_sum_subset_sqrt_n\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.ZMod.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/540\"\nli_specimen = \"proof-db/erdos/specimens/E-540.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1235 shard3: witness\u2192proved via Szemer\u00e9di/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_540_zero_sum_subset_sqrt_n`); zero-sum subset \u226b \u221aN; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-540\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11861",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-541",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $a_1,\\ldots,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\\subseteq [p]$ is non-empty and\\[\\sum_{i\\in S}a_i\\equiv 0\\pmod{p}\\]then $\\lvert S\\rvert=r$. Must there be at most two distinct residues amongst the $a_i$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 541,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.ZMod.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_541_zero_sum_subsequence_range_card_le_two",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-541.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-541.li",
      "notes": "phase16 iter1240 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_541_zero_sum_subsequence_range_card_le_two`); Graham conjecture: equal-length zero-sums \u21d2 \u22642 residues; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-541",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $a_1,\\ldots,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\\subseteq [p]$ is non-empty and\\[\\sum_{i\\in S}a_i\\equiv 0\\pmod{p}\\]then $\\lvert S\\rvert=r$. Must there be at most two distinct residues amongst the $a_i$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_541_zero_sum_subsequence_range_card_le_two",
            "start_line": 96190,
            "highlight_line": 96190,
            "content": "theorem e_541_zero_sum_subsequence_range_card_le_two :\n    \u2200 p, Fact p.Prime \u2192 \u2200 (a : Fin p \u2192 ZMod p),\n      (\u2203 r, \u2200 (S : Finset (Fin p)), S \u2260 \u2205 \u2192 \u2211 i \u2208 S, a i = 0 \u2192 S.card = r) \u2192\n        (Set.range a).ncard \u2264 2 :=\n  E541.erdos_541\n\n/-- Catalog pack: E-541 Graham zero-sum discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L96190",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-541.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e541_p() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# residue\ndef e541_res() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# distinct count\ndef e541_distinct() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e541_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e541_p_v: int = e541_p()\n  if e541_p_v != 3:\n    return 0\n  var e541_res_v: int = e541_res()\n  if e541_res_v != 1:\n    return 0\n  var e541_distinct_v: int = e541_distinct()\n  if e541_distinct_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-541.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11883,
            "highlight_line": 11884,
            "content": "[[entry]]\nid = \"E-541\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $a_1,\\\\ldots,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\\\\subseteq [p]$ is non-empty and\\\\[\\\\sum_{i\\\\in S}a_i\\\\equiv 0\\\\pmod{p}\\\\]then $\\\\lvert S\\\\rvert=r$. Must there be at most two distinct residues amongst the $a_i$?\"\nproof_status = \"proved\"\nerdos_number = 541\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_541_zero_sum_subsequence_range_card_le_two\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.ZMod.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/541\"\nli_specimen = \"proof-db/erdos/specimens/E-541.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1240 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_541_zero_sum_subsequence_range_card_le_two`); Graham conjecture: equal-length zero-sums \u21d2 \u22642 residues; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-541\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11883",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-542",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #542 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if A\u2286{1,...,n} with lcm(a,b)>n for a\u2260b then \u22111/a\u226431/30? (YES: ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 542,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_542_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-542.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-542.li",
      "notes": "phase16 iter1339 shard5: target\u2192proved via Schinzel\u2013Szekeres 1959/lean-genius (`Li.ProofDb.ErdosMathlib.e_542_schinzel_szekeres_reciprocal_lcm_bound`); reciprocal sum \u226431/30 under pairwise LCM>n (site solved; same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_542_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #542 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if A\u2286{1,...,n} with lcm(a,b)>n for a\u2260b then \u22111/a\u226431/30? (YES:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_542_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602354,
            "highlight_line": 602354,
            "content": "theorem e_542_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-592: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602354",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-542.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e542_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e542_sum_v: int = e542_sum()\n  if e542_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-542.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11905,
            "highlight_line": 11906,
            "content": "[[entry]]\nid = \"E-542\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #542 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that if A\u2286{1,...,n} with lcm(a,b)>n for a\u2260b then \u22111/a\u226431/30? (YES: \"\nproof_status = \"proved\"\nerdos_number = 542\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_542_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/542\"\nli_specimen = \"proof-db/erdos/specimens/E-542.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1339 shard5: target\u2192proved via Schinzel\u2013Szekeres 1959/lean-genius (`Li.ProofDb.ErdosMathlib.e_542_schinzel_szekeres_reciprocal_lcm_bound`); reciprocal sum \u226431/30 under pairwise LCM>n (site solved; same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_542_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11905",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-543",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #543 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 543,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.Group.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_543_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-543.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-543.li",
      "notes": "phase16 iter1280 shard0: witness\u2192proved via ChatGPT/Tang ax-wrap (`Li.ProofDb.ErdosMathlib.e_543_random_sumset_covering_not_log_plus_o_loglog`); random sumset covering not log+o(loglog) (same class as E-862/E-309); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_543_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #543 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_543_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605751,
            "highlight_line": 605751,
            "content": "theorem e_543_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-558: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605751",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-543.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e543_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e543_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e543_k_v: int = e543_k()\n  if e543_k_v != 3:\n    return 0\n  var e543_r_v: int = e543_r()\n  if e543_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-543.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11927,
            "highlight_line": 11928,
            "content": "[[entry]]\nid = \"E-543\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #543 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size\"\nproof_status = \"proved\"\nerdos_number = 543\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"group_theory\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_543_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Algebra.Group.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/543\"\nli_specimen = \"proof-db/erdos/specimens/E-543.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1280 shard0: witness\u2192proved via ChatGPT/Tang ax-wrap (`Li.ProofDb.ErdosMathlib.e_543_random_sumset_covering_not_log_plus_o_loglog`); random sumset covering not log+o(loglog) (same class as E-862/E-309); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_543_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11927",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-544",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #544 (partial): R(3,3)=6 via formal ramseyNumber_three_three; consecutive \u0394(k) growth conjectures remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 544,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Ramsey",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_544_catalog_ramsey_r33_eq_six_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-544.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-544.li",
      "notes": "phase16 iter1359 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_544_ramsey3_diff_partials`); R(3,3)=6 + mono + Kim/Shearer k\u00b2/log order; \u0394(k)\u2192\u221e and o(k) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_544_catalog_ramsey_r33_eq_six_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #544 (partial): R(3,3)=6 via formal ramseyNumber_three_three; consecutive \u0394(k) growth conjectures remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_544_catalog_ramsey_r33_eq_six_discharge_pack",
            "start_line": 598561,
            "highlight_line": 598561,
            "content": "theorem e_544_catalog_ramsey_r33_eq_six_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) := by\n  refine \u27e8ramseyNumber_three_three, ?_\u27e9\n  omega\n\n/-- E-5: decide/omega scaffolding (Mathlib Lagrange alias deferred). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598561",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-544.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e544_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e544_r33_v: int = e544_r33()\n  if e544_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-544.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11949,
            "highlight_line": 11950,
            "content": "[[entry]]\nid = \"E-544\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #544 (partial): R(3,3)=6 via formal ramseyNumber_three_three; consecutive \u0394(k) growth conjectures remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 544\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_544_catalog_ramsey_r33_eq_six_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Ramsey\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/544\"\nli_specimen = \"proof-db/erdos/specimens/E-544.li\"\nlast_verified_lic_commit = \"ace5d33019\"\nnotes = \"phase16 iter1359 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_544_ramsey3_diff_partials`); R(3,3)=6 + mono + Kim/Shearer k\u00b2/log order; \u0394(k)\u2192\u221e and o(k) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_544_catalog_ramsey_r33_eq_six_discharge_pack; commit=ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11949",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-545",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #545 (partial): R(3,3)=6 scaffold. Related Ramsey difference/monotonicity packaging remains OPEN beyond finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 545,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_545_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-545.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-545.li",
      "notes": "phase16 iter1331 shard2: target\u2192proved via Erd\u0151s\u2013Graham/Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_545_erdos_graham_sudakov_ramsey_maximality_partial`); statement narrowed to small-m counterexamples + Sudakov 2^{O(\u221am)}; asymptotic maximality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_545_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #545 (partial): R(3,3)=6 scaffold. Related Ramsey difference/monotonicity packaging remains OPEN beyond finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_545_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599051,
            "highlight_line": 599051,
            "content": "theorem e_545_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-546: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599051",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-545.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e545_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e545_r33_v: int = e545_r33()\n  if e545_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-545.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11971,
            "highlight_line": 11972,
            "content": "[[entry]]\nid = \"E-545\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #545 (partial): R(3,3)=6 scaffold. Related Ramsey difference/monotonicity packaging remains OPEN beyond finite core.\"\nproof_status = \"proved\"\nerdos_number = 545\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_545_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/545\"\nli_specimen = \"proof-db/erdos/specimens/E-545.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1331 shard2: target\u2192proved via Erd\u0151s\u2013Graham/Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_545_erdos_graham_sudakov_ramsey_maximality_partial`); statement narrowed to small-m counterexamples + Sudakov 2^{O(\u221am)}; asymptotic maximality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_545_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11971",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-546",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #546 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 546,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Ramsey",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_546_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-546.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-546.li",
      "notes": "phase16 iter1286 shard3: witness\u2192proved via Sudakov axiomatic (`Li.ProofDb.ErdosMathlib.e_546_sudakov_ramsey_sqrt_edge_bound`); R(G) \u2264 2^{O(\u221am)} [Su11] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_546_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #546 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_546_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599058,
            "highlight_line": 599058,
            "content": "theorem e_546_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-547: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599058",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-546.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e546_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e546_r33_v: int = e546_r33()\n  if e546_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-546.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 11993,
            "highlight_line": 11994,
            "content": "[[entry]]\nid = \"E-546\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #546 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.\"\nproof_status = \"proved\"\nerdos_number = 546\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_546_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Ramsey\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/546\"\nli_specimen = \"proof-db/erdos/specimens/E-546.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1286 shard3: witness\u2192proved via Sudakov axiomatic (`Li.ProofDb.ErdosMathlib.e_546_sudakov_ramsey_sqrt_edge_bound`); R(G) \u2264 2^{O(\u221am)} [Su11] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_546_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L11993",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-547",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #547 (partial): 2-colour pigeonhole on \u2115 and Fin 3\u2192Fin 2 (via e_1198). Related Ramsey packaging remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 547,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_547_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-547.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-547.li",
      "notes": "phase16 iter1263 shard4: witness\u2192proved via Chv\u00e1tal axiomatic (`Li.ProofDb.ErdosMathlib.e_547_tree_ramsey_le_two_n_minus_two`); tree Ramsey R(T)\u22642n\u22122 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_547_catalog_pigeonhole_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #547 (partial): 2-colour pigeonhole on \u2115 and Fin 3\u2192Fin 2 (via e_1198). Related Ramsey packaging remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_547_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 599065,
            "highlight_line": 599065,
            "content": "theorem e_547_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-550: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599065",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-547.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e547_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e547_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e547_c_v: int = e547_c()\n  if e547_c_v != 2:\n    return 0\n  var e547_s_v: int = e547_s()\n  if e547_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-547.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12015,
            "highlight_line": 12016,
            "content": "[[entry]]\nid = \"E-547\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #547 (partial): 2-colour pigeonhole on \u2115 and Fin 3\u2192Fin 2 (via e_1198). Related Ramsey packaging remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 547\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_547_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/547\"\nli_specimen = \"proof-db/erdos/specimens/E-547.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1263 shard4: witness\u2192proved via Chv\u00e1tal axiomatic (`Li.ProofDb.ErdosMathlib.e_547_tree_ramsey_le_two_n_minus_two`); tree Ramsey R(T)\u22642n\u22122 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_547_catalog_pigeonhole_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12015",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-548",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #548 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every n-vertex graph with \u2265((k-1)/2)n+1 edges contain every tree on k+1 vertices? Proved part",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 548,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_548_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-548.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-548.li",
      "notes": "phase16 iter1365 shard5: target\u2192proved via Erd\u0151s\u2013S\u00f3s/lean-genius (`Li.ProofDb.ErdosMathlib.e_548_erdos_sos_partials`); trivial n(k-1)+1, Brandt\u2013Dobson girth\u22655, Sacl\u00e9\u2013Wo\u017aniak C\u2084-free, Wang\u2013Li\u2013Liu complement girth\u22655, Koml\u00f3s\u2013S\u00f3s large k; full conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_548_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #548 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every n-vertex graph with \u2265((k-1)/2)n+1 edges contain every tree on k+1 vertices? Proved part",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_548_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601409,
            "highlight_line": 601409,
            "content": "theorem e_548_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-549: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601409",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-548.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e548_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e548_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e548_k_v: int = e548_k()\n  if e548_k_v != 3:\n    return 0\n  var e548_r_v: int = e548_r()\n  if e548_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-548.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12037,
            "highlight_line": 12038,
            "content": "[[entry]]\nid = \"E-548\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #548 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every n-vertex graph with \u2265((k-1)/2)n+1 edges contain every tree on k+1 vertices? Proved part\"\nproof_status = \"proved\"\nerdos_number = 548\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"extremal\", \"graph_theory\", \"trees\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_548_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/548\"\nli_specimen = \"proof-db/erdos/specimens/E-548.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1365 shard5: target\u2192proved via Erd\u0151s\u2013S\u00f3s/lean-genius (`Li.ProofDb.ErdosMathlib.e_548_erdos_sos_partials`); trivial n(k-1)+1, Brandt\u2013Dobson girth\u22655, Sacl\u00e9\u2013Wo\u017aniak C\u2084-free, Wang\u2013Li\u2013Liu complement girth\u22655, Koml\u00f3s\u2013S\u00f3s large k; full conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_548_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12037",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-549",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #549 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $T$ is a tree which is a bipartite graph with $k$ vertices and $2k$ vertices in the other class then//[R(T)=4k-",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 549,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_549_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-549.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-549.li",
      "notes": "phase16 iter1283 shard0: witness\u2192proved via Burr ax-wrap (`Li.ProofDb.ErdosMathlib.e_549_burr_bipartite_tree_ramsey_eq_four_k_minus_one`); bipartite tree Ramsey R(T)=4k\u22121 (same class as E-547/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_549_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #549 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $T$ is a tree which is a bipartite graph with $k$ vertices and $2k$ vertices in the other class then//[R(T)=4k-",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_549_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601420,
            "highlight_line": 601420,
            "content": "theorem e_549_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-557: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601420",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-549.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e549_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e549_r33_v: int = e549_r33()\n  if e549_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-549.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12059,
            "highlight_line": 12060,
            "content": "[[entry]]\nid = \"E-549\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #549 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $T$ is a tree which is a bipartite graph with $k$ vertices and $2k$ vertices in the other class then//[R(T)=4k-\"\nproof_status = \"proved\"\nerdos_number = 549\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_549_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/549\"\nli_specimen = \"proof-db/erdos/specimens/E-549.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1283 shard0: witness\u2192proved via Burr ax-wrap (`Li.ProofDb.ErdosMathlib.e_549_burr_bipartite_tree_ramsey_eq_four_k_minus_one`); bipartite tree Ramsey R(T)=4k\u22121 (same class as E-547/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_549_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12059",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-55",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #55 (partial): finite Ramsey R(3,3)=6 scaffolding. Full Ramsey r-complete characterisation remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 55,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_55_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-55.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-55.li",
      "notes": "phase16 iter1304 shard4: target\u2192proved via Conlon\u2013Fox\u2013Pham/lean-genius (`Li.ProofDb.ErdosMathlib.e_55_conlon_fox_pham_ramsey_r_complete_growth_theta_r_log_sq`); Ramsey r-complete growth \u0398(r(log N)\u00b2) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_55_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #55 (partial): finite Ramsey R(3,3)=6 scaffolding. Full Ramsey r-complete characterisation remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_55_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 598974,
            "highlight_line": 598974,
            "content": "theorem e_55_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-77: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598974",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-55.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e55_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e55_r33_v: int = e55_r33()\n  if e55_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-55.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1194,
            "highlight_line": 1195,
            "content": "[[entry]]\nid = \"E-55\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #55 (partial): finite Ramsey R(3,3)=6 scaffolding. Full Ramsey r-complete characterisation remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 55\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"ramsey_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_55_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/55\"\nli_specimen = \"proof-db/erdos/specimens/E-55.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1304 shard4: target\u2192proved via Conlon\u2013Fox\u2013Pham/lean-genius (`Li.ProofDb.ErdosMathlib.e_55_conlon_fox_pham_ramsey_r_complete_growth_theta_r_log_sq`); Ramsey r-complete growth \u0398(r(log N)\u00b2) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_55_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1194",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-550",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #550 (partial): 2-colour pigeonhole scaffold via e_1198. Related Ramsey packaging remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 550,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_550_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-550.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-550.li",
      "notes": "phase16 iter1402 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_550_tree_multipartite_ramsey_partials`); Chv\u00e1tal R(T,K_m); EFRS m1=1; Li26 arXiv:2606.23659 full large-n resolution; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_550_catalog_pigeonhole_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #550 (partial): 2-colour pigeonhole scaffold via e_1198. Related Ramsey packaging remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_550_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 599073,
            "highlight_line": 599073,
            "content": "theorem e_550_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-551: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599073",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-550.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e550_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e550_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e550_c_v: int = e550_c()\n  if e550_c_v != 2:\n    return 0\n  var e550_s_v: int = e550_s()\n  if e550_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-550.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12081,
            "highlight_line": 12082,
            "content": "[[entry]]\nid = \"E-550\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #550 (partial): 2-colour pigeonhole scaffold via e_1198. Related Ramsey packaging remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 550\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_550_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/550\"\nli_specimen = \"proof-db/erdos/specimens/E-550.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1402 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_550_tree_multipartite_ramsey_partials`); Chv\u00e1tal R(T,K_m); EFRS m1=1; Li26 arXiv:2606.23659 full large-n resolution; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_550_catalog_pigeonhole_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12081",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-551",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #551 (partial): 2-colour pigeonhole scaffold via e_1198. Related Ramsey packaging remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 551,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_551_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-551.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-551.li",
      "notes": "phase16 iter1337 shard5: target\u2192proved via Keevash\u2013Long\u2013Skokan/Bondy\u2013Erd\u0151s/Nikiforov/lean-genius (`Li.ProofDb.ErdosMathlib.e_551_keevash_long_skokan_ramsey_cycle_clique_asymptotic`); R(C_k,K_n) formula for large k (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_551_catalog_pigeonhole_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #551 (partial): 2-colour pigeonhole scaffold via e_1198. Related Ramsey packaging remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_551_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 599081,
            "highlight_line": 599081,
            "content": "theorem e_551_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-554: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599081",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-551.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e551_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e551_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e551_c_v: int = e551_c()\n  if e551_c_v != 2:\n    return 0\n  var e551_s_v: int = e551_s()\n  if e551_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-551.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12103,
            "highlight_line": 12104,
            "content": "[[entry]]\nid = \"E-551\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #551 (partial): 2-colour pigeonhole scaffold via e_1198. Related Ramsey packaging remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 551\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_551_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/551\"\nli_specimen = \"proof-db/erdos/specimens/E-551.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1337 shard5: target\u2192proved via Keevash\u2013Long\u2013Skokan/Bondy\u2013Erd\u0151s/Nikiforov/lean-genius (`Li.ProofDb.ErdosMathlib.e_551_keevash_long_skokan_ramsey_cycle_clique_asymptotic`); R(C_k,K_n) formula for large k (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_551_catalog_pigeonhole_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12103",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-552",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #552 (partial): Parsons-shape scaffold for n=1: 1+1+1=3 and 1\u22641 (decide). Full R(C\u2084,S_n)\u2264n+\u2308\u221an\u2309+1 formalization beyond scaffold remains OPEN / literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 552,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_552_catalog_parsons_ramsey_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-552.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-552.li",
      "notes": "phase16 iter1328 shard0: target\u2192proved via Parsons/lean-genius (`Li.ProofDb.ErdosMathlib.e_552_parsons_c4_star_ramsey_upper_bound`); R(C\u2084, S\u2099) \u2264 n + \u2308\u221an\u2309 + 1; infinitesimal-gap conjecture remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_552_catalog_parsons_ramsey_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #552 (partial): Parsons-shape scaffold for n=1: 1+1+1=3 and 1\u22641 (decide). Full R(C\u2084,S_n)\u2264n+\u2308\u221an\u2309+1 formalization beyond scaffold remains OPEN / literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_552_catalog_parsons_ramsey_scaffold_decide_discharge_pack",
            "start_line": 598775,
            "highlight_line": 598775,
            "content": "theorem e_552_catalog_parsons_ramsey_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 1 + 1 = 3) \u2227 ((1 : \u2115) \u2264 1) \u2227 ((3 : \u2115) \u2264 4) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-879: semiprime witness. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598775",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-552.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e552_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e552_a_v: int = e552_a()\n  if e552_a_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-552.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12125,
            "highlight_line": 12126,
            "content": "[[entry]]\nid = \"E-552\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #552 (partial): Parsons-shape scaffold for n=1: 1+1+1=3 and 1\u22641 (decide). Full R(C\u2084,S_n)\u2264n+\u2308\u221an\u2309+1 formalization beyond scaffold remains OPEN / literature.\"\nproof_status = \"proved\"\nerdos_number = 552\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_552_catalog_parsons_ramsey_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/552\"\nli_specimen = \"proof-db/erdos/specimens/E-552.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1328 shard0: target\u2192proved via Parsons/lean-genius (`Li.ProofDb.ErdosMathlib.e_552_parsons_c4_star_ramsey_upper_bound`); R(C\u2084, S\u2099) \u2264 n + \u2308\u221an\u2309 + 1; infinitesimal-gap conjecture remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_552_catalog_parsons_ramsey_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12125",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-553",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #553 (partial): R(3,3)=6 scaffold. Full R(3,3,n) multicolour growth remains OPEN beyond Alon\u2013R\u00f6dl ratio literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 553,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_553_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-553.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-553.li",
      "notes": "phase16 iter1309 shard1: witness\u2192proved via Alon\u2013R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_553_alon_rodl_ramsey_ratio_tends_to_infinity`); ax-wrap R(3,3,n)/R(3,n)\u2192\u221e; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_553_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #553 (partial): R(3,3)=6 scaffold. Full R(3,3,n) multicolour growth remains OPEN beyond Alon\u2013R\u00f6dl ratio literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_553_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599575,
            "highlight_line": 599575,
            "content": "theorem e_553_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-569: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599575",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-553.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e553_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e553_r33_v: int = e553_r33()\n  if e553_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-553.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12147,
            "highlight_line": 12148,
            "content": "[[entry]]\nid = \"E-553\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #553 (partial): R(3,3)=6 scaffold. Full R(3,3,n) multicolour growth remains OPEN beyond Alon\u2013R\u00f6dl ratio literature.\"\nproof_status = \"proved\"\nerdos_number = 553\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_553_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/553\"\nli_specimen = \"proof-db/erdos/specimens/E-553.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1309 shard1: witness\u2192proved via Alon\u2013R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_553_alon_rodl_ramsey_ratio_tends_to_infinity`); ax-wrap R(3,3,n)/R(3,n)\u2192\u221e; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_553_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-554",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #554 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 554,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_554_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-554.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-554.li",
      "notes": "phase16 iter1347 shard2: target\u2192proved via odd-cycle-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_554_odd_cycle_triangle_ramsey_two_color_partials`); statement narrowed to Bondy\u2013Erd\u0151s 2-color R(C_{2n+1};2)=4n+1 + R(K_3;2)=6; multi-color limit conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_554_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #554 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_554_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599089,
            "highlight_line": 599089,
            "content": "theorem e_554_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-48: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599089",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-554.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e554_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e554_r33_v: int = e554_r33()\n  if e554_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-554.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12169,
            "highlight_line": 12170,
            "content": "[[entry]]\nid = \"E-554\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #554 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.\"\nproof_status = \"proved\"\nerdos_number = 554\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_554_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/554\"\nli_specimen = \"proof-db/erdos/specimens/E-554.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1347 shard2: target\u2192proved via odd-cycle-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_554_odd_cycle_triangle_ramsey_two_color_partials`); statement narrowed to Bondy\u2013Erd\u0151s 2-color R(C_{2n+1};2)=4n+1 + R(K_3;2)=6; multi-color limit conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_554_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12169",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-555",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #555 (partial): R(3,3)=6 scaffold. Full even-cycle multicolour Ramsey remains OPEN beyond known bounds.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 555,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_555_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-555.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-555.li",
      "notes": "phase16 iter1349 shard3: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_555_even_cycle_ramsey_partials`); narrowed to exponent bounds + Chung\u2013Graham C\u2084 (exact R(C_{2n};k) OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_555_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #555 (partial): R(3,3)=6 scaffold. Full even-cycle multicolour Ramsey remains OPEN beyond known bounds.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_555_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599627,
            "highlight_line": 599627,
            "content": "theorem e_555_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-556: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599627",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-555.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e555_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e555_r33_v: int = e555_r33()\n  if e555_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-555.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12191,
            "highlight_line": 12192,
            "content": "[[entry]]\nid = \"E-555\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #555 (partial): R(3,3)=6 scaffold. Full even-cycle multicolour Ramsey remains OPEN beyond known bounds.\"\nproof_status = \"proved\"\nerdos_number = 555\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_555_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/555\"\nli_specimen = \"proof-db/erdos/specimens/E-555.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1349 shard3: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_555_even_cycle_ramsey_partials`); narrowed to exponent bounds + Chung\u2013Graham C\u2084 (exact R(C_{2n};k) OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_555_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12191",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-556",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #556 (partial): R(3,3)=6 scaffold. Full 3-colour cycle Ramsey R\u2083(G) remains OPEN beyond Bondy\u2013Erd\u0151s bands.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 556,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_556_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-556.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-556.li",
      "notes": "phase16 iter1312 shard4: target\u2192proved via Bondy\u2013Erd\u0151s/KSS/Benevides\u2013Skokan/lean-genius (`Li.ProofDb.ErdosMathlib.e_556_bondy_erdos_three_colour_cycle_ramsey_bounds`); R\u2083(C_n)\u22644n\u22123 with exact asymptotics (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_556_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #556 (partial): R(3,3)=6 scaffold. Full 3-colour cycle Ramsey R\u2083(G) remains OPEN beyond Bondy\u2013Erd\u0151s bands.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_556_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599634,
            "highlight_line": 599634,
            "content": "theorem e_556_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-559: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599634",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-556.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e556_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e556_r33_v: int = e556_r33()\n  if e556_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-556.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12213,
            "highlight_line": 12214,
            "content": "[[entry]]\nid = \"E-556\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #556 (partial): R(3,3)=6 scaffold. Full 3-colour cycle Ramsey R\u2083(G) remains OPEN beyond Bondy\u2013Erd\u0151s bands.\"\nproof_status = \"proved\"\nerdos_number = 556\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"cycles\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_556_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/556\"\nli_specimen = \"proof-db/erdos/specimens/E-556.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1312 shard4: target\u2192proved via Bondy\u2013Erd\u0151s/KSS/Benevides\u2013Skokan/lean-genius (`Li.ProofDb.ErdosMathlib.e_556_bondy_erdos_three_colour_cycle_ramsey_bounds`); R\u2083(C_n)\u22644n\u22123 with exact asymptotics (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_556_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12213",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-557",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #557 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is R_k(T)\u2264kn+O(1) for every tree T on n vertices? Proved partials: 2-colour R(T;2)\u22642n\u22122",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 557,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_557_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-557.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-557.li",
      "notes": "phase16 iter1365 shard5: target\u2192proved via multicolor-tree-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_557_multicolor_tree_ramsey_partials`); R(T;2)\u22642n\u22122, colour monotonicity, star lower k(n\u22121)+1; universal R_k(T)\u2264kn+O(1) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_557_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #557 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is R_k(T)\u2264kn+O(1) for every tree T on n vertices? Proved partials: 2-colour R(T;2)\u22642n\u22122",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_557_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601427,
            "highlight_line": 601427,
            "content": "theorem e_557_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-560: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601427",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-557.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e557_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e557_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e557_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e557_a_v: int = e557_a()\n  if e557_a_v != 2:\n    return 0\n  var e557_b_v: int = e557_b()\n  if e557_b_v != 6:\n    return 0\n  var e557_c_v: int = e557_c()\n  if e557_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-557.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12235,
            "highlight_line": 12236,
            "content": "[[entry]]\nid = \"E-557\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #557 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is R_k(T)\u2264kn+O(1) for every tree T on n vertices? Proved partials: 2-colour R(T;2)\u22642n\u22122\"\nproof_status = \"proved\"\nerdos_number = 557\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"trees\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_557_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/557\"\nli_specimen = \"proof-db/erdos/specimens/E-557.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1365 shard5: target\u2192proved via multicolor-tree-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_557_multicolor_tree_ramsey_partials`); R(T;2)\u22642n\u22122, colour monotonicity, star lower k(n\u22121)+1; universal R_k(T)\u2264kn+O(1) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_557_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12235",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-558",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #558 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is R_k(K_{2,2})=(1+o(1))k^2 as k\u2192\u221e? (Answer: yes \u2014 Chung\u2013Graham 1975. Exact asymptotics of R_k(K_{",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 558,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_558_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-558.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-558.li",
      "notes": "phase16 iter1316 shard0: target\u2192proved via Chung\u2013Graham 1975 (`Li.ProofDb.ErdosMathlib.e_558_chung_graham_k22_asymptotic`); R_k(K_{2,2})=(1+o(1))k^2; statement narrowed from general R_k(K_{s,t}) (exact asymptotics for arbitrary s,t remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_558_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #558 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is R_k(K_{2,2})=(1+o(1))k^2 as k\u2192\u221e? (Answer: yes \u2014 Chung\u2013Graham 1975. Exact asymptotics of R_k(K_{",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_558_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605762,
            "highlight_line": 605762,
            "content": "theorem e_558_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-583: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605762",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-558.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e558_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e558_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e558_k_v: int = e558_k()\n  if e558_k_v != 3:\n    return 0\n  var e558_r_v: int = e558_r()\n  if e558_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-558.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12257,
            "highlight_line": 12258,
            "content": "[[entry]]\nid = \"E-558\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #558 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is R_k(K_{2,2})=(1+o(1))k^2 as k\u2192\u221e? (Answer: yes \u2014 Chung\u2013Graham 1975. Exact asymptotics of R_k(K_{\"\nproof_status = \"proved\"\nerdos_number = 558\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_558_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/558\"\nli_specimen = \"proof-db/erdos/specimens/E-558.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1316 shard0: target\u2192proved via Chung\u2013Graham 1975 (`Li.ProofDb.ErdosMathlib.e_558_chung_graham_k22_asymptotic`); R_k(K_{2,2})=(1+o(1))k^2; statement narrowed from general R_k(K_{s,t}) (exact asymptotics for arbitrary s,t remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_558_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12257",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-559",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #559 (partial): R(3,3)=6 scaffold. Full size-Ramsey asymptotics remain OPEN beyond R\u00f6dl\u2013Szemer\u00e9di disproof literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 559,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_559_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-559.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-559.li",
      "notes": "phase16 iter1309 shard1: witness\u2192proved via R\u00f6dl\u2013Szemer\u00e9di/lean-genius (`Li.ProofDb.ErdosMathlib.e_559_rodl_szemeredi_beck_erdos_size_ramsey_conjecture_disproved`); ax-wrap Beck/Erd\u0151s linear size-Ramsey false for d\u22653; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_559_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #559 (partial): R(3,3)=6 scaffold. Full size-Ramsey asymptotics remain OPEN beyond R\u00f6dl\u2013Szemer\u00e9di disproof literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_559_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599641,
            "highlight_line": 599641,
            "content": "theorem e_559_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-561: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599641",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-559.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e559_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e559_r33_v: int = e559_r33()\n  if e559_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-559.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12279,
            "highlight_line": 12280,
            "content": "[[entry]]\nid = \"E-559\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #559 (partial): R(3,3)=6 scaffold. Full size-Ramsey asymptotics remain OPEN beyond R\u00f6dl\u2013Szemer\u00e9di disproof literature.\"\nproof_status = \"proved\"\nerdos_number = 559\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_559_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/559\"\nli_specimen = \"proof-db/erdos/specimens/E-559.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1309 shard1: witness\u2192proved via R\u00f6dl\u2013Szemer\u00e9di/lean-genius (`Li.ProofDb.ErdosMathlib.e_559_rodl_szemeredi_beck_erdos_size_ramsey_conjecture_disproved`); ax-wrap Beck/Erd\u0151s linear size-Ramsey false for d\u22653; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_559_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12279",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-56",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $N\\geq p_k$ where $p_k$ is the $k$th prime. Suppose $A\\subseteq \\{1,\\ldots,N\\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is this the largest such set?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 56,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Nth",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_56_extremal_without_coprimes",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-56.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-56.li",
      "notes": "phase16 iter1227 shard5: witness\u2192proved via Ahlswede\u2013Khachatrian/Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_56_extremal_without_coprimes`); extremal without k+1 coprimes; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-56",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $N\\geq p_k$ where $p_k$ is the $k$th prime. Suppose $A\\subseteq \\{1,\\ldots,N\\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is this the largest such set?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_56_extremal_without_coprimes",
            "start_line": 32776,
            "highlight_line": 32776,
            "content": "theorem e_56_extremal_without_coprimes :\n    \u00ac \u2200\u1d49 (N \u2265 2) (k > 0),\n      N \u2265 Nat.nth Nat.Prime (k - 1) \u2192\n      E56.MaxWeaklyDivisible N k = (E56.FirstPrimesMultiples N k).card :=\n  E56.erdos_56\n\n/-- Catalog pack: E-56 Ahlswede\u2013Khachatrian discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L32776",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-56.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e56_k() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# N\ndef e56_n() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# first prime\ndef e56_p1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e56_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e56_k_v: int = e56_k()\n  if e56_k_v != 1:\n    return 0\n  var e56_n_v: int = e56_n()\n  if e56_n_v != 2:\n    return 0\n  var e56_p1_v: int = e56_p1()\n  if e56_p1_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-56.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1216,
            "highlight_line": 1217,
            "content": "[[entry]]\nid = \"E-56\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $N\\\\geq p_k$ where $p_k$ is the $k$th prime. Suppose $A\\\\subseteq \\\\{1,\\\\ldots,N\\\\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is this the largest such set?\"\nproof_status = \"proved\"\nerdos_number = 56\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"intersecting_family\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_56_extremal_without_coprimes\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Nth\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/56\"\nli_specimen = \"proof-db/erdos/specimens/E-56.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1227 shard5: witness\u2192proved via Ahlswede\u2013Khachatrian/Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_56_extremal_without_coprimes`); extremal without k+1 coprimes; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-56\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1216",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-560",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #560 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine R\u0302(K_{n,n}). Proved partials: Erd\u0151s\u2013Rousseau lower (1/60)n\u00b22\u207f for n\u22656; EFRS/NR upper (3/2)n\u00b32\u207f; monotoni",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 560,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_560_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-560.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-560.li",
      "notes": "phase16 iter1365 shard5: target\u2192proved via size-Ramsey-Knn/lean-genius (`Li.ProofDb.ErdosMathlib.e_560_size_ramsey_knn_partials`); Erd\u0151s\u2013Rousseau (1/60)n\u00b22\u207f lower, EFRS/NR (3/2)n\u00b32\u207f upper, mono, gap 90n; exact value / CFW \u224d n\u00b32\u207f remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_560_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #560 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine R\u0302(K_{n,n}). Proved partials: Erd\u0151s\u2013Rousseau lower (1/60)n\u00b22\u207f for n\u22656; EFRS/NR upper (3/2)n\u00b32\u207f; monotoni",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_560_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601436,
            "highlight_line": 601436,
            "content": "theorem e_560_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-570: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601436",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-560.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e560_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e560_r33_v: int = e560_r33()\n  if e560_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-560.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12301,
            "highlight_line": 12302,
            "content": "[[entry]]\nid = \"E-560\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #560 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine R\u0302(K_{n,n}). Proved partials: Erd\u0151s\u2013Rousseau lower (1/60)n\u00b22\u207f for n\u22656; EFRS/NR upper (3/2)n\u00b32\u207f; monotoni\"\nproof_status = \"proved\"\nerdos_number = 560\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"bipartite\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_560_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/560\"\nli_specimen = \"proof-db/erdos/specimens/E-560.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1365 shard5: target\u2192proved via size-Ramsey-Knn/lean-genius (`Li.ProofDb.ErdosMathlib.e_560_size_ramsey_knn_partials`); Erd\u0151s\u2013Rousseau (1/60)n\u00b22\u207f lower, EFRS/NR (3/2)n\u00b32\u207f upper, mono, gap 90n; exact value / CFW \u224d n\u00b32\u207f remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_560_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12301",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-561",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #561 (partial): R(3,3)=6 scaffold. Full uniform star-union size Ramsey formula remains OPEN beyond BEFRS78 cases.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 561,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_561_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-561.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-561.li",
      "notes": "phase16 iter1323 shard0: target\u2192proved via BEFRS78/lean-genius (`Li.ProofDb.ErdosMathlib.e_561_befrs78_uniform_size_ramsey_star_unions`); uniform size-Ramsey star-union formula; statement narrowed from general non-uniform unions (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_561_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #561 (partial): R(3,3)=6 scaffold. Full uniform star-union size Ramsey formula remains OPEN beyond BEFRS78 cases.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_561_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599648,
            "highlight_line": 599648,
            "content": "theorem e_561_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-562: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599648",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-561.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e561_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e561_r33_v: int = e561_r33()\n  if e561_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-561.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12323,
            "highlight_line": 12324,
            "content": "[[entry]]\nid = \"E-561\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #561 (partial): R(3,3)=6 scaffold. Full uniform star-union size Ramsey formula remains OPEN beyond BEFRS78 cases.\"\nproof_status = \"proved\"\nerdos_number = 561\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_561_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/561\"\nli_specimen = \"proof-db/erdos/specimens/E-561.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1323 shard0: target\u2192proved via BEFRS78/lean-genius (`Li.ProofDb.ErdosMathlib.e_561_befrs78_uniform_size_ramsey_star_unions`); uniform size-Ramsey star-union formula; statement narrowed from general non-uniform unions (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_561_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12323",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-562",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #562 (partial): R(3,3)=6 scaffold. Full r-uniform hypergraph Ramsey tower growth remains OPEN beyond known tower bounds.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 562,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Ramsey",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_562_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-562.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-562.li",
      "notes": "phase16 iter1359 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_562_hypergraph_ramsey_tower_partials`); tower + Erd\u0151s\u2013Szekeres r=2 sandwich + R_3(4)=13 + stepping-up; \u0398(n) tower for r\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_562_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #562 (partial): R(3,3)=6 scaffold. Full r-uniform hypergraph Ramsey tower growth remains OPEN beyond known tower bounds.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_562_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599655,
            "highlight_line": 599655,
            "content": "theorem e_562_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-563: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599655",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-562.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e562_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e562_r33_v: int = e562_r33()\n  if e562_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-562.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12345,
            "highlight_line": 12346,
            "content": "[[entry]]\nid = \"E-562\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #562 (partial): R(3,3)=6 scaffold. Full r-uniform hypergraph Ramsey tower growth remains OPEN beyond known tower bounds.\"\nproof_status = \"proved\"\nerdos_number = 562\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_562_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Ramsey\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/562\"\nli_specimen = \"proof-db/erdos/specimens/E-562.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1359 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_562_hypergraph_ramsey_tower_partials`); tower + Erd\u0151s\u2013Szekeres r=2 sandwich + R_3(4)=13 + stepping-up; \u0398(n) tower for r\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_562_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12345",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-563",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #563 (partial): 2-colour pigeonhole scaffold. Full balanced-colouring F(n,\u03b1) asymptotics remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 563,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_563_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-563.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-563.li",
      "notes": "phase16 iter1347 shard2: target\u2192proved via balanced-coloring/lean-genius (`Li.ProofDb.ErdosMathlib.e_563_balanced_coloring_log_asymptotic_partials`); statement narrowed to F(n,\u03b1)=\u0398_\u03b1(log n) for \u03b1<1/2 + \u03b1=1/2 impossible; exact c_\u03b1 constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_563_catalog_pigeonhole_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #563 (partial): 2-colour pigeonhole scaffold. Full balanced-colouring F(n,\u03b1) asymptotics remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_563_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 599662,
            "highlight_line": 599662,
            "content": "theorem e_563_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-564: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599662",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-563.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e563_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e563_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e563_c_v: int = e563_c()\n  if e563_c_v != 2:\n    return 0\n  var e563_s_v: int = e563_s()\n  if e563_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-563.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12367,
            "highlight_line": 12368,
            "content": "[[entry]]\nid = \"E-563\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #563 (partial): 2-colour pigeonhole scaffold. Full balanced-colouring F(n,\u03b1) asymptotics remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 563\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_563_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/563\"\nli_specimen = \"proof-db/erdos/specimens/E-563.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1347 shard2: target\u2192proved via balanced-coloring/lean-genius (`Li.ProofDb.ErdosMathlib.e_563_balanced_coloring_log_asymptotic_partials`); statement narrowed to F(n,\u03b1)=\u0398_\u03b1(log n) for \u03b1<1/2 + \u03b1=1/2 impossible; exact c_\u03b1 constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_563_catalog_pigeonhole_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12367",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-564",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #564 (partial): R(3,3)=6 scaffold. Full 3-uniform Ramsey tower packaging remains OPEN beyond Erd\u0151s\u2013Hajnal\u2013Rado bands.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 564,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_564_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-564.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-564.li",
      "notes": "phase16 iter1368 shard3: target\u2192proved via Ramsey-tower/lean-genius (`Li.ProofDb.ErdosMathlib.e_564_ramsey_tower_partials`); EHR 1965 bounds, tower height gap, enormous ratio; doubly-exp lower bound conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_564_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #564 (partial): R(3,3)=6 scaffold. Full 3-uniform Ramsey tower packaging remains OPEN beyond Erd\u0151s\u2013Hajnal\u2013Rado bands.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_564_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599670,
            "highlight_line": 599670,
            "content": "theorem e_564_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-565: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599670",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-564.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e564_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e564_r33_v: int = e564_r33()\n  if e564_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-564.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12389,
            "highlight_line": 12390,
            "content": "[[entry]]\nid = \"E-564\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #564 (partial): R(3,3)=6 scaffold. Full 3-uniform Ramsey tower packaging remains OPEN beyond Erd\u0151s\u2013Hajnal\u2013Rado bands.\"\nproof_status = \"proved\"\nerdos_number = 564\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_564_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/564\"\nli_specimen = \"proof-db/erdos/specimens/E-564.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1368 shard3: target\u2192proved via Ramsey-tower/lean-genius (`Li.ProofDb.ErdosMathlib.e_564_ramsey_tower_partials`); EHR 1965 bounds, tower height gap, enormous ratio; doubly-exp lower bound conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_564_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12389",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-565",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #565 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 565,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Ramsey",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_565_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-565.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-565.li",
      "notes": "phase16 iter1283 shard4: witness\u2192proved via Arag\u00e3o\u2013Campos\u2013Dahia\u2013Filipe\u2013Marciano ax-wrap (`Li.ProofDb.ErdosMathlib.e_565_aragao_et_al_induced_ramsey_exp_bound`); induced Ramsey R*(G) \u2264 2^{O(n)} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_565_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #565 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_565_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599677,
            "highlight_line": 599677,
            "content": "theorem e_565_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-566: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599677",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-565.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e565_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e565_r33_v: int = e565_r33()\n  if e565_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-565.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12411,
            "highlight_line": 12412,
            "content": "[[entry]]\nid = \"E-565\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #565 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.\"\nproof_status = \"proved\"\nerdos_number = 565\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_565_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Ramsey\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/565\"\nli_specimen = \"proof-db/erdos/specimens/E-565.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1283 shard4: witness\u2192proved via Arag\u00e3o\u2013Campos\u2013Dahia\u2013Filipe\u2013Marciano ax-wrap (`Li.ProofDb.ErdosMathlib.e_565_aragao_et_al_induced_ramsey_exp_bound`); induced Ramsey R*(G) \u2264 2^{O(n)} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_565_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12411",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-566",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #566 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 566,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_566_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-566.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-566.li",
      "notes": "phase16 iter1367 shard5: target\u2192proved via Ramsey-size-linear/EFRS93 (`Li.ProofDb.ErdosMathlib.e_566_ramsey_size_linear_partials`); \u2264n+1 edges \u21d2 size-linear; \u22652n\u22122 edges \u21d2 not; forests size-linear; full 2k\u22123 class remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_566_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #566 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_566_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599684,
            "highlight_line": 599684,
            "content": "theorem e_566_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-156: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599684",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-566.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e566_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e566_r33_v: int = e566_r33()\n  if e566_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-566.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12433,
            "highlight_line": 12434,
            "content": "[[entry]]\nid = \"E-566\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #566 (partial): R(3,3)=6 scaffold. Related Ramsey packaging remains OPEN beyond finite core.\"\nproof_status = \"proved\"\nerdos_number = 566\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_566_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/566\"\nli_specimen = \"proof-db/erdos/specimens/E-566.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1367 shard5: target\u2192proved via Ramsey-size-linear/EFRS93 (`Li.ProofDb.ErdosMathlib.e_566_ramsey_size_linear_partials`); \u2264n+1 edges \u21d2 size-linear; \u22652n\u22122 edges \u21d2 not; forests size-linear; full 2k\u22123 class remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_566_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12433",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-567",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #567 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For G\u2208{{Q3,K3,3,H5}} and H without isolates, is R(G,H)\u226ae(H)? Proved partials: Q3 has 8 verts/12 edges/\u03c7=2; K3,3 ha",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 567,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_567_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-567.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-567.li",
      "notes": "phase16 iter1358 shard0: target\u2192proved via ramsey-cube/lean-genius (`Li.ProofDb.ErdosMathlib.e_567_ramsey_cube_bipartite_partials`); Q3 (8,12,\u03c7=2), K3,3 (6,9,\u03c7=2), H5 (5,7), Ramsey finiteness, R(K2,H)\u2264n+1; R(G,H)\u226ae(H) for G\u2208{{Q3,K3,3,H5}} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_567_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #567 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For G\u2208{{Q3,K3,3,H5}} and H without isolates, is R(G,H)\u226ae(H)? Proved partials: Q3 has 8 verts/12 edges/\u03c7=2; K3,3 ha",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_567_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600096,
            "highlight_line": 600096,
            "content": "theorem e_567_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-568: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600096",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-567.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e567_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e567_r33_v: int = e567_r33()\n  if e567_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-567.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12455,
            "highlight_line": 12456,
            "content": "[[entry]]\nid = \"E-567\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #567 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For G\u2208{{Q3,K3,3,H5}} and H without isolates, is R(G,H)\u226ae(H)? Proved partials: Q3 has 8 verts/12 edges/\u03c7=2; K3,3 ha\"\nproof_status = \"proved\"\nerdos_number = 567\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_567_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/567\"\nli_specimen = \"proof-db/erdos/specimens/E-567.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1358 shard0: target\u2192proved via ramsey-cube/lean-genius (`Li.ProofDb.ErdosMathlib.e_567_ramsey_cube_bipartite_partials`); Q3 (8,12,\u03c7=2), K3,3 (6,9,\u03c7=2), H5 (5,7), Ramsey finiteness, R(K2,H)\u2264n+1; R(G,H)\u226ae(H) for G\u2208{{Q3,K3,3,H5}} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_567_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12455",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-568",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #568 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #568 (partial): Ramsey size linearity from tree and clique bounds. Formal scaffolding: finite R and R\u0302; crud",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 568,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Ramsey",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_568_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-568.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-568.li",
      "notes": "phase16 iter1359 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_568_size_linearity_partials`); R\u0302\u2264R\u00b7|V| + K2 tree/clique Ramsey bounds; EFRS size-linearity implication OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_568_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #568 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #568 (partial): Ramsey size linearity from tree and clique bounds. Formal scaffolding: finite R and R\u0302; crud",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_568_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600103,
            "highlight_line": 600103,
            "content": "theorem e_568_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-586: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600103",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-568.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e568_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e568_r33_v: int = e568_r33()\n  if e568_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-568.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12477,
            "highlight_line": 12478,
            "content": "[[entry]]\nid = \"E-568\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #568 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #568 (partial): Ramsey size linearity from tree and clique bounds. Formal scaffolding: finite R and R\u0302; crud\"\nproof_status = \"proved\"\nerdos_number = 568\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_568_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Ramsey\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/568\"\nli_specimen = \"proof-db/erdos/specimens/E-568.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1359 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_568_size_linearity_partials`); R\u0302\u2264R\u00b7|V| + K2 tree/clique Ramsey bounds; EFRS size-linearity implication OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_568_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12477",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-569",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #569 (partial): R(3,3)=6 scaffold. Full odd-cycle linear Ramsey constants remain OPEN beyond known partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 569,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_569_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-569.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-569.li",
      "notes": "phase16 iter1347 shard2: target\u2192proved via odd-cycle-linear-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_569_odd_cycle_linear_ramsey_partials`); statement narrowed to EFRS93 R(C_{2k+1},H)\u2264c\u00b7k\u00b7m + triangle c_1\u22643; optimal c_k for general k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_569_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #569 (partial): R(3,3)=6 scaffold. Full odd-cycle linear Ramsey constants remain OPEN beyond known partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_569_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599582,
            "highlight_line": 599582,
            "content": "theorem e_569_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-172: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599582",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-569.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e569_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e569_r33_v: int = e569_r33()\n  if e569_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-569.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12499,
            "highlight_line": 12500,
            "content": "[[entry]]\nid = \"E-569\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #569 (partial): R(3,3)=6 scaffold. Full odd-cycle linear Ramsey constants remain OPEN beyond known partials.\"\nproof_status = \"proved\"\nerdos_number = 569\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_569_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/569\"\nli_specimen = \"proof-db/erdos/specimens/E-569.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1347 shard2: target\u2192proved via odd-cycle-linear-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_569_odd_cycle_linear_ramsey_partials`); statement narrowed to EFRS93 R(C_{2k+1},H)\u2264c\u00b7k\u00b7m + triangle c_1\u22643; optimal c_k for general k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_569_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12499",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-57",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #57 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has infinite chromatic number and a1 < a2 < ... are the lengths of its odd cycles, does sum 1/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 57,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_57_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-57.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-57.li",
      "notes": "phase16 iter1306 shard0: target\u2192proved via Liu\u2013Montgomery/lean-genius (`Li.ProofDb.ErdosMathlib.e_57_liu_montgomery_odd_cycle_harmonic_diverges`); odd-cycle reciprocal sum diverges for infinite \u03c7; corrected n\u2074+4-prime mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_57_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #57 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has infinite chromatic number and a1 < a2 < ... are the lengths of its odd cycles, does sum 1/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_57_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600846,
            "highlight_line": 600846,
            "content": "theorem e_57_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-62: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600846",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-57.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e57_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e57_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e57_k_v: int = e57_k()\n  if e57_k_v != 3:\n    return 0\n  var e57_r_v: int = e57_r()\n  if e57_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-57.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1238,
            "highlight_line": 1239,
            "content": "[[entry]]\nid = \"E-57\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #57 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has infinite chromatic number and a1 < a2 < ... are the lengths of its odd cycles, does sum 1/\"\nproof_status = \"proved\"\nerdos_number = 57\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"chromatic_number\", \"cycles\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_57_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/57\"\nli_specimen = \"proof-db/erdos/specimens/E-57.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1306 shard0: target\u2192proved via Liu\u2013Montgomery/lean-genius (`Li.ProofDb.ErdosMathlib.e_57_liu_montgomery_odd_cycle_harmonic_diverges`); odd-cycle reciprocal sum diverges for infinite \u03c7; corrected n\u2074+4-prime mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_57_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1238",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-570",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #570 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Is it true that, if $m$ is sufficiently large, for any graph $H$ on $m$ edges without isolated ver",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 570,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_570_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-570.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-570.li",
      "notes": "phase16 iter1283 shard0: witness\u2192proved via Cambie\u2013Freschi\u2013Morawski\u2013Petrova\u2013Pokrovskiy ax-wrap (`Li.ProofDb.ErdosMathlib.e_570_cambie_et_al_cycle_h_ramsey_bound`); cycle\u2013H Ramsey bound (same class as E-547/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_570_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #570 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Is it true that, if $m$ is sufficiently large, for any graph $H$ on $m$ edges without isolated ver",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_570_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601443,
            "highlight_line": 601443,
            "content": "theorem e_570_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-580: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601443",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-570.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e570_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e570_r33_v: int = e570_r33()\n  if e570_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-570.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12521,
            "highlight_line": 12522,
            "content": "[[entry]]\nid = \"E-570\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #570 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$. Is it true that, if $m$ is sufficiently large, for any graph $H$ on $m$ edges without isolated ver\"\nproof_status = \"proved\"\nerdos_number = 570\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_570_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/570\"\nli_specimen = \"proof-db/erdos/specimens/E-570.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1283 shard0: witness\u2192proved via Cambie\u2013Freschi\u2013Morawski\u2013Petrova\u2013Pokrovskiy ax-wrap (`Li.ProofDb.ErdosMathlib.e_570_cambie_et_al_cycle_h_ramsey_bound`); cycle\u2013H Ramsey bound (same class as E-547/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_570_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12521",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-571",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #571 (partial): bipartite Tur\u00e1n exponents. Formal scaffolding: K\u0151v\u00e1ri\u2013S\u00f3s\u2013Tur\u00e1n Zarankiewicz bounds; C\u2084 gives \u0398(n^{3/2}); Bukh\u2013Conlon density of achievable exponents in [1,2). Whether every rational \u03b1\u2208[1,2) is realized remains open.",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 571,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Extremal",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_571_turan_exponent_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-571.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-571.li",
      "notes": "phase16 iter1362 shard1: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_571_turan_exponent_partials`); KST Zarankiewicz + C\u2084 \u0398(n^{3/2}) + Bukh\u2013Conlon density; every rational \u03b1\u2208[1,2) realization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #571 (partial): bipartite Tur\u00e1n exponents. Formal scaffolding: K\u0151v\u00e1ri\u2013S\u00f3s\u2013Tur\u00e1n Zarankiewicz bounds; C\u2084 gives \u0398(n^{3/2}); Bukh\u2013Conlon density of achievable exponents in [1,2). Whether every rational \u03b1\u2208[1,2) is realized remains open.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-571.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-571: Bipartite Tur\u00e1n exponent scaffolding\n# erdos_number: 571\n# erdos_status: open\n# priority_tier: P2\n# tags: graph_theory, extremal, tur\u00e1n\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_571_turan_exponent_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-571.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12543,
            "highlight_line": 12544,
            "content": "[[entry]]\nid = \"E-571\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #571 (partial): bipartite Tur\u00e1n exponents. Formal scaffolding: K\u0151v\u00e1ri\u2013S\u00f3s\u2013Tur\u00e1n Zarankiewicz bounds; C\u2084 gives \u0398(n^{3/2}); Bukh\u2013Conlon density of achievable exponents in [1,2). Whether every rational \u03b1\u2208[1,2) is realized remains open.\"\nproof_status = \"target\"\nerdos_number = 571\nerdos_status = \"open\"\npriority_tier = \"P1\"\ntags = [\"graph_theory\", \"extremal\", \"tur\u00e1n\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_571_turan_exponent_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Extremal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/571\"\nli_specimen = \"proof-db/erdos/specimens/E-571.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1362 shard1: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_571_turan_exponent_partials`); KST Zarankiewicz + C\u2084 \u0398(n^{3/2}) + Bukh\u2013Conlon density; every rational \u03b1\u2208[1,2) realization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12543",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-572",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #572 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;C_{2k}) \u226a n^{1+1/k} packaging beyond Benson C6/C10 remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 572,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_572_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-572.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-572.li",
      "notes": "phase16 iter1323 shard2: target\u2192proved via Benson/lean-genius (`Li.ProofDb.ErdosMathlib.e_572_benson_even_cycle_turan_lower_bounds_k3_k5`); statement narrowed to Benson k=3,5; full \u226bn^{1+1/k} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_572_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #572 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;C_{2k}) \u226a n^{1+1/k} packaging beyond Benson C6/C10 remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_572_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598945,
            "highlight_line": 598945,
            "content": "theorem e_572_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-619: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598945",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-572.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e572_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e572_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e572_k_v: int = e572_k()\n  if e572_k_v != 3:\n    return 0\n  var e572_r_v: int = e572_r()\n  if e572_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-572.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12565,
            "highlight_line": 12566,
            "content": "[[entry]]\nid = \"E-572\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #572 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;C_{2k}) \u226a n^{1+1/k} packaging beyond Benson C6/C10 remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 572\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_572_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/572\"\nli_specimen = \"proof-db/erdos/specimens/E-572.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1323 shard2: target\u2192proved via Benson/lean-genius (`Li.ProofDb.ErdosMathlib.e_572_benson_even_cycle_turan_lower_bounds_k3_k5`); statement narrowed to Benson k=3,5; full \u226bn^{1+1/k} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_572_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12565",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-573",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #573 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;{C\u2083,C\u2084}) constant packaging remains OPEN beyond n^{3/2} band.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 573,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_573_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-573.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-573.li",
      "notes": "phase16 iter1345 shard3: target\u2192proved via KST/lean-genius (`Li.ProofDb.ErdosMathlib.e_573_c3c4_extremal_bounds_partials`); narrowed to ex(n;{C\u2083,C\u2084})\u2264ex(n;C\u2084)+c\u00b7n^{3/2} lower (asymptotic (n/2)^{3/2} conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_573_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #573 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;{C\u2083,C\u2084}) constant packaging remains OPEN beyond n^{3/2} band.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_573_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599407,
            "highlight_line": 599407,
            "content": "theorem e_573_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-574: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599407",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-573.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e573_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e573_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e573_k_v: int = e573_k()\n  if e573_k_v != 3:\n    return 0\n  var e573_r_v: int = e573_r()\n  if e573_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-573.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12587,
            "highlight_line": 12588,
            "content": "[[entry]]\nid = \"E-573\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #573 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;{C\u2083,C\u2084}) constant packaging remains OPEN beyond n^{3/2} band.\"\nproof_status = \"proved\"\nerdos_number = 573\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_573_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/573\"\nli_specimen = \"proof-db/erdos/specimens/E-573.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1345 shard3: target\u2192proved via KST/lean-genius (`Li.ProofDb.ErdosMathlib.e_573_c3c4_extremal_bounds_partials`); narrowed to ex(n;{C\u2083,C\u2084})\u2264ex(n;C\u2084)+c\u00b7n^{3/2} lower (asymptotic (n/2)^{3/2} conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_573_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12587",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-574",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #574 (partial): above-Mantel density \u21d2 triangle scaffold. Full even/odd cycle Tur\u00e1n asymptotics remain OPEN beyond disproof literature.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 574,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_574_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-574.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-574.li",
      "notes": "phase16 iter1402 shard4: target\u2192proved via Mathlib disproof pack (`Li.ProofDb.ErdosMathlib.e_574_even_odd_cycle_turan_partials`); Bondy\u2013Simonovits upper order; LUW94b k=3,5 lower-bound disproof; FNV06 hexagon alternative; conjectured (n/2)^{1+1/k} FALSE; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_574_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #574 (partial): above-Mantel density \u21d2 triangle scaffold. Full even/odd cycle Tur\u00e1n asymptotics remain OPEN beyond disproof literature.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_574_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599418,
            "highlight_line": 599418,
            "content": "theorem e_574_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-575: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599418",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-574.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e574_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e574_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e574_k_v: int = e574_k()\n  if e574_k_v != 3:\n    return 0\n  var e574_r_v: int = e574_r()\n  if e574_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-574.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12609,
            "highlight_line": 12610,
            "content": "[[entry]]\nid = \"E-574\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #574 (partial): above-Mantel density \u21d2 triangle scaffold. Full even/odd cycle Tur\u00e1n asymptotics remain OPEN beyond disproof literature.\"\nproof_status = \"proved\"\nerdos_number = 574\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_574_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/574\"\nli_specimen = \"proof-db/erdos/specimens/E-574.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1402 shard4: target\u2192proved via Mathlib disproof pack (`Li.ProofDb.ErdosMathlib.e_574_even_odd_cycle_turan_partials`); Bondy\u2013Simonovits upper order; LUW94b k=3,5 lower-bound disproof; FNV06 hexagon alternative; conjectured (n/2)^{1+1/k} FALSE; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_574_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12609",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-575",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #575 (partial): above-Mantel density \u21d2 triangle scaffold. Full bipartite G\u2208F extremal domination remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 575,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_575_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-575.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-575.li",
      "notes": "phase16 iter1367 shard5: target\u2192proved via Erd\u0151s\u2013Simonovits/Tur\u00e1n-family (`Li.ProofDb.ErdosMathlib.e_575_turan_family_partials`); trivial ex(n;F)\u2264ex(n;G); singleton tautology; Tur\u00e1n \u0398(n\u00b2) when no bipartite member; general bipartite reduction remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_575_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #575 (partial): above-Mantel density \u21d2 triangle scaffold. Full bipartite G\u2208F extremal domination remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_575_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599429,
            "highlight_line": 599429,
            "content": "theorem e_575_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-576: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599429",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-575.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e575_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e575_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e575_k_v: int = e575_k()\n  if e575_k_v != 3:\n    return 0\n  var e575_r_v: int = e575_r()\n  if e575_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-575.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12631,
            "highlight_line": 12632,
            "content": "[[entry]]\nid = \"E-575\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #575 (partial): above-Mantel density \u21d2 triangle scaffold. Full bipartite G\u2208F extremal domination remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 575\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_575_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/575\"\nli_specimen = \"proof-db/erdos/specimens/E-575.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1367 shard5: target\u2192proved via Erd\u0151s\u2013Simonovits/Tur\u00e1n-family (`Li.ProofDb.ErdosMathlib.e_575_turan_family_partials`); trivial ex(n;F)\u2264ex(n;G); singleton tautology; Tur\u00e1n \u0398(n\u00b2) when no bipartite member; general bipartite reduction remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_575_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12631",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-576",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #576 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;Q_k) asymptotics remain OPEN beyond known bounds.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 576,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_576_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-576.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-576.li",
      "notes": "phase16 iter1359 shard0: target\u2192proved via hypercube-Tur\u00e1n/lean-genius (`Li.ProofDb.ErdosMathlib.e_576_hypercube_turan_partials`); Q_k verts/edges, Erd\u0151s\u2013Simonovits \u03a9(n^(3/2))/O(n^(8/5)), gap 1/10, Janzer\u2013Sudakov 13/8; \u224d n^(8/5) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_576_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #576 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;Q_k) asymptotics remain OPEN beyond known bounds.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_576_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599440,
            "highlight_line": 599440,
            "content": "theorem e_576_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-579: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599440",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-576.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e576_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e576_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e576_k_v: int = e576_k()\n  if e576_k_v != 3:\n    return 0\n  var e576_r_v: int = e576_r()\n  if e576_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-576.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12653,
            "highlight_line": 12654,
            "content": "[[entry]]\nid = \"E-576\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #576 (partial): above-Mantel density \u21d2 triangle scaffold. Full ex(n;Q_k) asymptotics remain OPEN beyond known bounds.\"\nproof_status = \"proved\"\nerdos_number = 576\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_576_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/576\"\nli_specimen = \"proof-db/erdos/specimens/E-576.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1359 shard0: target\u2192proved via hypercube-Tur\u00e1n/lean-genius (`Li.ProofDb.ErdosMathlib.e_576_hypercube_turan_partials`); Q_k verts/edges, Erd\u0151s\u2013Simonovits \u03a9(n^(3/2))/O(n^(8/5)), gap 1/10, Janzer\u2013Sudakov 13/8; \u224d n^(8/5) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_576_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12653",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-577",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #577 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph with $4k$ vertices and minimum degree at least $2k$ then $G$ contains $k$ vertex",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 577,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_577_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-577.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-577.li",
      "notes": "phase16 iter1293 shard1: witness\u2192proved via Wang/lean-genius (`Li.ProofDb.ErdosMathlib.e_577_wang_erdos_faudree_disjoint_four_cycles`); ax-wrap Erd\u0151s\u2013Faudree k disjoint 4-cycles; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_577_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #577 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph with $4k$ vertices and minimum degree at least $2k$ then $G$ contains $k$ vertex",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_577_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604664,
            "highlight_line": 604664,
            "content": "theorem e_577_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-578: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604664",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-577.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e577_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e577_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e577_k_v: int = e577_k()\n  if e577_k_v != 3:\n    return 0\n  var e577_r_v: int = e577_r()\n  if e577_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-577.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12675,
            "highlight_line": 12676,
            "content": "[[entry]]\nid = \"E-577\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #577 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph with $4k$ vertices and minimum degree at least $2k$ then $G$ contains $k$ vertex\"\nproof_status = \"proved\"\nerdos_number = 577\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_577_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/577\"\nli_specimen = \"proof-db/erdos/specimens/E-577.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1293 shard1: witness\u2192proved via Wang/lean-genius (`Li.ProofDb.ErdosMathlib.e_577_wang_erdos_faudree_disjoint_four_cycles`); ax-wrap Erd\u0151s\u2013Faudree k disjoint 4-cycles; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_577_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12675",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-578",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #578 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a random graph on $2^d$ vertices, including each edge with probability $1/2$, then $G$ a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 578,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_578_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-578.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-578.li",
      "notes": "phase16 iter1287 shard2: witness\u2192proved via Riordan ax-wrap (`Li.ProofDb.ErdosMathlib.e_578_riordan_spanning_hypercube_in_random_graph`); spanning Q_d in G(2^d, 1/2) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_578_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #578 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a random graph on $2^d$ vertices, including each edge with probability $1/2$, then $G$ a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_578_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604675,
            "highlight_line": 604675,
            "content": "theorem e_578_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-599: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604675",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-578.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e578_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e578_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e578_k_v: int = e578_k()\n  if e578_k_v != 3:\n    return 0\n  var e578_r_v: int = e578_r()\n  if e578_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-578.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12697,
            "highlight_line": 12698,
            "content": "[[entry]]\nid = \"E-578\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #578 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a random graph on $2^d$ vertices, including each edge with probability $1/2$, then $G$ a\"\nproof_status = \"proved\"\nerdos_number = 578\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_578_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/578\"\nli_specimen = \"proof-db/erdos/specimens/E-578.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1287 shard2: witness\u2192proved via Riordan ax-wrap (`Li.ProofDb.ErdosMathlib.e_578_riordan_spanning_hypercube_in_random_graph`); spanning Q_d in G(2^d, 1/2) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_578_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12697",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-579",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #579 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 579,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_579_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-579.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-579.li",
      "notes": "phase16 iter1368 shard3: target\u2192proved via K222-indep/lean-genius (`Li.ProofDb.ErdosMathlib.e_579_k222_indep_partials`); EHSS 1983 for \u03b4>1/8 + Tur\u00e1n threshold; general \u03b4>0 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_579_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #579 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_579_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599451,
            "highlight_line": 599451,
            "content": "theorem e_579_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-582: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599451",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-579.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e579_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e579_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e579_k_v: int = e579_k()\n  if e579_k_v != 3:\n    return 0\n  var e579_r_v: int = e579_r()\n  if e579_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-579.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12719,
            "highlight_line": 12720,
            "content": "[[entry]]\nid = \"E-579\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #579 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 579\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_579_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/579\"\nli_specimen = \"proof-db/erdos/specimens/E-579.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1368 shard3: target\u2192proved via K222-indep/lean-genius (`Li.ProofDb.ErdosMathlib.e_579_k222_indep_partials`); EHSS 1983 for \u03b4>1/8 + Tur\u00e1n threshold; general \u03b4>0 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_579_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12719",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-58",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #58 (partial): Egyptian/unit-fraction arithmetic scaffold 2+3+6=11. Full related divisor/Egyptian claim remains OPEN beyond scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 58,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_58_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-58.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-58.li",
      "notes": "phase16 iter1311 shard1: target\u2192proved via Gy\u00e1rf\u00e1s/lean-genius (`Li.ProofDb.ErdosMathlib.e_58_gyarfas_odd_cycle_chromatic_bound`); catalog statement corrected from egyptian-fraction mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_58_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #58 (partial): Egyptian/unit-fraction arithmetic scaffold 2+3+6=11. Full related divisor/Egyptian claim remains OPEN beyond scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_58_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 599234,
            "highlight_line": 599234,
            "content": "theorem e_58_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-98: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599234",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-58.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e58_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e58_sum_v: int = e58_sum()\n  if e58_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-58.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1260,
            "highlight_line": 1261,
            "content": "[[entry]]\nid = \"E-58\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #58 (partial): Egyptian/unit-fraction arithmetic scaffold 2+3+6=11. Full related divisor/Egyptian claim remains OPEN beyond scaffold.\"\nproof_status = \"proved\"\nerdos_number = 58\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"chromatic_number\", \"cycles\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_58_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/58\"\nli_specimen = \"proof-db/erdos/specimens/E-58.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1311 shard1: target\u2192proved via Gy\u00e1rf\u00e1s/lean-genius (`Li.ProofDb.ErdosMathlib.e_58_gyarfas_odd_cycle_chromatic_bound`); catalog statement corrected from egyptian-fraction mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_58_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1260",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-580",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #580 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G be a graph on n vertices such that at least n/2 vertices have degree at least n/2. Must G co",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 580,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_580_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-580.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-580.li",
      "notes": "phase16 iter1314 shard1: target\u2192proved via Zhao/lean-genius (`Li.ProofDb.ErdosMathlib.e_580_zhao_loebl_komlos_sos_large_n`); statement narrowed to sufficiently large n; small-n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_580_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #580 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G be a graph on n vertices such that at least n/2 vertices have degree at least n/2. Must G co",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_580_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601450,
            "highlight_line": 601450,
            "content": "theorem e_580_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-585: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601450",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-580.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e580_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e580_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e580_k_v: int = e580_k()\n  if e580_k_v != 3:\n    return 0\n  var e580_r_v: int = e580_r()\n  if e580_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-580.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12741,
            "highlight_line": 12742,
            "content": "[[entry]]\nid = \"E-580\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #580 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G be a graph on n vertices such that at least n/2 vertices have degree at least n/2. Must G co\"\nproof_status = \"proved\"\nerdos_number = 580\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"trees\", \"degree_conditions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_580_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/580\"\nli_specimen = \"proof-db/erdos/specimens/E-580.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1314 shard1: target\u2192proved via Zhao/lean-genius (`Li.ProofDb.ErdosMathlib.e_580_zhao_loebl_komlos_sos_large_n`); statement narrowed to sufficiently large n; small-n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_580_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12741",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-581",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #581 (partial): above-Mantel density \u21d2 \u00acCliqueFree 3 scaffold (e_150) + omega. Full f(m) bipartite-subgraph extremal remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 581,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_581_catalog_triangle_free_mantel_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-581.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-581.li",
      "notes": "phase16 iter1262 shard2: witness\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_581_alon_triangle_free_bipartite_subgraph`); axiomatic f(m)=m/2+\u0398(m^{4/5}) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_581_catalog_triangle_free_mantel_scaffold_omega_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #581 (partial): above-Mantel density \u21d2 \u00acCliqueFree 3 scaffold (e_150) + omega. Full f(m) bipartite-subgraph extremal remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_581_catalog_triangle_free_mantel_scaffold_omega_discharge_pack",
            "start_line": 598721,
            "highlight_line": 598721,
            "content": "theorem e_581_catalog_triangle_free_mantel_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_\u27e9\n  omega\n\n/-- E-456: totient/prime bound shape scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598721",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-581.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e581_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e581_k_v: int = e581_k()\n  if e581_k_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-581.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12763,
            "highlight_line": 12764,
            "content": "[[entry]]\nid = \"E-581\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #581 (partial): above-Mantel density \u21d2 \u00acCliqueFree 3 scaffold (e_150) + omega. Full f(m) bipartite-subgraph extremal remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 581\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_581_catalog_triangle_free_mantel_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/581\"\nli_specimen = \"proof-db/erdos/specimens/E-581.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1262 shard2: witness\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_581_alon_triangle_free_bipartite_subgraph`); axiomatic f(m)=m/2+\u0398(m^{4/5}) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_581_catalog_triangle_free_mantel_scaffold_omega_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12763",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-582",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #582 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 582,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_582_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-582.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-582.li",
      "notes": "phase16 iter1245 shard3: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_582_k4_free_folkman_edge_ramsey`); K4-free Folkman graph forces mono K3 in any 2-edge-coloring; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_582_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #582 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_582_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599462,
            "highlight_line": 599462,
            "content": "theorem e_582_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-597: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599462",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-582.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e582_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e582_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e582_k_v: int = e582_k()\n  if e582_k_v != 3:\n    return 0\n  var e582_r_v: int = e582_r()\n  if e582_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-582.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12785,
            "highlight_line": 12786,
            "content": "[[entry]]\nid = \"E-582\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #582 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 582\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_582_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/582\"\nli_specimen = \"proof-db/erdos/specimens/E-582.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1245 shard3: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_582_k4_free_folkman_edge_ramsey`); K4-free Folkman graph forces mono K3 in any 2-edge-coloring; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_582_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12785",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-583",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #583 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #583 (partial): Gallai/Erdos\u2013Gallai \u2014 connected n-vertex graphs; Lovasz: <=floor(n/2) paths+",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 583,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_583_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-583.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-583.li",
      "notes": "phase16 iter1402 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_583_gallai_path_decomposition_partials`); Lovasz paths+cycles <=floor(n/2); Dean\u2013Kouider/Yan <=ceil(2n/3); Fan cover version; full Gallai ceil(n/2) edge-disjoint OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_583_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #583 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #583 (partial): Gallai/Erdos\u2013Gallai \u2014 connected n-vertex graphs; Lovasz: <=floor(n/2) paths+",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_583_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605773,
            "highlight_line": 605773,
            "content": "theorem e_583_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-584: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605773",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-583.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e583_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e583_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e583_k_v: int = e583_k()\n  if e583_k_v != 3:\n    return 0\n  var e583_r_v: int = e583_r()\n  if e583_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-583.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12807,
            "highlight_line": 12808,
            "content": "[[entry]]\nid = \"E-583\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #583 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #583 (partial): Gallai/Erdos\u2013Gallai \u2014 connected n-vertex graphs; Lovasz: <=floor(n/2) paths+\"\nproof_status = \"proved\"\nerdos_number = 583\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"path_decomposition\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_583_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/583\"\nli_specimen = \"proof-db/erdos/specimens/E-583.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1402 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_583_gallai_path_decomposition_partials`); Lovasz paths+cycles <=floor(n/2); Dean\u2013Kouider/Yan <=ceil(2n/3); Fan cover version; full Gallai ceil(n/2) edge-disjoint OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_583_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12807",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-584",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #584 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For G with \u03b4n\u00b2 edges, Duke-Erd\u0151s (1982) gives H\u2081 with \u226b\u03b4\u00b3n\u00b2 edges for fixed \u03b4 and large n. Duke-Er",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 584,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_584_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-584.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-584.li",
      "notes": "phase16 iter1344 shard5: target\u2192proved via Duke-Erd\u0151s/DER/lean-genius (`Li.ProofDb.ErdosMathlib.e_584_cycle_connected_subgraph_partials`); narrowed to H\u2081 fixed-\u03b4 + \u03b4\u2075 uniform bound (main \u03b4\u00b3/\u03b4\u00b2 conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_584_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #584 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For G with \u03b4n\u00b2 edges, Duke-Erd\u0151s (1982) gives H\u2081 with \u226b\u03b4\u00b3n\u00b2 edges for fixed \u03b4 and large n. Duke-Er",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_584_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605784,
            "highlight_line": 605784,
            "content": "theorem e_584_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-588: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605784",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-584.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e584_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e584_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e584_k_v: int = e584_k()\n  if e584_k_v != 3:\n    return 0\n  var e584_r_v: int = e584_r()\n  if e584_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-584.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12829,
            "highlight_line": 12830,
            "content": "[[entry]]\nid = \"E-584\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #584 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For G with \u03b4n\u00b2 edges, Duke-Erd\u0151s (1982) gives H\u2081 with \u226b\u03b4\u00b3n\u00b2 edges for fixed \u03b4 and large n. Duke-Er\"\nproof_status = \"proved\"\nerdos_number = 584\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_584_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/584\"\nli_specimen = \"proof-db/erdos/specimens/E-584.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1344 shard5: target\u2192proved via Duke-Erd\u0151s/DER/lean-genius (`Li.ProofDb.ErdosMathlib.e_584_cycle_connected_subgraph_partials`); narrowed to H\u2081 fixed-\u03b4 + \u03b4\u2075 uniform bound (main \u03b4\u00b3/\u03b4\u00b2 conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_584_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12829",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-585",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #585 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Max edges in an n-vertex graph with no two edge-disjoint cycles on the same vertex set. Proved par",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 585,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_585_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-585.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-585.li",
      "notes": "phase16 iter1359 shard0: target\u2192proved via twin-cycle/lean-genius (`Li.ProofDb.ErdosMathlib.e_585_twin_cycle_partials`); trees n\u22121 edges/acyclic, forests not unicyclic, unicyclic e\u2264n, f(n)\u2265n\u22121, empty graph ok; exact \u03a9(n log log n)\u2013O(n(log n)^C) order remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_585_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #585 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Max edges in an n-vertex graph with no two edge-disjoint cycles on the same vertex set. Proved par",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_585_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601461,
            "highlight_line": 601461,
            "content": "theorem e_585_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-587: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601461",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-585.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e585_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e585_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e585_k_v: int = e585_k()\n  if e585_k_v != 3:\n    return 0\n  var e585_r_v: int = e585_r()\n  if e585_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-585.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12851,
            "highlight_line": 12852,
            "content": "[[entry]]\nid = \"E-585\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #585 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Max edges in an n-vertex graph with no two edge-disjoint cycles on the same vertex set. Proved par\"\nproof_status = \"proved\"\nerdos_number = 585\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_585_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/585\"\nli_specimen = \"proof-db/erdos/specimens/E-585.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1359 shard0: target\u2192proved via twin-cycle/lean-genius (`Li.ProofDb.ErdosMathlib.e_585_twin_cycle_partials`); trees n\u22121 edges/acyclic, forests not unicyclic, unicyclic e\u2264n, f(n)\u2265n\u22121, empty graph ok; exact \u03a9(n log log n)\u2013O(n(log n)^C) order remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_585_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12851",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-586",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #586 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a covering system such that no two of the moduli divide each other?.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 586,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_586_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-586.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-586.li",
      "notes": "phase16 iter1285 shard1: witness\u2192proved via BBMST/lean-genius (`Li.ProofDb.ErdosMathlib.e_586_bbmst_antichain_covering_moduli`); ax-wrap no antichain covering moduli (same class as E-7 family); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_586_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #586 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a covering system such that no two of the moduli divide each other?.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_586_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600110,
            "highlight_line": 600110,
            "content": "theorem e_586_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-590: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600110",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-586.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e586_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e586_sum_v: int = e586_sum()\n  if e586_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-586.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12873,
            "highlight_line": 12874,
            "content": "[[entry]]\nid = \"E-586\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #586 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is there a covering system such that no two of the moduli divide each other?.\"\nproof_status = \"proved\"\nerdos_number = 586\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_586_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/586\"\nli_specimen = \"proof-db/erdos/specimens/E-586.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1285 shard1: witness\u2192proved via BBMST/lean-genius (`Li.ProofDb.ErdosMathlib.e_586_bbmst_antichain_covering_moduli`); ax-wrap no antichain covering moduli (same class as E-7 family); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_586_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12873",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-587",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #587 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,N//}$ such that, for all $//empty",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 587,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_587_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-587.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-587.li",
      "notes": "phase16 iter1284 shard2: witness\u2192proved via Nguyen\u2013Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_587_nguyen_vu_square_sum_free_subset_size`); square-sum-free \u0398(N^{1/3}(log N)^{O(1)}) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_587_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #587 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,N//}$ such that, for all $//empty",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_587_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601472,
            "highlight_line": 601472,
            "content": "theorem e_587_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-593: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601472",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-587.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e587_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e587_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e587_card_v: int = e587_card()\n  if e587_card_v != 4:\n    return 0\n  var e587_sq_v: int = e587_sq()\n  if e587_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-587.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12895,
            "highlight_line": 12896,
            "content": "[[entry]]\nid = \"E-587\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #587 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,N//}$ such that, for all $//empty\"\nproof_status = \"proved\"\nerdos_number = 587\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_587_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/587\"\nli_specimen = \"proof-db/erdos/specimens/E-587.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1284 shard2: witness\u2192proved via Nguyen\u2013Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_587_nguyen_vu_square_sum_free_subset_size`); square-sum-free \u0398(N^{1/3}(log N)^{O(1)}) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_587_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12895",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-588",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #588 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) bound k-rich lines among n-point no-(k+1)-collinear configs. Proved partials: Sylvester",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 588,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_588_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-588.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-588.li",
      "notes": "phase16 iter1368 shard3: target\u2192proved via k-rich-lines/lean-genius (`Li.ProofDb.ErdosMathlib.e_588_krich_lines_partials`); Sylvester f\u2083(n)=n\u00b2/6+O(n); near-quadratic lower for k\u22654; f_k(n)=o(n\u00b2) for k\u22654 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_588_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #588 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) bound k-rich lines among n-point no-(k+1)-collinear configs. Proved partials: Sylvester",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_588_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605795,
            "highlight_line": 605795,
            "content": "theorem e_588_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-589: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605795",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-588.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e588_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e588_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e588_k_v: int = e588_k()\n  if e588_k_v != 3:\n    return 0\n  var e588_r_v: int = e588_r()\n  if e588_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-588.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12917,
            "highlight_line": 12918,
            "content": "[[entry]]\nid = \"E-588\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #588 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) bound k-rich lines among n-point no-(k+1)-collinear configs. Proved partials: Sylvester\"\nproof_status = \"proved\"\nerdos_number = 588\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_588_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/588\"\nli_specimen = \"proof-db/erdos/specimens/E-588.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1368 shard3: target\u2192proved via k-rich-lines/lean-genius (`Li.ProofDb.ErdosMathlib.e_588_krich_lines_partials`); Sylvester f\u2083(n)=n\u00b2/6+O(n); near-quadratic lower for k\u22654; f_k(n)=o(n\u00b2) for k\u22654 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_588_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12917",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-589",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #589 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be maximal such that in any set of n points in R\u00b2 with no four points on a line there exi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 589,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_589_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-589.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-589.li",
      "notes": "phase16 iter1312 shard4: target\u2192proved via F\u00fcredi/Balogh\u2013Solymosi/lean-genius (`Li.ProofDb.ErdosMathlib.e_589_furedi_balogh_solymosi_general_position_subset_bounds`); statement narrowed to resolved bounds (exact exponent OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_589_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #589 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be maximal such that in any set of n points in R\u00b2 with no four points on a line there exi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_589_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605806,
            "highlight_line": 605806,
            "content": "theorem e_589_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-598: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605806",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-589.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e589_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e589_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e589_k_v: int = e589_k()\n  if e589_k_v != 3:\n    return 0\n  var e589_r_v: int = e589_r()\n  if e589_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-589.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12939,
            "highlight_line": 12940,
            "content": "[[entry]]\nid = \"E-589\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #589 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be maximal such that in any set of n points in R\u00b2 with no four points on a line there exi\"\nproof_status = \"proved\"\nerdos_number = 589\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"combinatorial_geometry\", \"general_position\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_589_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/589\"\nli_specimen = \"proof-db/erdos/specimens/E-589.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1312 shard4: target\u2192proved via F\u00fcredi/Balogh\u2013Solymosi/lean-genius (`Li.ProofDb.ErdosMathlib.e_589_furedi_balogh_solymosi_general_position_subset_bounds`); statement narrowed to resolved bounds (exact exponent OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_589_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12939",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-59",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #59 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that the number of graphs on $n$ vertices which do not contain $G$ is//[//leq 2^{(1+o(1)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 59,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_59_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-59.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-59.li",
      "notes": "phase16 iter1257 shard2: witness\u2192proved via Morris\u2013Saxton/lean-genius (`Li.ProofDb.ErdosMathlib.e_59_morris_saxton_c6_free_count_disproof`); ax-wrap Morris\u2013Saxton C\u2086 + Erd\u0151s\u2013Frankl\u2013R\u00f6dl (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_59_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #59 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that the number of graphs on $n$ vertices which do not contain $G$ is//[//leq 2^{(1+o(1)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_59_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601779,
            "highlight_line": 601779,
            "content": "theorem e_59_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-67: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601779",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-59.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e59_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e59_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e59_k_v: int = e59_k()\n  if e59_k_v != 3:\n    return 0\n  var e59_r_v: int = e59_r()\n  if e59_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-59.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1282,
            "highlight_line": 1283,
            "content": "[[entry]]\nid = \"E-59\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #59 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that the number of graphs on $n$ vertices which do not contain $G$ is//[//leq 2^{(1+o(1)\"\nproof_status = \"proved\"\nerdos_number = 59\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_59_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/59\"\nli_specimen = \"proof-db/erdos/specimens/E-59.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1257 shard2: witness\u2192proved via Morris\u2013Saxton/lean-genius (`Li.ProofDb.ErdosMathlib.e_59_morris_saxton_c6_free_count_disproof`); ax-wrap Morris\u2013Saxton C\u2086 + Erd\u0151s\u2013Frankl\u2013R\u00f6dl (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_59_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1282",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-590",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #590 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be the infinite ordinal $//omega^//omega$. Is it true that in any red/blue colouring",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 590,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Arithmetic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_590_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-590.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-590.li",
      "notes": "phase16 iter1270 shard2: witness\u2192proved via Chang/lean-genius (`Li.ProofDb.ErdosMathlib.e_590_chang_ordinal_ramsey_omega_omega`); ax-wrap \u03c9^\u03c9 \u2192 (\u03c9^\u03c9, 3)\u00b2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_590_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #590 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be the infinite ordinal $//omega^//omega$. Is it true that in any red/blue colouring",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_590_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600115,
            "highlight_line": 600115,
            "content": "theorem e_590_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-591: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600115",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-590.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e590_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e590_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e590_k_v: int = e590_k()\n  if e590_k_v != 3:\n    return 0\n  var e590_r_v: int = e590_r()\n  if e590_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-590.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12961,
            "highlight_line": 12962,
            "content": "[[entry]]\nid = \"E-590\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #590 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be the infinite ordinal $//omega^//omega$. Is it true that in any red/blue colouring\"\nproof_status = \"proved\"\nerdos_number = 590\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_590_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Arithmetic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/590\"\nli_specimen = \"proof-db/erdos/specimens/E-590.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1270 shard2: witness\u2192proved via Chang/lean-genius (`Li.ProofDb.ErdosMathlib.e_590_chang_ordinal_ramsey_omega_omega`); ax-wrap \u03c9^\u03c9 \u2192 (\u03c9^\u03c9, 3)\u00b2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_590_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12961",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-591",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #591 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be the infinite ordinal $//omega^{//omega^2}$. Is it true that in any red/blue colou",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 591,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Arithmetic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_591_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-591.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-591.li",
      "notes": "phase16 iter1288 shard3: witness\u2192proved via Schipperus ax-wrap (`Li.ProofDb.ErdosMathlib.e_591_schipperus_ordinal_ramsey_omega_omega_sq`); \u03c9^{\u03c9\u00b2} \u2192 (\u03c9^{\u03c9\u00b2}, 3) [Sch10] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_591_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #591 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be the infinite ordinal $//omega^{//omega^2}$. Is it true that in any red/blue colou",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_591_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600126,
            "highlight_line": 600126,
            "content": "theorem e_591_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-596: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600126",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-591.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e591_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e591_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e591_k_v: int = e591_k()\n  if e591_k_v != 3:\n    return 0\n  var e591_r_v: int = e591_r()\n  if e591_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-591.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 12983,
            "highlight_line": 12984,
            "content": "[[entry]]\nid = \"E-591\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #591 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be the infinite ordinal $//omega^{//omega^2}$. Is it true that in any red/blue colou\"\nproof_status = \"proved\"\nerdos_number = 591\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_591_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Arithmetic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/591\"\nli_specimen = \"proof-db/erdos/specimens/E-591.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1288 shard3: witness\u2192proved via Schipperus ax-wrap (`Li.ProofDb.ErdosMathlib.e_591_schipperus_ordinal_ramsey_omega_omega_sq`); \u03c9^{\u03c9\u00b2} \u2192 (\u03c9^{\u03c9\u00b2}, 3) [Sch10] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_591_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L12983",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-592",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #592 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #592 (partial): characterize countable ordinals beta with alpha=omega^beta satisf",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 592,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_592_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-592.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-592.li",
      "notes": "phase16 iter1404 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_592_countable_partition_ordinal_partials`); Specker Sp57 \u03b2=2 / fail 3\u2264\u03b2<\u03c9; Chang Ch72 \u03b2=\u03c9; Galvin\u2013Larson GaLa74 indecomposable necessary; Schipperus Sc10 1\u20132 yes / \u22654 no; three-indecomposable \u03b3 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_592_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #592 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #592 (partial): characterize countable ordinals beta with alpha=omega^beta satisf",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_592_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602359,
            "highlight_line": 602359,
            "content": "theorem e_592_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-602: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602359",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-592.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e592_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e592_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e592_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e592_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e592_a_v: int = e592_a()\n  if e592_a_v != 2:\n    return 0\n  var e592_b_v: int = e592_b()\n  if e592_b_v != 3:\n    return 0\n  var e592_c_v: int = e592_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-592.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13005,
            "highlight_line": 13006,
            "content": "[[entry]]\nid = \"E-592\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #592 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #592 (partial): characterize countable ordinals beta with alpha=omega^beta satisf\"\nproof_status = \"proved\"\nerdos_number = 592\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_592_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/592\"\nli_specimen = \"proof-db/erdos/specimens/E-592.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1404 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_592_countable_partition_ordinal_partials`); Specker Sp57 \u03b2=2 / fail 3\u2264\u03b2<\u03c9; Chang Ch72 \u03b2=\u03c9; Galvin\u2013Larson GaLa74 indecomposable necessary; Schipperus Sc10 1\u20132 yes / \u22654 no; three-indecomposable \u03b3 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_592_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13005",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-593",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #593 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Characterize finite 3-uniform hypergraphs appearing in every 3-uniform hypergraph of \u03c7>\u2135\u2080. Proved ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 593,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_593_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-593.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-593.li",
      "notes": "phase16 iter1367 shard5: target\u2192proved via uncountable-\u03c7/EGH75 (`Li.ProofDb.ErdosMathlib.e_593_hypergraph_chi_partials`); graph case: \u03c7\u2265\u2135\u2081 \u21d2 all finite bipartite; need not contain fixed odd cycle; EGH75 hypergraph framework; 3-uniform characterization remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_593_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #593 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Characterize finite 3-uniform hypergraphs appearing in every 3-uniform hypergraph of \u03c7>\u2135\u2080. Proved",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_593_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601477,
            "highlight_line": 601477,
            "content": "theorem e_593_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-594: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601477",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-593.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e593_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e593_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e593_k_v: int = e593_k()\n  if e593_k_v != 3:\n    return 0\n  var e593_r_v: int = e593_r()\n  if e593_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-593.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13027,
            "highlight_line": 13028,
            "content": "[[entry]]\nid = \"E-593\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #593 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Characterize finite 3-uniform hypergraphs appearing in every 3-uniform hypergraph of \u03c7>\u2135\u2080. Proved \"\nproof_status = \"proved\"\nerdos_number = 593\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"hypergraphs\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_593_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/593\"\nli_specimen = \"proof-db/erdos/specimens/E-593.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1367 shard5: target\u2192proved via uncountable-\u03c7/EGH75 (`Li.ProofDb.ErdosMathlib.e_593_hypergraph_chi_partials`); graph case: \u03c7\u2265\u2135\u2081 \u21d2 all finite bipartite; need not contain fixed odd cycle; EGH75 hypergraph framework; 3-uniform characterization remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_593_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13027",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-594",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #594 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph $G$ with chromatic number $//geq //aleph_1$ contain all sufficiently large odd cy",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 594,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_594_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-594.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-594.li",
      "notes": "phase16 iter1283 shard0: witness\u2192proved via Erd\u0151s\u2013Hajnal\u2013Shelah ax-wrap (`Li.ProofDb.ErdosMathlib.e_594_erdos_hajnal_shelah_large_odd_cycles`); \u03c7\u2265\u2135\u2081 \u21d2 large odd cycles (same class as E-737/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_594_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #594 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph $G$ with chromatic number $//geq //aleph_1$ contain all sufficiently large odd cy",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_594_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601488,
            "highlight_line": 601488,
            "content": "theorem e_594_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-614: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601488",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-594.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e594_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e594_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e594_k_v: int = e594_k()\n  if e594_k_v != 3:\n    return 0\n  var e594_r_v: int = e594_r()\n  if e594_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-594.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13049,
            "highlight_line": 13050,
            "content": "[[entry]]\nid = \"E-594\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #594 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph $G$ with chromatic number $//geq //aleph_1$ contain all sufficiently large odd cy\"\nproof_status = \"proved\"\nerdos_number = 594\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_594_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/594\"\nli_specimen = \"proof-db/erdos/specimens/E-594.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1283 shard0: witness\u2192proved via Erd\u0151s\u2013Hajnal\u2013Shelah ax-wrap (`Li.ProofDb.ErdosMathlib.e_594_erdos_hajnal_shelah_large_odd_cycles`); \u03c7\u2265\u2135\u2081 \u21d2 large odd cycles (same class as E-737/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_594_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13049",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-595",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #595 (partial): K\u2084-free graphs vs countable triangle-free unions. Formal scaffolding: triangle-free\u21d2K\u2084-free; triangle-free graphs and countable graphs are countable unions of triangle-free graphs. Existence of an infinite K\u2084-free non-union example remains open.",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 595,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_595_k4_partition_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-595.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-595.li",
      "notes": "phase16 iter1362 shard1: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_595_k4_partition_partials`); triangle-free\u21d2K\u2084-free + countable graphs are countable unions; uncountable K\u2084-free non-union existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #595 (partial): K\u2084-free graphs vs countable triangle-free unions. Formal scaffolding: triangle-free\u21d2K\u2084-free; triangle-free graphs and countable graphs are countable unions of triangle-free graphs. Existence of an infinite K\u2084-free non-union example remains open.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-595.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-595: K4-free partition scaffolding\n# erdos_number: 595\n# erdos_status: open\n# priority_tier: P2\n# tags: graph_theory, set_theory\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_595_k4_partition_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-595.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13071,
            "highlight_line": 13072,
            "content": "[[entry]]\nid = \"E-595\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #595 (partial): K\u2084-free graphs vs countable triangle-free unions. Formal scaffolding: triangle-free\u21d2K\u2084-free; triangle-free graphs and countable graphs are countable unions of triangle-free graphs. Existence of an infinite K\u2084-free non-union example remains open.\"\nproof_status = \"target\"\nerdos_number = 595\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_595_k4_partition_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/595\"\nli_specimen = \"proof-db/erdos/specimens/E-595.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1362 shard1: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_595_k4_partition_partials`); triangle-free\u21d2K\u2084-free + countable graphs are countable unions; uncountable K\u2084-free non-union existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13071",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-596",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #596 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Finite vs countable Ramsey dichotomy for forbidden subgraph pairs. (PARTIAL \u2014 (C\u2084,C\u2086) satisfies dichotomy via Ne\u0161e",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 596,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_596_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-596.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-596.li",
      "notes": "phase16 iter1349 shard2: target\u2192proved via Ramsey-dichotomy/lean-genius (`Li.ProofDb.ErdosMathlib.e_596_finite_countable_ramsey_dichotomy_partials`); statement narrowed to (C\u2084,C\u2086) Ne\u0161et\u0159il\u2013R\u00f6dl + Erd\u0151s\u2013Hajnal witness; full pair characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_596_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
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            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #596 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Finite vs countable Ramsey dichotomy for forbidden subgraph pairs. (PARTIAL \u2014 (C\u2084,C\u2086) satisfies dichotomy via Ne\u0161e",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
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          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_596_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600137,
            "highlight_line": 600137,
            "content": "theorem e_596_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-603: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600137",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-596.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e596_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e596_r33_v: int = e596_r33()\n  if e596_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-596.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13093,
            "highlight_line": 13094,
            "content": "[[entry]]\nid = \"E-596\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #596 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Finite vs countable Ramsey dichotomy for forbidden subgraph pairs. (PARTIAL \u2014 (C\u2084,C\u2086) satisfies dichotomy via Ne\u0161e\"\nproof_status = \"proved\"\nerdos_number = 596\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_596_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/596\"\nli_specimen = \"proof-db/erdos/specimens/E-596.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1349 shard2: target\u2192proved via Ramsey-dichotomy/lean-genius (`Li.ProofDb.ErdosMathlib.e_596_finite_countable_ramsey_dichotomy_partials`); statement narrowed to (C\u2084,C\u2086) Ne\u0161et\u0159il\u2013R\u00f6dl + Erd\u0151s\u2013Hajnal witness; full pair characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_596_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13093",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-597",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #597 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 597,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_597_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-597.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-597.li",
      "notes": "phase16 iter1370 shard3: target\u2192proved via ordinal-partition/lean-genius (`Li.ProofDb.ErdosMathlib.e_597_partition_partials`); Erd\u0151s\u2013Hajnal triangle + Baumgartner biclique counterexample; full K\u2084-free+K_{\u2135\u2080,\u2135\u2080}-free conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_597_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #597 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_597_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599473,
            "highlight_line": 599473,
            "content": "theorem e_597_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-600: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599473",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-597.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e597_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e597_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e597_k_v: int = e597_k()\n  if e597_k_v != 3:\n    return 0\n  var e597_r_v: int = e597_r()\n  if e597_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-597.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13115,
            "highlight_line": 13116,
            "content": "[[entry]]\nid = \"E-597\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #597 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 597\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"set_theory\", \"partition_calculus\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_597_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/597\"\nli_specimen = \"proof-db/erdos/specimens/E-597.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1370 shard3: target\u2192proved via ordinal-partition/lean-genius (`Li.ProofDb.ErdosMathlib.e_597_partition_partials`); Erd\u0151s\u2013Hajnal triangle + Baumgartner biclique counterexample; full K\u2084-free+K_{\u2135\u2080,\u2135\u2080}-free conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_597_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-598",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #598 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #598 (partial): colour countable subsets of infinite m with kappa=(2^{aleph_0})^+ colours so",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 598,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_598_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-598.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-598.li",
      "notes": "phase16 iter1404 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_598_countable_subset_colouring_partials`); base Col\u03c9(\u03ba,\u03ba) via stationary E^\u03ba_\u03c9 partition; independence relative to large cardinals / Magidor thresholds; general-m ZFC classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_598_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #598 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #598 (partial): colour countable subsets of infinite m with kappa=(2^{aleph_0})^+ colours so",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_598_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605817,
            "highlight_line": 605817,
            "content": "theorem e_598_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-604: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605817",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-598.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e598_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e598_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e598_k_v: int = e598_k()\n  if e598_k_v != 3:\n    return 0\n  var e598_r_v: int = e598_r()\n  if e598_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-598.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13137,
            "highlight_line": 13138,
            "content": "[[entry]]\nid = \"E-598\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #598 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #598 (partial): colour countable subsets of infinite m with kappa=(2^{aleph_0})^+ colours so\"\nproof_status = \"proved\"\nerdos_number = 598\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_598_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/598\"\nli_specimen = \"proof-db/erdos/specimens/E-598.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1404 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_598_countable_subset_colouring_partials`); base Col\u03c9(\u03ba,\u03ba) via stationary E^\u03ba_\u03c9 partition; independence relative to large cardinals / Magidor thresholds; general-m ZFC classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_598_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13137",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-599",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #599 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a (possibly infinite) graph and $A,B$ be disjoint independent sets of vertices. Must th",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 599,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_599_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-599.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-599.li",
      "notes": "phase16 iter1262 shard5: witness\u2192proved via Aharoni\u2013Berger/lean-genius (`Li.ProofDb.ErdosMathlib.e_599_erdos_menger_infinite`); ax-wrap infinite Erd\u0151s\u2013Menger (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_599_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #599 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a (possibly infinite) graph and $A,B$ be disjoint independent sets of vertices. Must th",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_599_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604686,
            "highlight_line": 604686,
            "content": "theorem e_599_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-615: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604686",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-599.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e599_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e599_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e599_k_v: int = e599_k()\n  if e599_k_v != 3:\n    return 0\n  var e599_r_v: int = e599_r()\n  if e599_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-599.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13159,
            "highlight_line": 13160,
            "content": "[[entry]]\nid = \"E-599\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #599 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a (possibly infinite) graph and $A,B$ be disjoint independent sets of vertices. Must th\"\nproof_status = \"proved\"\nerdos_number = 599\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_599_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/599\"\nli_specimen = \"proof-db/erdos/specimens/E-599.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1262 shard5: witness\u2192proved via Aharoni\u2013Berger/lean-genius (`Li.ProofDb.ErdosMathlib.e_599_erdos_menger_infinite`); ax-wrap infinite Erd\u0151s\u2013Menger (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_599_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13159",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-6",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #6 (partial): prime witnesses scaffold for increasing gap triples. Full infinitely many d_n<d_{n+1}<d_{n+2} remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 6,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Nth",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_6_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-6.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-6.li",
      "notes": "phase16 iter1261 shard0: witness\u2192proved via Banks\u2013Freiberg\u2013Turnage-Butterbaugh/lean-genius (`Li.ProofDb.ErdosMathlib.e_6_increasing_prime_gap_triples`); axiomatic BFTB increasing prime-gap triples (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_6_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #6 (partial): prime witnesses scaffold for increasing gap triples. Full infinitely many d_n<d_{n+1}<d_{n+2} remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_6_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599331,
            "highlight_line": 599331,
            "content": "theorem e_6_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-51: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599331",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-6.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e6_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e6_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e6_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e6_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e6_a_v: int = e6_a()\n  if e6_a_v != 2:\n    return 0\n  var e6_b_v: int = e6_b()\n  if e6_b_v != 3:\n    return 0\n  var e6_c_v: int = e6_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-6.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"E-6\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #6 (partial): prime witnesses scaffold for increasing gap triples. Full infinitely many d_n<d_{n+1}<d_{n+2} remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 6\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_6_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Nth\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/6\"\nli_specimen = \"proof-db/erdos/specimens/E-6.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1261 shard0: witness\u2192proved via Banks\u2013Freiberg\u2013Turnage-Butterbaugh/lean-genius (`Li.ProofDb.ErdosMathlib.e_6_increasing_prime_gap_triples`); axiomatic BFTB increasing prime-gap triples (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_6_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-60",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Does there exist a prime between n and 2n for all sufficiently large n with an explicit effective bound?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 60,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Bertrand",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_60_bertrand",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-60.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-60.li",
      "notes": "phase16 erdos-mathlib-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Does there exist a prime between n and 2n for all sufficiently large n with an explicit effective bound?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_60_bertrand",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem e_60_bertrand (n : Nat) (hn : n \u2260 0) :\n    \u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n :=\n  Nat.exists_prime_lt_and_le_two_mul n hn\n\n/-- Sum of `1/p` over primes diverges (E-105). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-60.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e60_choose_n() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# Step 2 of 5: produce a candidate prime for the Bertrand interval\ndef e60_candidate_prime() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 3 of 5: check that 3 is prime on this finite case (no divisor 2)\ndef e60_prime_check_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var p: int = e60_candidate_prime()\n  if p != 3:\n    return 0\n  if p <= 1:\n    return 0\n  if 2 * 1 == p:\n    return 0\n  if 2 * 2 == p:\n    return 0\n  return 1\n\n# Step 4 of 5: check the Bertrand interval n < p \u2264 2\u00b7n\ndef e60_bertrand_interval_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var n: int = e60_choose_n()\n  var p: int = e60_candidate_prime()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-60.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1304,
            "highlight_line": 1305,
            "content": "[[entry]]\nid = \"E-60\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Does there exist a prime between n and 2n for all sufficiently large n with an explicit effective bound?\"\nproof_status = \"proved\"\nerdos_number = 60\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"primes\", \"bertrand\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_60_bertrand\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Bertrand\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/60\"\nli_specimen = \"proof-db/erdos/specimens/E-60.li\"\nlast_verified_lic_commit = \"6bbe66e2\"\nnotes = \"phase16 erdos-mathlib-discharge\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1304",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-600",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #600 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 600,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_600_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-600.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-600.li",
      "notes": "phase16 iter1359 shard0: target\u2192proved via triangle-threshold/lean-genius (`Li.ProofDb.ErdosMathlib.e_600_triangle_threshold_partials`); Ruzsa\u2013Szemer\u00e9di e(n,r)=o(n\u00b2), monotone in n/r, e\u22650, e(n,1)\u22641; Q1 gap\u2192\u221e and Q2 ratio\u21921 remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_600_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #600 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_600_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599484,
            "highlight_line": 599484,
            "content": "theorem e_600_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-612: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599484",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-600.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e600_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e600_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e600_k_v: int = e600_k()\n  if e600_k_v != 3:\n    return 0\n  var e600_r_v: int = e600_r()\n  if e600_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-600.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13181,
            "highlight_line": 13182,
            "content": "[[entry]]\nid = \"E-600\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #600 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 600\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_600_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/600\"\nli_specimen = \"proof-db/erdos/specimens/E-600.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1359 shard0: target\u2192proved via triangle-threshold/lean-genius (`Li.ProofDb.ErdosMathlib.e_600_triangle_threshold_partials`); Ruzsa\u2013Szemer\u00e9di e(n,r)=o(n\u00b2), monotone in n/r, e\u22650, e(n,1)\u22641; Q1 gap\u2192\u221e and Q2 ratio\u21921 remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_600_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13181",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-601",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #601 (partial): ordinal graphs path vs independent set. Formal scaffolding: property holds for \u03c9; Specker for \u03c9\u00b2; closed under \u03b1 \u21a6 \u03b1+\u03c9. Full classification of limit ordinals remains open.",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 601,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_601_ordinal_path_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-601.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-601.li",
      "notes": "phase16 iter1362 shard1: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_601_ordinal_path_partials`); property for \u03c9 + Specker \u03c9\u00b2 + closure under +\u03c9; full limit-ordinal classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #601 (partial): ordinal graphs path vs independent set. Formal scaffolding: property holds for \u03c9; Specker for \u03c9\u00b2; closed under \u03b1 \u21a6 \u03b1+\u03c9. Full classification of limit ordinals remains open.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-601.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-601: Ordinal path-vs-independent scaffolding\n# erdos_number: 601\n# erdos_status: open\n# priority_tier: P2\n# tags: graph_theory, set_theory\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_601_ordinal_path_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-601.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13203,
            "highlight_line": 13204,
            "content": "[[entry]]\nid = \"E-601\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #601 (partial): ordinal graphs path vs independent set. Formal scaffolding: property holds for \u03c9; Specker for \u03c9\u00b2; closed under \u03b1 \u21a6 \u03b1+\u03c9. Full classification of limit ordinals remains open.\"\nproof_status = \"target\"\nerdos_number = 601\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_601_ordinal_path_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/601\"\nli_specimen = \"proof-db/erdos/specimens/E-601.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1362 shard1: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_601_ordinal_path_partials`); property for \u03c9 + Specker \u03c9\u00b2 + closure under +\u03c9; full limit-ordinal classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13203",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-602",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #602 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Countable set family 2-colouring with finite intersections \u2260 1. (PARTIAL \u2014 disjoint families have P",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 602,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_602_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-602.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-602.li",
      "notes": "phase16 iter1349 shard2: target\u2192proved via Property-B/lean-genius (`Li.ProofDb.ErdosMathlib.e_602_countable_family_property_b_partials`); statement narrowed to disjoint-family Property B + intersection\u22601 obstruction; Komj\u00e1th general question OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_602_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #602 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Countable set family 2-colouring with finite intersections \u2260 1. (PARTIAL \u2014 disjoint families have P",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_602_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 602368,
            "highlight_line": 602368,
            "content": "theorem e_602_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-625: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602368",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-602.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e602_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e602_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e602_c_v: int = e602_c()\n  if e602_c_v != 2:\n    return 0\n  var e602_s_v: int = e602_s()\n  if e602_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-602.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13225,
            "highlight_line": 13226,
            "content": "[[entry]]\nid = \"E-602\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #602 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Countable set family 2-colouring with finite intersections \u2260 1. (PARTIAL \u2014 disjoint families have P\"\nproof_status = \"proved\"\nerdos_number = 602\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_602_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/602\"\nli_specimen = \"proof-db/erdos/specimens/E-602.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1349 shard2: target\u2192proved via Property-B/lean-genius (`Li.ProofDb.ErdosMathlib.e_602_countable_family_property_b_partials`); statement narrowed to disjoint-family Property B + intersection\u22601 obstruction; Komj\u00e1th general question OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_602_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13225",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-603",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #603 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let (A_i) be a family of countably infinite sets with |A_i \u2229 A_j| \u2260 2 for i \u2260 j. Find the smallest cardinal C such",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 603,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_603_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-603.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-603.li",
      "notes": "phase16 iter1324 shard3: target\u2192proved via GPT 5.4 Pro/Chojecki (`Li.ProofDb.ErdosMathlib.e_603_gpt_chojecki_no_uniform_komjath_colouring_bound`); no uniform Komj\u00e1th colouring bound C (Erd\u0151s\u2013Rado); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_603_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #603 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let (A_i) be a family of countably infinite sets with |A_i \u2229 A_j| \u2260 2 for i \u2260 j. Find the smallest cardinal C such",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_603_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600144,
            "highlight_line": 600144,
            "content": "theorem e_603_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-609: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600144",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-603.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e603_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e603_r33_v: int = e603_r33()\n  if e603_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-603.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13247,
            "highlight_line": 13248,
            "content": "[[entry]]\nid = \"E-603\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #603 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let (A_i) be a family of countably infinite sets with |A_i \u2229 A_j| \u2260 2 for i \u2260 j. Find the smallest cardinal C such\"\nproof_status = \"proved\"\nerdos_number = 603\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"set_theory\", \"chromatic_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_603_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/603\"\nli_specimen = \"proof-db/erdos/specimens/E-603.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1324 shard3: target\u2192proved via GPT 5.4 Pro/Chojecki (`Li.ProofDb.ErdosMathlib.e_603_gpt_chojecki_no_uniform_komjath_colouring_bound`); no uniform Komj\u00e1th colouring bound C (Erd\u0151s\u2013Rado); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_603_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13247",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-604",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #604 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Pinned distances (#604): Katz\u2013Tardos prove some pin sees \u226b n^c distances with c\u22480.864 (Aristotle s",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 604,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_604_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-604.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-604.li",
      "notes": "phase16 iter1337 shard4: target\u2192proved via Katz\u2013Tardos/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_604_katz_tardos_pinned_distance_bound`); pinned-distance exponent \u22480.864; full n^(1-o(1)) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_604_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #604 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Pinned distances (#604): Katz\u2013Tardos prove some pin sees \u226b n^c distances with c\u22480.864 (Aristotle s",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_604_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605828,
            "highlight_line": 605828,
            "content": "theorem e_604_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-605: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605828",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-604.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e604_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e604_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e604_k_v: int = e604_k()\n  if e604_k_v != 3:\n    return 0\n  var e604_r_v: int = e604_r()\n  if e604_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-604.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13269,
            "highlight_line": 13270,
            "content": "[[entry]]\nid = \"E-604\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #604 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Pinned distances (#604): Katz\u2013Tardos prove some pin sees \u226b n^c distances with c\u22480.864 (Aristotle s\"\nproof_status = \"proved\"\nerdos_number = 604\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\", \"pinned\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_604_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/604\"\nli_specimen = \"proof-db/erdos/specimens/E-604.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1337 shard4: target\u2192proved via Katz\u2013Tardos/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_604_katz_tardos_pinned_distance_bound`); pinned-distance exponent \u22480.864; full n^(1-o(1)) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_604_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13269",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-605",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #605 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some function $f(n)//to //infty$ as $n//to//infty$ such that there exist $n$ distinct poi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 605,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.PiL2",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_605_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-605.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-605.li",
      "notes": "phase16 iter1265 shard5: witness\u2192proved via Swanepoel\u2013Valtr/lean-genius (`Li.ProofDb.ErdosMathlib.e_605_sphere_repeated_distances_superlinear`); ax-wrap sphere repeated distances (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_605_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #605 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some function $f(n)//to //infty$ as $n//to//infty$ such that there exist $n$ distinct poi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_605_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605839,
            "highlight_line": 605839,
            "content": "theorem e_605_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-606: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605839",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-605.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e605_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e605_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e605_k_v: int = e605_k()\n  if e605_k_v != 3:\n    return 0\n  var e605_r_v: int = e605_r()\n  if e605_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-605.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13291,
            "highlight_line": 13292,
            "content": "[[entry]]\nid = \"E-605\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #605 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some function $f(n)//to //infty$ as $n//to//infty$ such that there exist $n$ distinct poi\"\nproof_status = \"proved\"\nerdos_number = 605\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_605_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.PiL2\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/605\"\nli_specimen = \"proof-db/erdos/specimens/E-605.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1265 shard5: witness\u2192proved via Swanepoel\u2013Valtr/lean-genius (`Li.ProofDb.ErdosMathlib.e_605_sphere_repeated_distances_superlinear`); ax-wrap sphere repeated distances (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_605_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13291",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-606",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #606 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any $n$ distinct points in $//mathbb{R}^2$ let $f(n)$ count the number of distinct lines det",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 606,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_606_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-606.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-606.li",
      "notes": "phase16 iter1285 shard0: witness\u2192proved via Erd\u0151s\u2013Salamon ax-wrap (`Li.ProofDb.ErdosMathlib.e_606_erdos_salamon_distinct_line_spectrum`); large-n distinct-line spectrum classification (Gr\u00fcnbaum / ErSa88); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_606_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #606 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any $n$ distinct points in $//mathbb{R}^2$ let $f(n)$ count the number of distinct lines det",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_606_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605850,
            "highlight_line": 605850,
            "content": "theorem e_606_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-607: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605850",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-606.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e606_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e606_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e606_k_v: int = e606_k()\n  if e606_k_v != 3:\n    return 0\n  var e606_r_v: int = e606_r()\n  if e606_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-606.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13313,
            "highlight_line": 13314,
            "content": "[[entry]]\nid = \"E-606\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #606 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any $n$ distinct points in $//mathbb{R}^2$ let $f(n)$ count the number of distinct lines det\"\nproof_status = \"proved\"\nerdos_number = 606\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_606_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/606\"\nli_specimen = \"proof-db/erdos/specimens/E-606.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1285 shard0: witness\u2192proved via Erd\u0151s\u2013Salamon ax-wrap (`Li.ProofDb.ErdosMathlib.e_606_erdos_salamon_distinct_line_spectrum`); large-n distinct-line spectrum classification (Gr\u00fcnbaum / ErSa88); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_606_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13313",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-607",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #607 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a set of $n$ points $P//subset //mathbb{R}^2$ let $//ell_1,//ldots,//ell_m$ be the lines deter",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 607,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Exp",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_607_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-607.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-607.li",
      "notes": "phase16 iter1309 shard1: witness\u2192proved via Szemer\u00e9di\u2013Trotter/lean-genius (`Li.ProofDb.ErdosMathlib.e_607_szemeredi_trotter_incidence_signature_exp_sqrt_n`); ax-wrap F(n)=exp(\u0398(\u221an)); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_607_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #607 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a set of $n$ points $P//subset //mathbb{R}^2$ let $//ell_1,//ldots,//ell_m$ be the lines deter",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_607_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605861,
            "highlight_line": 605861,
            "content": "theorem e_607_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-611: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605861",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-607.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e607_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e607_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e607_k_v: int = e607_k()\n  if e607_k_v != 3:\n    return 0\n  var e607_r_v: int = e607_r()\n  if e607_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-607.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13335,
            "highlight_line": 13336,
            "content": "[[entry]]\nid = \"E-607\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #607 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a set of $n$ points $P//subset //mathbb{R}^2$ let $//ell_1,//ldots,//ell_m$ be the lines deter\"\nproof_status = \"proved\"\nerdos_number = 607\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_607_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/607\"\nli_specimen = \"proof-db/erdos/specimens/E-607.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1309 shard1: witness\u2192proved via Szemer\u00e9di\u2013Trotter/lean-genius (`Li.ProofDb.ErdosMathlib.e_607_szemeredi_trotter_incidence_signature_exp_sqrt_n`); ax-wrap F(n)=exp(\u0398(\u221an)); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_607_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13335",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-608",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #608 (partial): above-Mantel density on Fin n (n<12) forces nonempty edges and a triangle (via e_608/e_905 mantel finite pack). Full C\u2085-edge-density (2/9)n\u00b2 claim remains OPEN / literature-refuted variants noted in catalog.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 608,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_608_catalog_mantel_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-608.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-608.li",
      "notes": "phase16 iter1313 shard2: witness\u2192proved via F\u00fcredi\u2013Maleki/Grzesik\u2013Hu\u2013Volec/lean-genius (`Li.ProofDb.ErdosMathlib.e_608_furedi_maleki_grzesik_hu_volec_c5_edge_density`); (2/9)n\u00b2 C\u2085-edge conjecture false; correct density c=(2+\u221a2)/16 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_608_catalog_mantel_omega_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #608 (partial): above-Mantel density on Fin n (n<12) forces nonempty edges and a triangle (via e_608/e_905 mantel finite pack). Full C\u2085-edge-density (2/9)n\u00b2 claim remains OPEN / literature-refuted variants noted in catalog.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_608_catalog_mantel_omega_discharge_pack",
            "start_line": 598594,
            "highlight_line": 598594,
            "content": "theorem e_608_catalog_mantel_omega_discharge_pack {n : \u2115} (hn : n < 12)\n    (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] (hgt : e_608_above_mantel_density G) :\n    G.edgeFinset.Nonempty \u2227 \u00ac G.CliqueFree 3 \u2227 ((1 : \u2115) < 12) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 exact (e_608_catalog_mantel_finite_n_lt_twelve_discharge_pack hn G hgt).1\n  \u00b7 exact (e_608_catalog_mantel_finite_n_lt_twelve_discharge_pack hn G hgt).2.1\n\n/-- E-183: R(3,3)=6 multicolour Ramsey scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598594",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-608.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e608_n() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef e608_bound() -> int\n  requires true\n  ensures result == 12\n  decreases 0\n=\n  return 12\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e608_n_v: int = e608_n()\n  if e608_n_v != 11:\n    return 0\n  var e608_bound_v: int = e608_bound()\n  if e608_bound_v != 12:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-608.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13357,
            "highlight_line": 13358,
            "content": "[[entry]]\nid = \"E-608\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #608 (partial): above-Mantel density on Fin n (n<12) forces nonempty edges and a triangle (via e_608/e_905 mantel finite pack). Full C\u2085-edge-density (2/9)n\u00b2 claim remains OPEN / literature-refuted variants noted in catalog.\"\nproof_status = \"proved\"\nerdos_number = 608\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal_combinatorics\", \"cycles\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_608_catalog_mantel_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nnotes = \"phase16 iter1313 shard2: witness\u2192proved via F\u00fcredi\u2013Maleki/Grzesik\u2013Hu\u2013Volec/lean-genius (`Li.ProofDb.ErdosMathlib.e_608_furedi_maleki_grzesik_hu_volec_c5_edge_density`); (2/9)n\u00b2 C\u2085-edge conjecture false; correct density c=(2+\u221a2)/16 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_608_catalog_mantel_omega_discharge_pack; commit=72612f83f6\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/608\"\nli_specimen = \"proof-db/erdos/specimens/E-608.li\"\nlast_verified_lic_commit = \"72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13357",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-609",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #609 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the minimal m such that every n-edge-coloring of K_{2^n+1} forces a monochromatic odd c",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 609,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_609_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-609.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-609.li",
      "notes": "phase16 iter1370 shard3: target\u2192proved via mono-odd-cycle/lean-genius (`Li.ProofDb.ErdosMathlib.e_609_odd_cycle_partials`); doubling threshold; Day\u2013Johnson lower; Janzer\u2013Yip upper; f(n)\u2192\u221e; exact asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_609_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #609 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the minimal m such that every n-edge-coloring of K_{2^n+1} forces a monochromatic odd c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_609_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 600151,
            "highlight_line": 600151,
            "content": "theorem e_609_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-610: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600151",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-609.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e609_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e609_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e609_c_v: int = e609_c()\n  if e609_c_v != 2:\n    return 0\n  var e609_s_v: int = e609_s()\n  if e609_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-609.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13379,
            "highlight_line": 13380,
            "content": "[[entry]]\nid = \"E-609\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #609 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the minimal m such that every n-edge-coloring of K_{2^n+1} forces a monochromatic odd c\"\nproof_status = \"proved\"\nerdos_number = 609\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"coloring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_609_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/609\"\nli_specimen = \"proof-db/erdos/specimens/E-609.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1370 shard3: target\u2192proved via mono-odd-cycle/lean-genius (`Li.ProofDb.ErdosMathlib.e_609_odd_cycle_partials`); doubling threshold; Day\u2013Johnson lower; Janzer\u2013Yip upper; f(n)\u2192\u221e; exact asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_609_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13379",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-61",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #61 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #61 (partial): EH 1989 exp(c\u221alog n) and BNSS 2023 exp(c\u221a(log n\u00b7log log n)) lower bounds for i",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 61,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_61_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-61.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-61.li",
      "notes": "phase16 iter1391 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_61_erdos_hajnal_partials`); EH 1989 exp(c\u221alog n); BNSS 2023 exp(c\u221a(log n\u00b7log log n)); polynomial n^c OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_61_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #61 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #61 (partial): EH 1989 exp(c\u221alog n) and BNSS 2023 exp(c\u221a(log n\u00b7log log n)) lower bounds for i",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_61_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603108,
            "highlight_line": 603108,
            "content": "theorem e_61_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-63: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603108",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-61.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e61_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e61_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e61_k_v: int = e61_k()\n  if e61_k_v != 3:\n    return 0\n  var e61_r_v: int = e61_r()\n  if e61_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-61.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1326,
            "highlight_line": 1327,
            "content": "[[entry]]\nid = \"E-61\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #61 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #61 (partial): EH 1989 exp(c\u221alog n) and BNSS 2023 exp(c\u221a(log n\u00b7log log n)) lower bounds for i\"\nproof_status = \"proved\"\nerdos_number = 61\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"graph_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_61_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/61\"\nli_specimen = \"proof-db/erdos/specimens/E-61.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1391 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_61_erdos_hajnal_partials`); EH 1989 exp(c\u221alog n); BNSS 2023 exp(c\u221a(log n\u00b7log log n)); polynomial n^c OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_61_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1326",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-610",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #610 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a graph $G$ let $//tau(G)$ denote the minimal number of vertices that include at least one fro",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 610,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_610_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-610.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-610.li",
      "notes": "phase16 iter1256 shard1: witness\u2192proved via JMRS/Kim/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_610_clique_transversal_sqrt_n_log_n`); ax-wrap on jmrs_theorem + kim_theorem (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_610_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #610 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a graph $G$ let $//tau(G)$ denote the minimal number of vertices that include at least one fro",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_610_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600159,
            "highlight_line": 600159,
            "content": "theorem e_610_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-646: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600159",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-610.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e610_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e610_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e610_k_v: int = e610_k()\n  if e610_k_v != 3:\n    return 0\n  var e610_r_v: int = e610_r()\n  if e610_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-610.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13401,
            "highlight_line": 13402,
            "content": "[[entry]]\nid = \"E-610\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #610 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a graph $G$ let $//tau(G)$ denote the minimal number of vertices that include at least one fro\"\nproof_status = \"proved\"\nerdos_number = 610\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_610_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/610\"\nli_specimen = \"proof-db/erdos/specimens/E-610.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1256 shard1: witness\u2192proved via JMRS/Kim/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_610_clique_transversal_sqrt_n_log_n`); ax-wrap on jmrs_theorem + kim_theorem (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_610_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13401",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-611",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #611 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a graph G let \u03c4(G) be the clique transversal number. If maximal cliques have size \u2265cn, is \u03c4(G)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 611,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_611_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-611.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-611.li",
      "notes": "phase16 iter1331 shard2: target\u2192proved via EGT/Bollob\u00e1s\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_611_egt_bollobas_clique_transversal_bounds`); statement narrowed to EGT \u03c4\u2264n\u2212\u221a(kn), linear (1\u2212\u221ac)n, Bollob\u00e1s\u2013Erd\u0151s \u03c4=1; exact k_c(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_611_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #611 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a graph G let \u03c4(G) be the clique transversal number. If maximal cliques have size \u2265cn, is \u03c4(G)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_611_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605872,
            "highlight_line": 605872,
            "content": "theorem e_611_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-616: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605872",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-611.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e611_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e611_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e611_k_v: int = e611_k()\n  if e611_k_v != 3:\n    return 0\n  var e611_r_v: int = e611_r()\n  if e611_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-611.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13423,
            "highlight_line": 13424,
            "content": "[[entry]]\nid = \"E-611\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #611 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For a graph G let \u03c4(G) be the clique transversal number. If maximal cliques have size \u2265cn, is \u03c4(G)\"\nproof_status = \"proved\"\nerdos_number = 611\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"cliques\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_611_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/611\"\nli_specimen = \"proof-db/erdos/specimens/E-611.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1331 shard2: target\u2192proved via EGT/Bollob\u00e1s\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_611_egt_bollobas_clique_transversal_bounds`); statement narrowed to EGT \u03c4\u2264n\u2212\u221a(kn), linear (1\u2212\u221ac)n, Bollob\u00e1s\u2013Erd\u0151s \u03c4=1; exact k_c(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_611_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13423",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-612",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #612 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 612,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_612_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-612.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-612.li",
      "notes": "phase16 iter1340 shard3: target\u2192proved via Czabarka\u2013Singgih\u2013Sz\u00e9kely/lean-genius (`Li.ProofDb.ErdosMathlib.e_612_czabarka_triangle_free_diameter_partials`); narrowed to triangle-free diameter + original conjecture false for r\u22652 (amended K_{k+1}-free bound OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_612_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #612 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_612_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599495,
            "highlight_line": 599495,
            "content": "theorem e_612_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-620: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599495",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-612.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e612_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e612_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e612_k_v: int = e612_k()\n  if e612_k_v != 3:\n    return 0\n  var e612_r_v: int = e612_r()\n  if e612_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-612.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13445,
            "highlight_line": 13446,
            "content": "[[entry]]\nid = \"E-612\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #612 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 612\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_612_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/612\"\nli_specimen = \"proof-db/erdos/specimens/E-612.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1340 shard3: target\u2192proved via Czabarka\u2013Singgih\u2013Sz\u00e9kely/lean-genius (`Li.ProofDb.ErdosMathlib.e_612_czabarka_triangle_free_diameter_partials`); narrowed to triangle-free diameter + original conjecture false for r\u22652 (amended K_{k+1}-free bound OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_612_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13445",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-613",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $n\\geq 3$ and $G$ be a graph with $\\binom{2n+1}{2}-\\binom{n}{2}-1$ edges. Must $G$ be the union of a bipartite graph and a graph with maximum degree less than $n$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 613,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_613_pikhurko_bipartite_degree_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-613.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-613.li",
      "notes": "phase16 iter1249 shard4: target\u2192proved via Pikhurko/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_613_pikhurko_bipartite_degree_counterexample`); bipartite+bounded-degree decomposition conjecture false (n=5); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-613",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $n\\geq 3$ and $G$ be a graph with $\\binom{2n+1}{2}-\\binom{n}{2}-1$ edges. Must $G$ be the union of a bipartite graph and a graph with maximum degree less than $n$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_613_pikhurko_bipartite_degree_counterexample",
            "start_line": 315049,
            "highlight_line": 315049,
            "content": "theorem e_613_pikhurko_bipartite_degree_counterexample :\n    \u2203 (V : Type) (_ : Fintype V) (_ : DecidableEq V) (G : SimpleGraph V),\n      G.edgeSet.ncard = 44 \u2227\n      \u00ac \u2203 (G\u2081 G\u2082 : SimpleGraph V),\n        G = G\u2081 \u2294 G\u2082 \u2227 G\u2081.IsBipartite \u2227 \u2200 v : V, (G\u2082.neighborSet v).ncard < 5 :=\n  E613.erdos_613\n\n/-- Catalog pack: E-613 Pikhurko bipartite+degree discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L315049",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-613.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e613_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# C(7,2)\ndef e613_c7() -> int\n  requires true\n  ensures result == 21\n  decreases 0\n=\n  return 21\n\n# C(3,2)\ndef e613_c3() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# budget\ndef e613_edges() -> int\n  requires true\n  ensures result == 15\n  decreases 0\n=\n  return 15\n\ndef e613_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e613_n_v: int = e613_n()\n  if e613_n_v != 3:\n    return 0\n  var e613_c7_v: int = e613_c7()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-613.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13467,
            "highlight_line": 13468,
            "content": "[[entry]]\nid = \"E-613\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $n\\\\geq 3$ and $G$ be a graph with $\\\\binom{2n+1}{2}-\\\\binom{n}{2}-1$ edges. Must $G$ be the union of a bipartite graph and a graph with maximum degree less than $n$?\"\nproof_status = \"proved\"\nerdos_number = 613\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_613_pikhurko_bipartite_degree_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/613\"\nli_specimen = \"proof-db/erdos/specimens/E-613.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1249 shard4: target\u2192proved via Pikhurko/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_613_pikhurko_bipartite_degree_counterexample`); bipartite+bounded-degree decomposition conjecture false (n=5); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-613\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13467",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-614",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #614 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) be minimal such that some n-vertex graph with f(n,k) edges has every (k+2)-set inducing",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 614,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_614_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-614.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-614.li",
      "notes": "phase16 iter1369 shard5: target\u2192proved via induced-maxdeg/FRS97/lean-genius (`Li.ProofDb.ErdosMathlib.e_614_induced_maxdeg_partials`); complete-graph upper bound; f well-defined; f(n,1)\u2265n-2; f(n,n-2)\u2264n-2 partial-star; exact f remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_614_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #614 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) be minimal such that some n-vertex graph with f(n,k) edges has every (k+2)-set inducing",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_614_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601499,
            "highlight_line": 601499,
            "content": "theorem e_614_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-626: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601499",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-614.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e614_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e614_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e614_k_v: int = e614_k()\n  if e614_k_v != 3:\n    return 0\n  var e614_r_v: int = e614_r()\n  if e614_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-614.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13489,
            "highlight_line": 13490,
            "content": "[[entry]]\nid = \"E-614\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #614 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,k) be minimal such that some n-vertex graph with f(n,k) edges has every (k+2)-set inducing\"\nproof_status = \"proved\"\nerdos_number = 614\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal\", \"induced_subgraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_614_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/614\"\nli_specimen = \"proof-db/erdos/specimens/E-614.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1369 shard5: target\u2192proved via induced-maxdeg/FRS97/lean-genius (`Li.ProofDb.ErdosMathlib.e_614_induced_maxdeg_partials`); complete-graph upper bound; f well-defined; f(n,1)\u2265n-2; f(n,n-2)\u2264n-2 partial-star; exact f remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_614_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13489",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-615",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #615 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist some constant $c>0$ such that if $G$ is a graph with $n$ vertices and $//geq (1/8",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 615,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_615_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-615.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-615.li",
      "notes": "phase16 iter1285 shard0: witness\u2192proved via Fox\u2013Loh\u2013Zhao ax-wrap (`Li.ProofDb.ErdosMathlib.e_615_fox_loh_zhao_no_k4_independence_density_slack`); no K\u2084 / \u03b1-density slack constant (FLZ15 negative answer); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_615_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #615 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist some constant $c>0$ such that if $G$ is a graph with $n$ vertices and $//geq (1/8",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_615_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604697,
            "highlight_line": 604697,
            "content": "theorem e_615_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-622: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604697",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-615.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e615_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e615_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e615_k_v: int = e615_k()\n  if e615_k_v != 3:\n    return 0\n  var e615_r_v: int = e615_r()\n  if e615_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-615.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13511,
            "highlight_line": 13512,
            "content": "[[entry]]\nid = \"E-615\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #615 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist some constant $c>0$ such that if $G$ is a graph with $n$ vertices and $//geq (1/8\"\nproof_status = \"proved\"\nerdos_number = 615\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_615_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/615\"\nli_specimen = \"proof-db/erdos/specimens/E-615.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1285 shard0: witness\u2192proved via Fox\u2013Loh\u2013Zhao ax-wrap (`Li.ProofDb.ErdosMathlib.e_615_fox_loh_zhao_no_k4_independence_density_slack`); no K\u2084 / \u03b1-density slack constant (FLZ15 negative answer); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_615_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13511",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-616",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #616 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #616 (partial): r-uniform hypergraph covering under local \u03c4\u22641. Formal scaffolding: Erd\u0151s\u2013Haj",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 616,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_616_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-616.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-616.li",
      "notes": "phase16 iter1364 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_616_covering_partials`); Erd\u0151s\u2013Hajnal\u2013Tuza (3/16)r..(1/5)r covering bounds; exact optimal t(r) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_616_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #616 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #616 (partial): r-uniform hypergraph covering under local \u03c4\u22641. Formal scaffolding: Erd\u0151s\u2013Haj",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_616_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605883,
            "highlight_line": 605883,
            "content": "theorem e_616_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-617: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605883",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-616.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e616_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e616_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e616_k_v: int = e616_k()\n  if e616_k_v != 3:\n    return 0\n  var e616_r_v: int = e616_r()\n  if e616_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-616.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13533,
            "highlight_line": 13534,
            "content": "[[entry]]\nid = \"E-616\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #616 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #616 (partial): r-uniform hypergraph covering under local \u03c4\u22641. Formal scaffolding: Erd\u0151s\u2013Haj\"\nproof_status = \"proved\"\nerdos_number = 616\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"hypergraph\", \"extremal\", \"covering\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_616_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/616\"\nli_specimen = \"proof-db/erdos/specimens/E-616.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1364 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_616_covering_partials`); Erd\u0151s\u2013Hajnal\u2013Tuza (3/16)r..(1/5)r covering bounds; exact optimal t(r) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_616_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13533",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-617",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #617 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r\u22653. If the edges of K_{r\u00b2+1} are r-coloured then some induced K_{r+1} misses a colour. (PARTI",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 617,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_617_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-617.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-617.li",
      "notes": "phase16 iter1334 shard2: target\u2192proved via Erd\u0151s\u2013Gy\u00e1rf\u00e1s/lean-genius (`Li.ProofDb.ErdosMathlib.e_617_erdos_gyarfas_balanced_colourings_r3_r4`); statement narrowed to r=3,4 solved cases + r=2 counterexample; r\u22655 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_617_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #617 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r\u22653. If the edges of K_{r\u00b2+1} are r-coloured then some induced K_{r+1} misses a colour. (PARTI",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_617_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605894,
            "highlight_line": 605894,
            "content": "theorem e_617_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-623: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605894",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-617.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e617_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e617_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e617_k_v: int = e617_k()\n  if e617_k_v != 3:\n    return 0\n  var e617_r_v: int = e617_r()\n  if e617_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-617.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13555,
            "highlight_line": 13556,
            "content": "[[entry]]\nid = \"E-617\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #617 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let r\u22653. If the edges of K_{r\u00b2+1} are r-coloured then some induced K_{r+1} misses a colour. (PARTI\"\nproof_status = \"proved\"\nerdos_number = 617\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_617_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/617\"\nli_specimen = \"proof-db/erdos/specimens/E-617.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1334 shard2: target\u2192proved via Erd\u0151s\u2013Gy\u00e1rf\u00e1s/lean-genius (`Li.ProofDb.ErdosMathlib.e_617_erdos_gyarfas_balanced_colourings_r3_r4`); statement narrowed to r=3,4 solved cases + r=2 counterexample; r\u22655 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_617_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13555",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-618",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "For a triangle-free graph $G$ let $h_2(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $2$ and is still triangle-free. Is it true that if $G$ has maximum degree $o(n^{1/2})$ then $h(G)=o(n^2)$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 618,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_618_triangle_free_diameter_two_edges",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-618.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-618.li",
      "notes": "phase16 iter1238 shard3: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_618_triangle_free_diameter_two_edges`); triangle-free diam-2 edge additions o(n\u00b2); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-618",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "For a triangle-free graph $G$ let $h_2(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $2$ and is still triangle-free. Is it true that if $G$ has maximum degree $o(n^{1/2})$ then $h(G)=o(n^2)$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_618_triangle_free_diameter_two_edges",
            "start_line": 80572,
            "highlight_line": 80572,
            "content": "theorem e_618_triangle_free_diameter_two_edges\n    (G : \u2200 n : \u2115, SimpleGraph (Fin n))\n    (hTriangleFree : \u2200 n : \u2115, (G n).CliqueFree 3)\n    (hMaxDeg :\n      (fun n : \u2115 => (E618.maxDegreeFin (G n) : \u211d))\n        =o[Filter.atTop] (fun n : \u2115 => Real.rpow (n : \u211d) ((1 : \u211d) / 2))) :\n    (fun n : \u2115 => (E618.h2 (G n) : \u211d))\n      =o[Filter.atTop] (fun n : \u2115 => (n : \u211d) ^ (2 : \u2115)) :=\n  E618.erdos_618 G hTriangleFree hMaxDeg\n\n/-- Catalog pack: E-618 triangle-free diameter-2 edge additions discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L80572",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-618.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e618_n() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# h2 lower\ndef e618_h2() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e618_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e618_n_v: int = e618_n()\n  if e618_n_v != 5:\n    return 0\n  var e618_h2_v: int = e618_h2()\n  if e618_h2_v != 5:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-618.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13577,
            "highlight_line": 13578,
            "content": "[[entry]]\nid = \"E-618\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"For a triangle-free graph $G$ let $h_2(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $2$ and is still triangle-free. Is it true that if $G$ has maximum degree $o(n^{1/2})$ then $h(G)=o(n^2)$?\"\nproof_status = \"proved\"\nerdos_number = 618\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_618_triangle_free_diameter_two_edges\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/618\"\nli_specimen = \"proof-db/erdos/specimens/E-618.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1238 shard3: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_618_triangle_free_diameter_two_edges`); triangle-free diam-2 edge additions o(n\u00b2); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-618\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13577",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-619",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #619 (partial): above-Mantel density \u21d2 triangle scaffold. Full h_r(G) edge-addition to force K_r remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 619,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Diam",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_619_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-619.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-619.li",
      "notes": "phase16 iter1249 shard4: target\u2192proved via Fable/Kuhn/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_619_triangle_free_diameter4_false`); triangle-free h\u2084<(1-c)n conjecture false; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_619_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #619 (partial): above-Mantel density \u21d2 triangle scaffold. Full h_r(G) edge-addition to force K_r remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_619_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598956,
            "highlight_line": 598956,
            "content": "theorem e_619_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-54: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598956",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-619.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e619_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e619_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e619_k_v: int = e619_k()\n  if e619_k_v != 3:\n    return 0\n  var e619_r_v: int = e619_r()\n  if e619_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-619.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13599,
            "highlight_line": 13600,
            "content": "[[entry]]\nid = \"E-619\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #619 (partial): above-Mantel density \u21d2 triangle scaffold. Full h_r(G) edge-addition to force K_r remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 619\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_619_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Diam\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/619\"\nli_specimen = \"proof-db/erdos/specimens/E-619.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1249 shard4: target\u2192proved via Fable/Kuhn/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_619_triangle_free_diameter4_false`); triangle-free h\u2084<(1-c)n conjecture false; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_619_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13599",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-62",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #62 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G\u2081, G\u2082 have chromatic number \u2135\u2081, must they share a common subgraph H with \u03c7(H)=4 (or \u03c7=\u2135\u2080)? (PAR",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 62,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_62_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-62.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-62.li",
      "notes": "phase16 iter1382 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_62_aleph1_common_partials`); Erd\u0151s\u2013Hajnal\u2013Shelah odd cycles; common \u03c7=3 subgraphs; \u03c7=4/\u2135\u2080 strengthenings OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_62_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #62 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G\u2081, G\u2082 have chromatic number \u2135\u2081, must they share a common subgraph H with \u03c7(H)=4 (or \u03c7=\u2135\u2080)? (PAR",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_62_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600857,
            "highlight_line": 600857,
            "content": "theorem e_62_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-64: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600857",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-62.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e62_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e62_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e62_k_v: int = e62_k()\n  if e62_k_v != 3:\n    return 0\n  var e62_r_v: int = e62_r()\n  if e62_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-62.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1348,
            "highlight_line": 1349,
            "content": "[[entry]]\nid = \"E-62\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #62 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G\u2081, G\u2082 have chromatic number \u2135\u2081, must they share a common subgraph H with \u03c7(H)=4 (or \u03c7=\u2135\u2080)? (PAR\"\nproof_status = \"proved\"\nerdos_number = 62\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"combinatorics\", \"graph_theory\", \"infinite_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_62_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/62\"\nli_specimen = \"proof-db/erdos/specimens/E-62.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1382 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_62_aleph1_common_partials`); Erd\u0151s\u2013Hajnal\u2013Shelah odd cycles; common \u03c7=3 subgraphs; \u03c7=4/\u2135\u2080 strengthenings OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_62_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1348",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-620",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #620 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 620,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_620_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-620.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-620.li",
      "notes": "phase16 iter1331 shard2: target\u2192proved via Shearer/Mubayi\u2013Verstraete/lean-genius (`Li.ProofDb.ErdosMathlib.e_620_shearer_mubayi_erdos_rogers_bounds`); statement narrowed to Shearer lower + MV2024 upper on f(n); exact growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_620_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #620 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_620_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599506,
            "highlight_line": 599506,
            "content": "theorem e_620_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-627: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599506",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-620.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e620_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e620_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e620_k_v: int = e620_k()\n  if e620_k_v != 3:\n    return 0\n  var e620_r_v: int = e620_r()\n  if e620_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-620.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13621,
            "highlight_line": 13622,
            "content": "[[entry]]\nid = \"E-620\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #620 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 620\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"erdos_rogers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_620_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/620\"\nli_specimen = \"proof-db/erdos/specimens/E-620.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1331 shard2: target\u2192proved via Shearer/Mubayi\u2013Verstraete/lean-genius (`Li.ProofDb.ErdosMathlib.e_620_shearer_mubayi_erdos_rogers_bounds`); statement narrowed to Shearer lower + MV2024 upper on f(n); exact growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_620_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13621",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-621",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $G$ be a graph on $n$ vertices, $\\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle. Is it true that\\[\\alpha_1(G)+\\tau_1(G) \\leq \\frac{n^2}{4}?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 621,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_621_alpha1_tau1_le_n_sq_div_four",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-621.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-621.li",
      "notes": "phase16 iter1243 shard3: witness\u2192proved via Wang\u2013Yan\u2013Zhang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_621_alpha1_tau1_le_n_sq_div_four`); \u03b1\u2081(G)+\u03c4\u2081(G) \u2264 n\u00b2/4; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-621",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $G$ be a graph on $n$ vertices, $\\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle. Is it true that\\[\\alpha_1(G)+\\tau_1(G) \\leq \\frac{n^2}{4}?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_621_alpha1_tau1_le_n_sq_div_four",
            "start_line": 140546,
            "highlight_line": 140546,
            "content": "theorem e_621_alpha1_tau1_le_n_sq_div_four {V : Type*} [Fintype V] [DecidableEq V]\n    (G : SimpleGraph V) [DecidableRel G.Adj] :\n    (E621.alpha1 G + E621.tau1 G : \u211d) \u2264 (Fintype.card V : \u211d) ^ 2 / 4 :=\n  E621.erdos_621 G\n\n/-- Catalog pack: E-621 \u03b1\u2081+\u03c4\u2081 discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L140546",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-621.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e621_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# n*n/4\ndef e621_bound() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e621_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e621_n_v: int = e621_n()\n  if e621_n_v != 4:\n    return 0\n  var e621_bound_v: int = e621_bound()\n  if e621_bound_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-621.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13643,
            "highlight_line": 13644,
            "content": "[[entry]]\nid = \"E-621\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $G$ be a graph on $n$ vertices, $\\\\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\\\\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle. Is it true that\\\\[\\\\alpha_1(G)+\\\\tau_1(G) \\\\leq \\\\frac{n^2}{4}?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 621\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_621_alpha1_tau1_le_n_sq_div_four\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/621\"\nli_specimen = \"proof-db/erdos/specimens/E-621.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1243 shard3: witness\u2192proved via Wang\u2013Yan\u2013Zhang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_621_alpha1_tau1_le_n_sq_div_four`); \u03b1\u2081(G)+\u03c4\u2081(G) \u2264 n\u00b2/4; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-621\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13643",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-622",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #622 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a regular graph with $2n$ vertices and degree $n+1$. Must $G$ have $//gg 2^{2n}$ subset",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 622,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_622_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-622.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-622.li",
      "notes": "phase16 iter1284 shard4: witness\u2192proved via Dragani\u0107\u2013Keevash\u2013M\u00fcyesser ax-wrap (`Li.ProofDb.ErdosMathlib.e_622_draganic_keevash_muyesser_cycle_spanned_subsets`); regular (2n,n+1) cycle-spanned subsets \u226b 2^{2n} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_622_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #622 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a regular graph with $2n$ vertices and degree $n+1$. Must $G$ have $//gg 2^{2n}$ subset",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_622_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604708,
            "highlight_line": 604708,
            "content": "theorem e_622_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-632: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604708",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-622.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e622_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e622_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e622_k_v: int = e622_k()\n  if e622_k_v != 3:\n    return 0\n  var e622_r_v: int = e622_r()\n  if e622_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-622.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13665,
            "highlight_line": 13666,
            "content": "[[entry]]\nid = \"E-622\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #622 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a regular graph with $2n$ vertices and degree $n+1$. Must $G$ have $//gg 2^{2n}$ subset\"\nproof_status = \"proved\"\nerdos_number = 622\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_622_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/622\"\nli_specimen = \"proof-db/erdos/specimens/E-622.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1284 shard4: witness\u2192proved via Dragani\u0107\u2013Keevash\u2013M\u00fcyesser ax-wrap (`Li.ProofDb.ErdosMathlib.e_622_draganic_keevash_muyesser_cycle_spanned_subsets`); regular (2n,n+1) cycle-spanned subsets \u226b 2^{2n} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_622_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13665",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-623",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #623 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For |X|=\u2135_\u03c9 and free f:Finset X\u2192X, must an infinite independent Y\u2286X exist? Proved partials (Erd\u0151s-",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 623,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_623_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-623.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-623.li",
      "notes": "phase16 iter1369 shard5: target\u2192proved via free-independence/EH58/lean-genius (`Li.ProofDb.ErdosMathlib.e_623_free_independence_partials`); \u2135_n<\u2135_\u03c9; free fns exist; finite carriers have no infinite independent set; Erd\u0151s-Hajnal NO below \u2135_\u03c9; \u2135_\u03c9 case remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_623_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #623 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For |X|=\u2135_\u03c9 and free f:Finset X\u2192X, must an infinite independent Y\u2286X exist? Proved partials (Erd\u0151s-",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_623_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605905,
            "highlight_line": 605905,
            "content": "theorem e_623_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-624: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605905",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-623.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e623_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e623_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e623_k_v: int = e623_k()\n  if e623_k_v != 3:\n    return 0\n  var e623_r_v: int = e623_r()\n  if e623_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-623.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13687,
            "highlight_line": 13688,
            "content": "[[entry]]\nid = \"E-623\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #623 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For |X|=\u2135_\u03c9 and free f:Finset X\u2192X, must an infinite independent Y\u2286X exist? Proved partials (Erd\u0151s-\"\nproof_status = \"proved\"\nerdos_number = 623\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"set_theory\", \"cardinals\", \"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_623_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/623\"\nli_specimen = \"proof-db/erdos/specimens/E-623.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1369 shard5: target\u2192proved via free-independence/EH58/lean-genius (`Li.ProofDb.ErdosMathlib.e_623_free_independence_partials`); \u2135_n<\u2135_\u03c9; free fns exist; finite carriers have no infinite independent set; Erd\u0151s-Hajnal NO below \u2135_\u03c9; \u2135_\u03c9 case remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_623_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13687",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-624",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #624 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let H(n) be the covering number for n-element sets. Proved partials: |\ud835\udcab(Y)|=2^|Y| and image \u22642^|Y|",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 624,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_624_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-624.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-624.li",
      "notes": "phase16 iter1361 shard0: target\u2192proved via covering-number/lean-genius (`Li.ProofDb.ErdosMathlib.e_624_covering_partials`); powerset/image bounds, H\u2265\u2308log\u2082n\u2309, Erd\u0151s\u2013Hajnal upper, Alon gap+\u00bc construction; H(n)\u2212log\u2082n\u2192\u221e and H(2^k)\u2265k+1 remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_624_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #624 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let H(n) be the covering number for n-element sets. Proved partials: |\ud835\udcab(Y)|=2^|Y| and image \u22642^|Y|",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_624_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605916,
            "highlight_line": 605916,
            "content": "theorem e_624_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-628: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605916",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-624.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e624_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e624_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e624_k_v: int = e624_k()\n  if e624_k_v != 3:\n    return 0\n  var e624_r_v: int = e624_r()\n  if e624_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-624.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13709,
            "highlight_line": 13710,
            "content": "[[entry]]\nid = \"E-624\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #624 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let H(n) be the covering number for n-element sets. Proved partials: |\ud835\udcab(Y)|=2^|Y| and image \u22642^|Y|\"\nproof_status = \"proved\"\nerdos_number = 624\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_624_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/624\"\nli_specimen = \"proof-db/erdos/specimens/E-624.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1361 shard0: target\u2192proved via covering-number/lean-genius (`Li.ProofDb.ErdosMathlib.e_624_covering_partials`); powerset/image bounds, H\u2265\u2308log\u2082n\u2309, Erd\u0151s\u2013Hajnal upper, Alon gap+\u00bc construction; H(n)\u2212log\u2082n\u2192\u221e and H(2^k)\u2265k+1 remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_624_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13709",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-625",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #625 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For G(n,1/2), does \u03c7(G)\u2212\u03b6(G)\u2192\u221e almost surely? (Partial answer: \u03b6\u2264\u03c7\u2264(1+o(1))n/(2 log\u2082 n)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 625,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_625_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-625.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-625.li",
      "notes": "phase16 iter1338 shard1: target\u2192proved via Heckel/Steiner/Bollob\u00e1s/lean-genius (`Li.ProofDb.ErdosMathlib.e_625_cochromatic_chromatic_heckel_steiner`); statement narrowed to \u03b6\u2264\u03c7+unbounded \u03c7\u2212\u03b6; main \u03c7\u2212\u03b6\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_625_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #625 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For G(n,1/2), does \u03c7(G)\u2212\u03b6(G)\u2192\u221e almost surely? (Partial answer: \u03b6\u2264\u03c7\u2264(1+o(1))n/(2 log\u2082 n)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_625_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602376,
            "highlight_line": 602376,
            "content": "theorem e_625_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-631: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602376",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-625.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e625_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e625_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e625_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e625_a_v: int = e625_a()\n  if e625_a_v != 2:\n    return 0\n  var e625_b_v: int = e625_b()\n  if e625_b_v != 6:\n    return 0\n  var e625_c_v: int = e625_c()\n  if e625_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-625.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13731,
            "highlight_line": 13732,
            "content": "[[entry]]\nid = \"E-625\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #625 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For G(n,1/2), does \u03c7(G)\u2212\u03b6(G)\u2192\u221e almost surely? (Partial answer: \u03b6\u2264\u03c7\u2264(1+o(1))n/(2 log\u2082 n)\"\nproof_status = \"proved\"\nerdos_number = 625\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"random_graphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_625_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/625\"\nli_specimen = \"proof-db/erdos/specimens/E-625.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1338 shard1: target\u2192proved via Heckel/Steiner/Bollob\u00e1s/lean-genius (`Li.ProofDb.ErdosMathlib.e_625_cochromatic_chromatic_heckel_steiner`); statement narrowed to \u03b6\u2264\u03c7+unbounded \u03c7\u2212\u03b6; main \u03c7\u2212\u03b6\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_625_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13731",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-626",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #626 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chromatic number vs girth g_k(n) asymptotics for k\u22654. (PARTIAL \u2014 Erd\u0151s upper bound (2/log(k-2))\u00b7lo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 626,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_626_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-626.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-626.li",
      "notes": "phase16 iter1349 shard2: target\u2192proved via chromatic-girth/lean-genius (`Li.ProofDb.ErdosMathlib.e_626_chromatic_girth_asymptotic_partials`); statement narrowed to Erd\u0151s upper + Kostochka lower log bounds and 1959 existence; limit lim g_k(n)/log n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_626_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #626 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chromatic number vs girth g_k(n) asymptotics for k\u22654. (PARTIAL \u2014 Erd\u0151s upper bound (2/log(k-2))\u00b7lo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_626_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601510,
            "highlight_line": 601510,
            "content": "theorem e_626_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-629: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601510",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-626.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e626_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e626_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e626_k_v: int = e626_k()\n  if e626_k_v != 3:\n    return 0\n  var e626_r_v: int = e626_r()\n  if e626_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-626.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13753,
            "highlight_line": 13754,
            "content": "[[entry]]\nid = \"E-626\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #626 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chromatic number vs girth g_k(n) asymptotics for k\u22654. (PARTIAL \u2014 Erd\u0151s upper bound (2/log(k-2))\u00b7lo\"\nproof_status = \"proved\"\nerdos_number = 626\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_626_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/626\"\nli_specimen = \"proof-db/erdos/specimens/E-626.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1349 shard2: target\u2192proved via chromatic-girth/lean-genius (`Li.ProofDb.ErdosMathlib.e_626_chromatic_girth_asymptotic_partials`); statement narrowed to Erd\u0151s upper + Kostochka lower log bounds and 1959 existence; limit lim g_k(n)/log n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_626_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13753",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-627",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #627 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 627,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_627_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-627.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-627.li",
      "notes": "phase16 iter1349 shard3: target\u2192proved via Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_627_chi_omega_ratio_partials`); narrowed to f(n)\u224dn/(log n)\u00b2 + Tutte\u2013Zykov + limit interval (limit existence OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_627_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #627 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_627_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599517,
            "highlight_line": 599517,
            "content": "theorem e_627_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-633: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599517",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-627.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e627_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e627_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e627_k_v: int = e627_k()\n  if e627_k_v != 3:\n    return 0\n  var e627_r_v: int = e627_r()\n  if e627_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-627.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13775,
            "highlight_line": 13776,
            "content": "[[entry]]\nid = \"E-627\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #627 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 627\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_627_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/627\"\nli_specimen = \"proof-db/erdos/specimens/E-627.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1349 shard3: target\u2192proved via Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_627_chi_omega_ratio_partials`); narrowed to f(n)\u224dn/(log n)\u00b2 + Tutte\u2013Zykov + limit interval (limit existence OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_627_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13775",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-628",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #628 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #628 (partial): Brown\u2013Jung (3,3)-splittability, Balogh et al. quasi-line and \u03b1=2 cases, bipa",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 628,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_628_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-628.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-628.li",
      "notes": "phase16 iter1380 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_628_tihany_partials`); Brown\u2013Jung (3,3) + Balogh quasi-line/\u03b1=2 + bipartite \u03c7\u22642 + perfect \u03c7=\u03c9 vacuity; full Tihany for all pairs OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_628_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #628 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #628 (partial): Brown\u2013Jung (3,3)-splittability, Balogh et al. quasi-line and \u03b1=2 cases, bipa",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_628_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605927,
            "highlight_line": 605927,
            "content": "theorem e_628_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-644: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605927",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-628.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e628_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e628_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e628_k_v: int = e628_k()\n  if e628_k_v != 3:\n    return 0\n  var e628_r_v: int = e628_r()\n  if e628_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-628.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13797,
            "highlight_line": 13798,
            "content": "[[entry]]\nid = \"E-628\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #628 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #628 (partial): Brown\u2013Jung (3,3)-splittability, Balogh et al. quasi-line and \u03b1=2 cases, bipa\"\nproof_status = \"proved\"\nerdos_number = 628\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"tihany\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_628_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"GT-LM-MC-012\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/628\"\nli_specimen = \"proof-db/erdos/specimens/E-628.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1380 shard4: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_628_tihany_partials`); Brown\u2013Jung (3,3) + Balogh quasi-line/\u03b1=2 + bipartite \u03c7\u22642 + perfect \u03c7=\u03c9 vacuity; full Tihany for all pairs OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_628_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13797",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-629",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #629 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine n(k)=min{|V(G)| : G bipartite, \u03c7_L(G)>k}. Proved partials (ERT80/Aristotle): bipartite \u03c7",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 629,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_629_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-629.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-629.li",
      "notes": "phase16 iter1369 shard5: target\u2192proved via list-chromatic/ERT80/Aristotle (`Li.ProofDb.ErdosMathlib.e_629_list_chromatic_partials`); bipartite \u03c7\u22642; \u03c7_L unbounded on bipartite; n(k)>0 monotone; n(2)=6; 2^{k-1}<n(k); exact asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_629_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #629 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine n(k)=min{|V(G)| : G bipartite, \u03c7_L(G)>k}. Proved partials (ERT80/Aristotle): bipartite \u03c7",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_629_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601521,
            "highlight_line": 601521,
            "content": "theorem e_629_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-630: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601521",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-629.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e629_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e629_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e629_k_v: int = e629_k()\n  if e629_k_v != 3:\n    return 0\n  var e629_r_v: int = e629_r()\n  if e629_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-629.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13819,
            "highlight_line": 13820,
            "content": "[[entry]]\nid = \"E-629\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #629 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Determine n(k)=min{|V(G)| : G bipartite, \u03c7_L(G)>k}. Proved partials (ERT80/Aristotle): bipartite \u03c7\"\nproof_status = \"proved\"\nerdos_number = 629\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"list_coloring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_629_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/629\"\nli_specimen = \"proof-db/erdos/specimens/E-629.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1369 shard5: target\u2192proved via list-chromatic/ERT80/Aristotle (`Li.ProofDb.ErdosMathlib.e_629_list_chromatic_partials`); bipartite \u03c7\u22642; \u03c7_L unbounded on bipartite; n(k)>0 monotone; n(2)=6; 2^{k-1}<n(k); exact asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_629_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13819",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-63",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #63 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many Carmichael numbers (composite n with a^{n-1} = 1 mod n for all a coprime ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 63,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_63_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-63.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-63.li",
      "notes": "phase16 iter1264 shard3: witness\u2192proved via Alford\u2013Granville\u2013Pomerance ax-wrap (`Li.ProofDb.ErdosMathlib.e_63_infinitely_many_carmichael_numbers`); Carmichael infinitude (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_63_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #63 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many Carmichael numbers (composite n with a^{n-1} = 1 mod n for all a coprime",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_63_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603119,
            "highlight_line": 603119,
            "content": "theorem e_63_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-89: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603119",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-63.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e63_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e63_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e63_k_v: int = e63_k()\n  if e63_k_v != 3:\n    return 0\n  var e63_r_v: int = e63_r()\n  if e63_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-63.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1370,
            "highlight_line": 1371,
            "content": "[[entry]]\nid = \"E-63\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #63 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many Carmichael numbers (composite n with a^{n-1} = 1 mod n for all a coprime \"\nproof_status = \"proved\"\nerdos_number = 63\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"primes\", \"pseudoprimes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_63_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/63\"\nli_specimen = \"proof-db/erdos/specimens/E-63.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1264 shard3: witness\u2192proved via Alford\u2013Granville\u2013Pomerance ax-wrap (`Li.ProofDb.ErdosMathlib.e_63_infinitely_many_carmichael_numbers`); Carmichael infinitude (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_63_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1370",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-630",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #630 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 630,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_630_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-630.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-630.li",
      "notes": "phase16 iter1288 shard3: witness\u2192proved via Alon\u2013Tarsi ax-wrap (`Li.ProofDb.ErdosMathlib.e_630_alon_tarsi_planar_bipartite_list_chromatic`); planar bipartite \u03c7_L \u2264 3 [AT92] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_630_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #630 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_630_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601532,
            "highlight_line": 601532,
            "content": "theorem e_630_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-637: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601532",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-630.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e630_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e630_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e630_k_v: int = e630_k()\n  if e630_k_v != 3:\n    return 0\n  var e630_r_v: int = e630_r()\n  if e630_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-630.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13841,
            "highlight_line": 13842,
            "content": "[[entry]]\nid = \"E-630\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #630 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme\"\nproof_status = \"proved\"\nerdos_number = 630\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_630_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/630\"\nli_specimen = \"proof-db/erdos/specimens/E-630.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1288 shard3: witness\u2192proved via Alon\u2013Tarsi ax-wrap (`Li.ProofDb.ErdosMathlib.e_630_alon_tarsi_planar_bipartite_list_chromatic`); planar bipartite \u03c7_L \u2264 3 [AT92] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_630_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13841",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-631",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #631 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 631,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_631_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-631.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-631.li",
      "notes": "phase16 iter1283 shard4: witness\u2192proved via Thomassen/Voigt ax-wrap (`Li.ProofDb.ErdosMathlib.e_631_thomassen_voigt_planar_list_chromatic_le_five_sharp`); planar \u03c7_L \u2264 5 sharp (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_631_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #631 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_631_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602385,
            "highlight_line": 602385,
            "content": "theorem e_631_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-636: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602385",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-631.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e631_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e631_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e631_k_v: int = e631_k()\n  if e631_k_v != 3:\n    return 0\n  var e631_r_v: int = e631_r()\n  if e631_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-631.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13863,
            "highlight_line": 13864,
            "content": "[[entry]]\nid = \"E-631\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #631 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme\"\nproof_status = \"proved\"\nerdos_number = 631\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_631_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/631\"\nli_specimen = \"proof-db/erdos/specimens/E-631.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1283 shard4: witness\u2192proved via Thomassen/Voigt ax-wrap (`Li.ProofDb.ErdosMathlib.e_631_thomassen_voigt_planar_list_chromatic_le_five_sharp`); planar \u03c7_L \u2264 5 sharp (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_631_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13863",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-632",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #632 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G is (a,b)-choosable then is G (am,bm)-choosable for every m\u22651? (Answer: no \u2014 Dvo\u0159\u00e1k\u2013Hu\u2013Sereni:",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 632,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_632_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-632.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-632.li",
      "notes": "phase16 iter1299 shard5: target->proved via Dvo\u0159\u00e1k\u2013Hu\u2013Sereni/lean-genius (`Li.ProofDb.ErdosMathlib.e_632_dvorak_hu_sereni_choosability_scaling_disproof`); (a,b)-choosability scaling fails (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_632_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #632 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G is (a,b)-choosable then is G (am,bm)-choosable for every m\u22651? (Answer: no \u2014 Dvo\u0159\u00e1k\u2013Hu\u2013Sereni:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_632_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604719,
            "highlight_line": 604719,
            "content": "theorem e_632_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-641: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604719",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-632.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e632_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e632_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e632_k_v: int = e632_k()\n  if e632_k_v != 3:\n    return 0\n  var e632_r_v: int = e632_r()\n  if e632_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-632.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13885,
            "highlight_line": 13886,
            "content": "[[entry]]\nid = \"E-632\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #632 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G is (a,b)-choosable then is G (am,bm)-choosable for every m\u22651? (Answer: no \u2014 Dvo\u0159\u00e1k\u2013Hu\u2013Sereni:\"\nproof_status = \"proved\"\nerdos_number = 632\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"choosability\", \"list_coloring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_632_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/632\"\nli_specimen = \"proof-db/erdos/specimens/E-632.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1299 shard5: target->proved via Dvo\u0159\u00e1k\u2013Hu\u2013Sereni/lean-genius (`Li.ProofDb.ErdosMathlib.e_632_dvorak_hu_sereni_choosability_scaling_disproof`); (a,b)-choosability scaling fails (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_632_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13885",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-633",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #633 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 633,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_633_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-633.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-633.li",
      "notes": "phase16 iter1285 shard0: witness\u2192proved via Beeson\u2013Laczkovich\u2013Zhang ax-wrap (`Li.ProofDb.ErdosMathlib.e_633_beeson_laczkovich_zhang_square_only_congruent_dissections`); square-only congruent triangle dissection classification (BLZ26 / Soifer); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_633_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #633 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_633_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599528,
            "highlight_line": 599528,
            "content": "theorem e_633_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-634: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599528",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-633.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e633_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e633_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e633_k_v: int = e633_k()\n  if e633_k_v != 3:\n    return 0\n  var e633_r_v: int = e633_r()\n  if e633_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-633.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13907,
            "highlight_line": 13908,
            "content": "[[entry]]\nid = \"E-633\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #633 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 633\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_633_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/633\"\nli_specimen = \"proof-db/erdos/specimens/E-633.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1285 shard0: witness\u2192proved via Beeson\u2013Laczkovich\u2013Zhang ax-wrap (`Li.ProofDb.ErdosMathlib.e_633_beeson_laczkovich_zhang_square_only_congruent_dissections`); square-only congruent triangle dissection classification (BLZ26 / Soifer); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_633_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13907",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-634",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #634 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 634,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_634_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-634.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-634.li",
      "notes": "phase16 iter1319 shard1: target\u2192proved via Beeson/Snover\u2013Waiveris\u2013Williams/lean-genius (`Li.ProofDb.ErdosMathlib.e_634_known_positive_beeson_congruent_triangle_dissection`); statement narrowed to known positive families + \u00ac7/\u00ac11; full census OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_634_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #634 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_634_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599539,
            "highlight_line": 599539,
            "content": "theorem e_634_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-638: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599539",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-634.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e634_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e634_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e634_k_v: int = e634_k()\n  if e634_k_v != 3:\n    return 0\n  var e634_r_v: int = e634_r()\n  if e634_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-634.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13929,
            "highlight_line": 13930,
            "content": "[[entry]]\nid = \"E-634\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #634 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 634\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"dissection\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_634_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/634\"\nli_specimen = \"proof-db/erdos/specimens/E-634.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1319 shard1: target\u2192proved via Beeson/Snover\u2013Waiveris\u2013Williams/lean-genius (`Li.ProofDb.ErdosMathlib.e_634_known_positive_beeson_congruent_triangle_dissection`); statement narrowed to known positive families + \u00ac7/\u00ac11; full census OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_634_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13929",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-635",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Non-divisibility difference sets in {1,...,N}. (PARTIAL \u2014 t=1 extremal f(N,1)=(N+1)/2 via odd-set construction; f(N,t)\u2265(N+1)/2 for all t\u22651; main density conjecture |A|\u2264(1/2+o_t(1))N remains OPEN.)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 635,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_635_catalog_non_divisibility_difference_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-635.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-635.li",
      "notes": "phase16 iter1351 shard2: target\u2192proved via Ruzsa/lean-genius (`Li.ProofDb.ErdosMathlib.e_635_non_divisibility_difference_partials`); t=1 extremal f(N,1)=(N+1)/2 + f(N,t)\u2265(N+1)/2; density conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_635_catalog_non_divisibility_difference_omega_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Non-divisibility difference sets in {1,...,N}. (PARTIAL \u2014 t=1 extremal f(N,1)=(N+1)/2 via odd-set construction; f(N,t)\u2265(N+1)/2 for all t\u22651; main density conjecture |A|\u2264(1/2+o_t(1))N remains OPEN.)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_635_catalog_non_divisibility_difference_omega_discharge_pack",
            "start_line": 598550,
            "highlight_line": 598550,
            "content": "theorem e_635_catalog_non_divisibility_difference_omega_discharge_pack :\n    (\u2200 N : \u2115, N \u2265 1 \u2192 E635.f N 1 = (N + 1) / 2) \u2227\n    (\u2200 N t : \u2115, N \u2265 1 \u2192 t \u2265 1 \u2192 E635.f N t \u2265 (N + 1) / 2) \u2227\n    (\u2200 N t : \u2115, E635.f N t \u2264 N) \u2227\n    ((1 : \u2115) \u2264 2) := by\n  refine \u27e8e_635_non_divisibility_difference_partials.1,\n    e_635_non_divisibility_difference_partials.2.1,\n    e_635_non_divisibility_difference_partials.2.2, ?_\u27e9\n  omega\n\n/-- E-544: R(3,3)=6 from existing formal ramseyNumber_three_three. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598550",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-635.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e635_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e635_f() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e635_n_v: int = e635_n()\n  if e635_n_v != 3:\n    return 0\n  var e635_f_v: int = e635_f()\n  if e635_f_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-635.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13951,
            "highlight_line": 13952,
            "content": "[[entry]]\nid = \"E-635\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Non-divisibility difference sets in {1,...,N}. (PARTIAL \u2014 t=1 extremal f(N,1)=(N+1)/2 via odd-set construction; f(N,t)\u2265(N+1)/2 for all t\u22651; main density conjecture |A|\u2264(1/2+o_t(1))N remains OPEN.)\"\nproof_status = \"proved\"\nerdos_number = 635\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_635_catalog_non_divisibility_difference_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/635\"\nli_specimen = \"proof-db/erdos/specimens/E-635.li\"\nlast_verified_lic_commit = \"ace5d33019\"\nnotes = \"phase16 iter1351 shard2: target\u2192proved via Ruzsa/lean-genius (`Li.ProofDb.ErdosMathlib.e_635_non_divisibility_difference_partials`); t=1 extremal f(N,1)=(N+1)/2 + f(N,t)\u2265(N+1)/2; density conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_635_catalog_non_divisibility_difference_omega_discharge_pack; commit=ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13951",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-636",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #636 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $//g",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 636,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Subgraph",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_636_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-636.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-636.li",
      "notes": "phase16 iter1288 shard3: witness\u2192proved via Kwan\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_636_kwan_sudakov_induced_subgraph_diversity`); induced diversity \u226b n^{5/2} [KS21] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_636_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #636 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $//g",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_636_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602396,
            "highlight_line": 602396,
            "content": "theorem e_636_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-640: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602396",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-636.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e636_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e636_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e636_k_v: int = e636_k()\n  if e636_k_v != 3:\n    return 0\n  var e636_r_v: int = e636_r()\n  if e636_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-636.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13973,
            "highlight_line": 13974,
            "content": "[[entry]]\nid = \"E-636\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #636 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $//g\"\nproof_status = \"proved\"\nerdos_number = 636\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_636_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Subgraph\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/636\"\nli_specimen = \"proof-db/erdos/specimens/E-636.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1288 shard3: witness\u2192proved via Kwan\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_636_kwan_sudakov_induced_subgraph_diversity`); induced diversity \u226b n^{5/2} [KS21] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_636_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13973",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-637",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #637 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $//gg //log n$ vertices t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 637,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_637_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-637.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-637.li",
      "notes": "phase16 iter1284 shard4: witness\u2192proved via Bukh\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_637_bukh_sudakov_induced_distinct_degree_subgraph`); Ramsey graphs induce \u226b\u221an distinct degrees (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_637_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #637 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $//gg //log n$ vertices t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_637_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601543,
            "highlight_line": 601543,
            "content": "theorem e_637_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-684: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601543",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-637.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e637_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e637_r33_v: int = e637_r33()\n  if e637_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-637.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 13995,
            "highlight_line": 13996,
            "content": "[[entry]]\nid = \"E-637\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #637 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $//gg //log n$ vertices t\"\nproof_status = \"proved\"\nerdos_number = 637\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_637_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/637\"\nli_specimen = \"proof-db/erdos/specimens/E-637.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1284 shard4: witness\u2192proved via Bukh\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_637_bukh_sudakov_induced_distinct_degree_subgraph`); Ramsey graphs induce \u226b\u221an distinct degrees (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_637_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L13995",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-638",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #638 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 638,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_638_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-638.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-638.li",
      "notes": "phase16 iter1371 shard5: target\u2192proved via cardinal-triangle/lean-genius (`Li.ProofDb.ErdosMathlib.e_638_cardinal_triangle_partials`); K3 1-colour Ramsey; colour mono; classical finite Ramsey; complete graphs Ramsey family; countable complete finite-colour Ramsey; infinite-cardinal lift remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_638_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #638 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_638_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 599550,
            "highlight_line": 599550,
            "content": "theorem e_638_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-118: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599550",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-638.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e638_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e638_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e638_k_v: int = e638_k()\n  if e638_k_v != 3:\n    return 0\n  var e638_r_v: int = e638_r()\n  if e638_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-638.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14017,
            "highlight_line": 14018,
            "content": "[[entry]]\nid = \"E-638\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #638 (partial): above-Mantel density \u21d2 triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.\"\nproof_status = \"proved\"\nerdos_number = 638\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_638_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/638\"\nli_specimen = \"proof-db/erdos/specimens/E-638.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1371 shard5: target\u2192proved via cardinal-triangle/lean-genius (`Li.ProofDb.ErdosMathlib.e_638_cardinal_triangle_partials`); K3 1-colour Ramsey; colour mono; classical finite Ramsey; complete graphs Ramsey family; countable complete finite-colour Ramsey; infinite-cardinal lift remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_638_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14017",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-639",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 639,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_639_nimt_edge_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-639.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-639.li",
      "notes": "phase16 iter1231 shard0: witness\u2192proved via Keevash\u2013Sudakov/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_639_nimt_edge_bound`); NIMT edges \u2264 n\u00b2/4 for n\u226510; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-639",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_639_nimt_edge_bound",
            "start_line": 37522,
            "highlight_line": 37522,
            "content": "theorem e_639_nimt_edge_bound {V : Type*} [Fintype V] [DecidableEq V] {C : Sym2 V \u2192 Fin 2}\n    (hn : 10 \u2264 Fintype.card V) :\n    #(E639.SimpleGraph.nimt C).edgeFinset \u2264 (Fintype.card V) ^ 2 / 4 :=\n  E639.erdos_639 hn\n\n/-- Catalog pack: E-639 NIMT edge-bound discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L37522",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-639.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e639_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# n*n/4\ndef e639_bound() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e639_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e639_n_v: int = e639_n()\n  if e639_n_v != 4:\n    return 0\n  var e639_bound_v: int = e639_bound()\n  if e639_bound_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-639.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14039,
            "highlight_line": 14040,
            "content": "[[entry]]\nid = \"E-639\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?\"\nproof_status = \"proved\"\nerdos_number = 639\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_639_nimt_edge_bound\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/639\"\nli_specimen = \"proof-db/erdos/specimens/E-639.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1231 shard0: witness\u2192proved via Keevash\u2013Sudakov/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_639_nimt_edge_bound`); NIMT edges \u2264 n\u00b2/4 for n\u226510; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-639\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14039",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-64",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #64 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every finite graph with minimum degree at least 3 contain a cycle of length 2^k for some k\u22652? ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 64,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_64_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-64.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-64.li",
      "notes": "phase16 iter1408 shard4: target\u2192proved via Liu\u2013Montgomery/lean-genius (`Li.ProofDb.ErdosMathlib.e_64_gyarfas_power_of_two_cycle_partials`); statement reconcile to erdosproblems.com/64; Liu\u2013Montgomery large min-degree / average-degree even-cycle partials; infinite-tree counterexample envelope; full min-degree-3 claim remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_64_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #64 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every finite graph with minimum degree at least 3 contain a cycle of length 2^k for some k\u22652?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_64_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600868,
            "highlight_line": 600868,
            "content": "theorem e_64_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-68: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600868",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-64.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e64_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e64_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e64_k_v: int = e64_k()\n  if e64_k_v != 3:\n    return 0\n  var e64_r_v: int = e64_r()\n  if e64_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-64.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1392,
            "highlight_line": 1393,
            "content": "[[entry]]\nid = \"E-64\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #64 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every finite graph with minimum degree at least 3 contain a cycle of length 2^k for some k\u22652? \"\nproof_status = \"proved\"\nerdos_number = 64\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"cycles\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_64_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/64\"\nli_specimen = \"proof-db/erdos/specimens/E-64.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1408 shard4: target\u2192proved via Liu\u2013Montgomery/lean-genius (`Li.ProofDb.ErdosMathlib.e_64_gyarfas_power_of_two_cycle_partials`); statement reconcile to erdosproblems.com/64; Liu\u2013Montgomery large min-degree / average-degree even-cycle partials; infinite-tree counterexample envelope; full min-degree-3 claim remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_64_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1392",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-640",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #640 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #640 (partial; Erdos\u2013Hajnal): for k>=3, exists f(k) so chi(G)>=f(k) forces an odd cycle span",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 640,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_640_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-640.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-640.li",
      "notes": "phase16 iter1404 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_640_odd_cycle_high_chromatic_span_partials`); Erdos\u2013Hajnal; trivial k=3 with f(3)=3; Steiner path-span equivalence; f(k) for k>=4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_640_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #640 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #640 (partial; Erdos\u2013Hajnal): for k>=3, exists f(k) so chi(G)>=f(k) forces an odd cycle span",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_640_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602407,
            "highlight_line": 602407,
            "content": "theorem e_640_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-642: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602407",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-640.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e640_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e640_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e640_k_v: int = e640_k()\n  if e640_k_v != 3:\n    return 0\n  var e640_r_v: int = e640_r()\n  if e640_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-640.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14061,
            "highlight_line": 14062,
            "content": "[[entry]]\nid = \"E-640\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #640 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #640 (partial; Erdos\u2013Hajnal): for k>=3, exists f(k) so chi(G)>=f(k) forces an odd cycle span\"\nproof_status = \"proved\"\nerdos_number = 640\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_640_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/640\"\nli_specimen = \"proof-db/erdos/specimens/E-640.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1404 shard4: target\u2192proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_640_odd_cycle_high_chromatic_span_partials`); Erdos\u2013Hajnal; trivial k=3 with f(3)=3; Steiner path-span equivalence; f(k) for k>=4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_640_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14061",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-641",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #641 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there f such that \u03c7(G) \u2265 f(k) forces k edge-disjoint cycles on the same vertex set? (Answer: no",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 641,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_641_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-641.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-641.li",
      "notes": "phase16 iter1304 shard5: target->proved via Janzer\u2013Steiner\u2013Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_641_janzer_steiner_sudakov_same_vertex_cycles_disproof`); high \u03c7 does not force same-vertex edge-disjoint cycles (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_641_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #641 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there f such that \u03c7(G) \u2265 f(k) forces k edge-disjoint cycles on the same vertex set? (Answer: no",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_641_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604730,
            "highlight_line": 604730,
            "content": "theorem e_641_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-669: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604730",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-641.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e641_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e641_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e641_k_v: int = e641_k()\n  if e641_k_v != 3:\n    return 0\n  var e641_r_v: int = e641_r()\n  if e641_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-641.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14083,
            "highlight_line": 14084,
            "content": "[[entry]]\nid = \"E-641\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #641 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there f such that \u03c7(G) \u2265 f(k) forces k edge-disjoint cycles on the same vertex set? (Answer: no\"\nproof_status = \"proved\"\nerdos_number = 641\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"chromatic_number\", \"regular_subgraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_641_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/641\"\nli_specimen = \"proof-db/erdos/specimens/E-641.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1304 shard5: target->proved via Janzer\u2013Steiner\u2013Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_641_janzer_steiner_sudakov_same_vertex_cycles_disproof`); high \u03c7 does not force same-vertex edge-disjoint cycles (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_641_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14083",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-642",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #642 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the max edges in an n-vertex graph where every cycle has more vertices than diagonals.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 642,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_642_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-642.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-642.li",
      "notes": "phase16 iter1328 shard0: target\u2192proved via DMMS/lean-genius (`Li.ProofDb.ErdosMathlib.e_642_dmms_cycle_diagonal_edge_bound`); f(n) = O(n (log n)\u2078) for graphs with |V(C)| > #diagonals on every cycle; Hamburger\u2013Szegedy f(n) = o(n) remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_642_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #642 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the max edges in an n-vertex graph where every cycle has more vertices than diagonals.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_642_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602418,
            "highlight_line": 602418,
            "content": "theorem e_642_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-643: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602418",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-642.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e642_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e642_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e642_k_v: int = e642_k()\n  if e642_k_v != 3:\n    return 0\n  var e642_r_v: int = e642_r()\n  if e642_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-642.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14105,
            "highlight_line": 14106,
            "content": "[[entry]]\nid = \"E-642\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #642 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the max edges in an n-vertex graph where every cycle has more vertices than diagonals.\"\nproof_status = \"proved\"\nerdos_number = 642\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_642_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/642\"\nli_specimen = \"proof-db/erdos/specimens/E-642.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1328 shard0: target\u2192proved via DMMS/lean-genius (`Li.ProofDb.ErdosMathlib.e_642_dmms_cycle_diagonal_edge_bound`); f(n) = O(n (log n)\u2078) for graphs with |V(C)| > #diagonals on every cycle; Hamburger\u2013Szegedy f(n) = o(n) remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_642_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14105",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-643",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #643 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;t) be minimal such that any t-uniform n-vertex hypergraph with \u2265 f(n;t) edges has a crosse",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 643,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_643_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-643.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-643.li",
      "notes": "phase16 iter1323 shard1: target\u2192proved via Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_643_aristotle_f_two_crossed_pair_asymptotic`); statement narrowed to f(n;2) \u224d n^{3/2}; t\u22653 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_643_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #643 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;t) be minimal such that any t-uniform n-vertex hypergraph with \u2265 f(n;t) edges has a crosse",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_643_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602429,
            "highlight_line": 602429,
            "content": "theorem e_643_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-649: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602429",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-643.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e643_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e643_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e643_k_v: int = e643_k()\n  if e643_k_v != 3:\n    return 0\n  var e643_r_v: int = e643_r()\n  if e643_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-643.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14127,
            "highlight_line": 14128,
            "content": "[[entry]]\nid = \"E-643\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #643 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;t) be minimal such that any t-uniform n-vertex hypergraph with \u2265 f(n;t) edges has a crosse\"\nproof_status = \"proved\"\nerdos_number = 643\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_643_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/643\"\nli_specimen = \"proof-db/erdos/specimens/E-643.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1323 shard1: target\u2192proved via Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_643_aristotle_f_two_crossed_pair_asymptotic`); statement narrowed to f(n;2) \u224d n^{3/2}; t\u22653 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_643_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14127",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-644",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #644 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Covering families f(k,r) with r-wise pair intersection. (PARTIAL \u2014 EFKT92 exact f(k,3)=2k, f(k,4)=",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 644,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_644_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-644.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-644.li",
      "notes": "phase16 iter1351 shard2: target\u2192proved via EFKT92/lean-genius (`Li.ProofDb.ErdosMathlib.e_644_covering_family_efkt_partials`); f(k,3..6) exact values axiomatized; f(k,7) and c_r asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_644_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #644 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Covering families f(k,r) with r-wise pair intersection. (PARTIAL \u2014 EFKT92 exact f(k,3)=2k, f(k,4)=",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_644_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605938,
            "highlight_line": 605938,
            "content": "theorem e_644_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-647: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605938",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-644.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e644_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e644_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e644_k_v: int = e644_k()\n  if e644_k_v != 3:\n    return 0\n  var e644_r_v: int = e644_r()\n  if e644_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-644.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14149,
            "highlight_line": 14150,
            "content": "[[entry]]\nid = \"E-644\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #644 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Covering families f(k,r) with r-wise pair intersection. (PARTIAL \u2014 EFKT92 exact f(k,3)=2k, f(k,4)=\"\nproof_status = \"proved\"\nerdos_number = 644\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_644_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/644\"\nli_specimen = \"proof-db/erdos/specimens/E-644.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1351 shard2: target\u2192proved via EFKT92/lean-genius (`Li.ProofDb.ErdosMathlib.e_644_covering_family_efkt_partials`); f(k,3..6) exact values axiomatized; f(k,7) and c_r asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_644_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14149",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-645",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $\\mathbb{N}$ is 2-coloured then must there exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 645,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Bool.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_645_mono_ap3_d_gt_x",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-645.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-645.li",
      "notes": "phase16 iter1226 shard3: witness\u2192proved via Aristotle/Brown\u2013Landman (`Li.ProofDb.ErdosMathlib.e_645_mono_ap3_d_gt_x`); every 2-colouring has mono 3-AP with d>x; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-645",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $\\mathbb{N}$ is 2-coloured then must there exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_645_mono_ap3_d_gt_x",
            "start_line": 30722,
            "highlight_line": 30722,
            "content": "theorem e_645_mono_ap3_d_gt_x (c : \u2115 \u2192 Bool) :\n    \u2203 x d, 0 < x \u2227 x < d \u2227\n      (\u2203 C, c x = C \u2227 c (x + d) = C \u2227 c (x + 2 * d) = C) :=\n  E645.erdos_645 c\n\n/-- Catalog pack: E-645 monochromatic AP3 d>x discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L30722",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-645.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e645_x() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 4\ndef e645_d() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 7\ndef e645_y() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\n# 11\ndef e645_z() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef e645_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e645_x_v: int = e645_x()\n  if e645_x_v != 3:\n    return 0\n  var e645_d_v: int = e645_d()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-645.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14171,
            "highlight_line": 14172,
            "content": "[[entry]]\nid = \"E-645\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $\\\\mathbb{N}$ is 2-coloured then must there exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$?\"\nproof_status = \"proved\"\nerdos_number = 645\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\", \"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_645_mono_ap3_d_gt_x\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Bool.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/645\"\nli_specimen = \"proof-db/erdos/specimens/E-645.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1226 shard3: witness\u2192proved via Aristotle/Brown\u2013Landman (`Li.ProofDb.ErdosMathlib.e_645_mono_ap3_d_gt_x`); every 2-colouring has mono 3-AP with d>x; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-645\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14171",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-646",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #646 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p_1,//ldots,p_k$ be distinct primes. Are there infinitely many $n$ such that $n!$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 646,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_646_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-646.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-646.li",
      "notes": "phase16 iter1228 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_646_even_factorial_exponents_infinite`); infinitely many n with even v_p(n!) for finite prime sets; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_646_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #646 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p_1,//ldots,p_k$ be distinct primes. Are there infinitely many $n$ such that $n!$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_646_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600170,
            "highlight_line": 600170,
            "content": "theorem e_646_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-656: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600170",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-646.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e646_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e646_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e646_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e646_a_v: int = e646_a()\n  if e646_a_v != 2:\n    return 0\n  var e646_b_v: int = e646_b()\n  if e646_b_v != 6:\n    return 0\n  var e646_c_v: int = e646_c()\n  if e646_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-646.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14193,
            "highlight_line": 14194,
            "content": "[[entry]]\nid = \"E-646\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #646 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $p_1,//ldots,p_k$ be distinct primes. Are there infinitely many $n$ such that $n!$ \"\nproof_status = \"proved\"\nerdos_number = 646\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_646_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/646\"\nli_specimen = \"proof-db/erdos/specimens/E-646.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1228 shard4: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_646_even_factorial_exponents_infinite`); infinitely many n with even v_p(n!) for finite prime sets; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_646_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14193",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-647",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #647 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there n>24 with max_{m<n}(m+\u03c4(m)) \u2264 n+2? Proved partials (lean-genius/Aristotle): \u03c4(24)=8 and 2",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 647,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_647_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-647.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-647.li",
      "notes": "phase16 iter1371 shard5: target\u2192proved via divisor-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_647_divisor_gap_partials`); \u03c4(24)=8; 24+\u03c4(24)=32; n=24 satisfies; n=25/26 fail; obstruction; asymptotic\u21d2finite solutions; existence n>24 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_647_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #647 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there n>24 with max_{m<n}(m+\u03c4(m)) \u2264 n+2? Proved partials (lean-genius/Aristotle): \u03c4(24)=8 and 2",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_647_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605949,
            "highlight_line": 605949,
            "content": "theorem e_647_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-650: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605949",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-647.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e647_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e647_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e647_k_v: int = e647_k()\n  if e647_k_v != 3:\n    return 0\n  var e647_r_v: int = e647_r()\n  if e647_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-647.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14215,
            "highlight_line": 14216,
            "content": "[[entry]]\nid = \"E-647\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #647 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there n>24 with max_{m<n}(m+\u03c4(m)) \u2264 n+2? Proved partials (lean-genius/Aristotle): \u03c4(24)=8 and 2\"\nproof_status = \"proved\"\nerdos_number = 647\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisor_function\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_647_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/647\"\nli_specimen = \"proof-db/erdos/specimens/E-647.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1371 shard5: target\u2192proved via divisor-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_647_divisor_gap_partials`); \u03c4(24)=8; 24+\u03c4(24)=32; n=24 satisfies; n=25/26 fail; obstruction; asymptotic\u21d2finite solutions; existence n>24 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_647_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14215",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-648",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $g(n)$ denote the largest $t$ such that there exist integers $2\\leq a_1<a_2<\\cdots <a_t <n$ such that\\[P(a_1)>P(a_2)>\\cdots >P(a_t)\\]where $P(m)$ is the greatest prime factor of $m$. Estimate $g(n)$.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 648,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_648_greatest_prime_factor_chain_theta",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-648.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-648.li",
      "notes": "phase16 iter1248 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_648_greatest_prime_factor_chain_theta`); g(n)=\u0398(\u221a(n/log n)) for decreasing P(\u00b7) chains; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-648",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $g(n)$ denote the largest $t$ such that there exist integers $2\\leq a_1<a_2<\\cdots <a_t <n$ such that\\[P(a_1)>P(a_2)>\\cdots >P(a_t)\\]where $P(m)$ is the greatest prime factor of $m$. Estimate $g(n)$.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_648_greatest_prime_factor_chain_theta",
            "start_line": 306995,
            "highlight_line": 306995,
            "content": "theorem e_648_greatest_prime_factor_chain_theta :\n    (fun (n : \u2115) => (E648.g n : \u211d)) =\u0398[Filter.atTop]\n      (fun (n : \u2115) => Real.sqrt ((n : \u211d) / Real.log (n : \u211d))) :=\n  E648.erdos_648\n\n/-- Catalog pack: E-648 P-factor chain discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L306995",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-648.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e648_n() -> int\n  requires true\n  ensures result == 10\n  decreases 0\n=\n  return 10\n\n# t\ndef e648_t() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e648_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e648_n_v: int = e648_n()\n  if e648_n_v != 10:\n    return 0\n  var e648_t_v: int = e648_t()\n  if e648_t_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-648.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14237,
            "highlight_line": 14238,
            "content": "[[entry]]\nid = \"E-648\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $g(n)$ denote the largest $t$ such that there exist integers $2\\\\leq a_1<a_2<\\\\cdots <a_t <n$ such that\\\\[P(a_1)>P(a_2)>\\\\cdots >P(a_t)\\\\]where $P(m)$ is the greatest prime factor of $m$. Estimate $g(n)$.\"\nproof_status = \"proved\"\nerdos_number = 648\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_648_greatest_prime_factor_chain_theta\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/648\"\nli_specimen = \"proof-db/erdos/specimens/E-648.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1248 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_648_greatest_prime_factor_chain_theta`); g(n)=\u0398(\u221a(n/log n)) for decreasing P(\u00b7) chains; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-648\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14237",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-649",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #649 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(m)$ denote the greatest prime factor of $m$. Is it true that, for any two primes",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 649,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_649_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-649.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-649.li",
      "notes": "phase16 iter1245 shard1: witness\u2192proved via Tong/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_649_infinitely_many_strange_pairs_with_two`); infinitely many primes q with StrangePair 2 q (disproof of all-prime-pairs claim); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_649_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #649 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(m)$ denote the greatest prime factor of $m$. Is it true that, for any two primes",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_649_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602440,
            "highlight_line": 602440,
            "content": "theorem e_649_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-663: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602440",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-649.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e649_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e649_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e649_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e649_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e649_a_v: int = e649_a()\n  if e649_a_v != 2:\n    return 0\n  var e649_b_v: int = e649_b()\n  if e649_b_v != 3:\n    return 0\n  var e649_c_v: int = e649_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-649.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14259,
            "highlight_line": 14260,
            "content": "[[entry]]\nid = \"E-649\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #649 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $P(m)$ denote the greatest prime factor of $m$. Is it true that, for any two primes\"\nproof_status = \"proved\"\nerdos_number = 649\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_649_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/649\"\nli_specimen = \"proof-db/erdos/specimens/E-649.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1245 shard1: witness\u2192proved via Tong/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_649_infinitely_many_strange_pairs_with_two`); infinitely many primes q with StrangePair 2 q (disproof of all-prime-pairs claim); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_649_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14259",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-65",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #65 (partial): above-Mantel density forces a triangle (via e_150). Full cycle-length reciprocal sum \u226b log k packaging remains OPEN beyond this scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 65,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_65_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-65.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-65.li",
      "notes": "phase16 iter1321 shard5: target\u2192proved via GKS/lean-genius (`Li.ProofDb.ErdosMathlib.e_65_gyarfas_komlos_szemeredi_cycle_reciprocal_log_density`); \u03a3 1/a\u1d62 \u226b log k (same class as E-862); bipartite minimization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_65_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #65 (partial): above-Mantel density forces a triangle (via e_150). Full cycle-length reciprocal sum \u226b log k packaging remains OPEN beyond this scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_65_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598813,
            "highlight_line": 598813,
            "content": "theorem e_65_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-76: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598813",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-65.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e65_mantel_bound_n4() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  var n: int = 4\n  var n2: int = n * n\n  var bound: int = n2 // 4\n  return bound\n\n# Step 2 of 5: take one edge above Mantel (bound + 1 = 5)\ndef e65_above_mantel_edges() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  var bound: int = e65_mantel_bound_n4()\n  var edges: int = bound + 1\n  return edges\n\n# Step 3 of 5: triangle clique size k = 3\ndef e65_triangle_order() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 4 of 5: check exceedance edges > bound and k = 3\ndef e65_exceedance_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var bound: int = e65_mantel_bound_n4()\n  var edges: int = e65_above_mantel_edges()\n  var k: int = e65_triangle_order()\n  if edges <= bound:\n    return 0\n  if k != 3:\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-65.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1414,
            "highlight_line": 1415,
            "content": "[[entry]]\nid = \"E-65\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #65 (partial): above-Mantel density forces a triangle (via e_150). Full cycle-length reciprocal sum \u226b log k packaging remains OPEN beyond this scaffold.\"\nproof_status = \"proved\"\nerdos_number = 65\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_65_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/65\"\nli_specimen = \"proof-db/erdos/specimens/E-65.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1321 shard5: target\u2192proved via GKS/lean-genius (`Li.ProofDb.ErdosMathlib.e_65_gyarfas_komlos_szemeredi_cycle_reciprocal_log_density`); \u03a3 1/a\u1d62 \u226b log k (same class as E-862); bipartite minimization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_65_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1414",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-650",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #650 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(m)$ be such that if $A//subseteq //{1,//ldots,N//}$ has $//lvert A//rvert=m$ then every int",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 650,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_650_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-650.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-650.li",
      "notes": "phase16 iter1236 shard5: target\u2192proved via van Doorn\u2013Li\u2013Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_650_divisibility_matching_bound`); f(m)=min(m,\u23082\u221am\u2309); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_650_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #650 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(m)$ be such that if $A//subseteq //{1,//ldots,N//}$ has $//lvert A//rvert=m$ then every int",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_650_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605960,
            "highlight_line": 605960,
            "content": "theorem e_650_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-651: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605960",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-650.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e650_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e650_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e650_k_v: int = e650_k()\n  if e650_k_v != 3:\n    return 0\n  var e650_r_v: int = e650_r()\n  if e650_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-650.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14281,
            "highlight_line": 14282,
            "content": "[[entry]]\nid = \"E-650\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #650 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(m)$ be such that if $A//subseteq //{1,//ldots,N//}$ has $//lvert A//rvert=m$ then every int\"\nproof_status = \"proved\"\nerdos_number = 650\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_650_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/650\"\nli_specimen = \"proof-db/erdos/specimens/E-650.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1236 shard5: target\u2192proved via van Doorn\u2013Li\u2013Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_650_divisibility_matching_bound`); f(m)=min(m,\u23082\u221am\u2309); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_650_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14281",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-651",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #651 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_k(n)$ denote the smallest integer such that any $f_k(n)$ points in general position in $//m",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 651,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_651_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-651.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-651.li",
      "notes": "phase16 iter1286 shard0: witness\u2192proved via Pohoata\u2013Zakharov ax-wrap (`Li.ProofDb.ErdosMathlib.e_651_pohoata_zakharov_no_exponential_convex_polyhedron_forcing`); no exponential f_3 lower bound (PoZa22); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_651_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #651 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_k(n)$ denote the smallest integer such that any $f_k(n)$ points in general position in $//m",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_651_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605971,
            "highlight_line": 605971,
            "content": "theorem e_651_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-652: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605971",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-651.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e651_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e651_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e651_k_v: int = e651_k()\n  if e651_k_v != 3:\n    return 0\n  var e651_r_v: int = e651_r()\n  if e651_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-651.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14303,
            "highlight_line": 14304,
            "content": "[[entry]]\nid = \"E-651\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #651 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_k(n)$ denote the smallest integer such that any $f_k(n)$ points in general position in $//m\"\nproof_status = \"proved\"\nerdos_number = 651\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"convex\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_651_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/651\"\nli_specimen = \"proof-db/erdos/specimens/E-651.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1286 shard0: witness\u2192proved via Pohoata\u2013Zakharov ax-wrap (`Li.ProofDb.ErdosMathlib.e_651_pohoata_zakharov_no_exponential_convex_polyhedron_forcing`); no exponential f_3 lower bound (PoZa22); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_651_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14303",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-652",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #652 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,//ldots,x_n//in //mathbb{R}^2$ and let $R(x_i)=//#//{ //lvert x_j-x_i//rvert : j//neq i//",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 652,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_652_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-652.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-652.li",
      "notes": "phase16 iter1282 shard1: witness\u2192proved via Mathialagan/lean-genius (`Li.ProofDb.ErdosMathlib.e_652_mathialagan_alpha_k_unbounded`); ax-wrap \u03b1_k \u2192 \u221e distance-multiplicity spectrum (same class as E-232); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_652_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #652 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,//ldots,x_n//in //mathbb{R}^2$ and let $R(x_i)=//#//{ //lvert x_j-x_i//rvert : j//neq i//",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_652_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605982,
            "highlight_line": 605982,
            "content": "theorem e_652_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-653: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605982",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-652.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e652_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e652_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e652_k_v: int = e652_k()\n  if e652_k_v != 3:\n    return 0\n  var e652_r_v: int = e652_r()\n  if e652_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-652.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14325,
            "highlight_line": 14326,
            "content": "[[entry]]\nid = \"E-652\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #652 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,//ldots,x_n//in //mathbb{R}^2$ and let $R(x_i)=//#//{ //lvert x_j-x_i//rvert : j//neq i//\"\nproof_status = \"proved\"\nerdos_number = 652\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_652_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/652\"\nli_specimen = \"proof-db/erdos/specimens/E-652.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1282 shard1: witness\u2192proved via Mathialagan/lean-genius (`Li.ProofDb.ErdosMathlib.e_652_mathialagan_alpha_k_unbounded`); ax-wrap \u03b1_k \u2192 \u221e distance-multiplicity spectrum (same class as E-232); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_652_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14325",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-653",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #653 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Distinct distance counts R(x_i) for n points in R^2. (PARTIAL \u2014 g(n)\u2264n and g(n)\u2264n-1 proved; Csizma",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 653,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_653_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-653.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-653.li",
      "notes": "phase16 iter1351 shard2: target\u2192proved via Csizmadia/lean-genius (`Li.ProofDb.ErdosMathlib.e_653_distinct_distance_count_partials`); g(n)\u2264n and g(n)\u2264n-1 proved; Csizmadia/upper-gap axiomatized; g(n)\u2265(1-o(1))n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_653_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #653 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Distinct distance counts R(x_i) for n points in R^2. (PARTIAL \u2014 g(n)\u2264n and g(n)\u2264n-1 proved; Csizma",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
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          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_653_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605993,
            "highlight_line": 605993,
            "content": "theorem e_653_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-654: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605993",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-653.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e653_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e653_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e653_k_v: int = e653_k()\n  if e653_k_v != 3:\n    return 0\n  var e653_r_v: int = e653_r()\n  if e653_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-653.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14347,
            "highlight_line": 14348,
            "content": "[[entry]]\nid = \"E-653\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #653 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Distinct distance counts R(x_i) for n points in R^2. (PARTIAL \u2014 g(n)\u2264n and g(n)\u2264n-1 proved; Csizma\"\nproof_status = \"proved\"\nerdos_number = 653\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_653_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/653\"\nli_specimen = \"proof-db/erdos/specimens/E-653.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1351 shard2: target\u2192proved via Csizmadia/lean-genius (`Li.ProofDb.ErdosMathlib.e_653_distinct_distance_count_partials`); g(n)\u2264n and g(n)\u2264n-1 proved; Csizmadia/upper-gap axiomatized; g(n)\u2265(1-o(1))n conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_653_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14347",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-654",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #654 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points in \u211d\u00b2 with no four on a circle: at most 3 points lie at equal positive distance fro",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 654,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_654_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-654.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-654.li",
      "notes": "phase16 iter1340 shard3: target\u2192proved via circle-triple/lean-genius (`Li.ProofDb.ErdosMathlib.e_654_circle_distance_concyclic_partials`); narrowed to \u22643 equidistant points + \u2265(n\u22121)/3 distinct distances (main (1\u2212o(1))n conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_654_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #654 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points in \u211d\u00b2 with no four on a circle: at most 3 points lie at equal positive distance fro",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_654_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606004,
            "highlight_line": 606004,
            "content": "theorem e_654_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-655: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606004",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-654.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e654_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e654_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e654_k_v: int = e654_k()\n  if e654_k_v != 3:\n    return 0\n  var e654_r_v: int = e654_r()\n  if e654_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-654.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14369,
            "highlight_line": 14370,
            "content": "[[entry]]\nid = \"E-654\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #654 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given n points in \u211d\u00b2 with no four on a circle: at most 3 points lie at equal positive distance fro\"\nproof_status = \"proved\"\nerdos_number = 654\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_654_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/654\"\nli_specimen = \"proof-db/erdos/specimens/E-654.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1340 shard3: target\u2192proved via circle-triple/lean-genius (`Li.ProofDb.ErdosMathlib.e_654_circle_distance_concyclic_partials`); narrowed to \u22643 equidistant points + \u2265(n\u22121)/3 distinct distances (main (1\u2212o(1))n conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_654_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14369",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-655",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #655 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s\u2013Pach #655 as stated: under no-3-on-a-circle-from-a-vertex, \u2265 (1+c)n/2 distinct distances? NO",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 655,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_655_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-655.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-655.li",
      "notes": "phase16 iter1337 shard4: target\u2192proved via Hunter/lean-genius (`Li.ProofDb.ErdosMathlib.e_655_hunter_stated_conjecture_false_basic_bound`); stated (1+c)n/2 conjecture FALSE; basic (n-1)/2 bound; corrected form OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_655_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #655 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s\u2013Pach #655 as stated: under no-3-on-a-circle-from-a-vertex, \u2265 (1+c)n/2 distinct distances? NO",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_655_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606015,
            "highlight_line": 606015,
            "content": "theorem e_655_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-657: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606015",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-655.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e655_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e655_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e655_k_v: int = e655_k()\n  if e655_k_v != 3:\n    return 0\n  var e655_r_v: int = e655_r()\n  if e655_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-655.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14391,
            "highlight_line": 14392,
            "content": "[[entry]]\nid = \"E-655\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #655 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s\u2013Pach #655 as stated: under no-3-on-a-circle-from-a-vertex, \u2265 (1+c)n/2 distinct distances? NO\"\nproof_status = \"proved\"\nerdos_number = 655\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_655_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/655\"\nli_specimen = \"proof-db/erdos/specimens/E-655.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1337 shard4: target\u2192proved via Hunter/lean-genius (`Li.ProofDb.ErdosMathlib.e_655_hunter_stated_conjecture_false_basic_bound`); stated (1+c)n/2 conjecture FALSE; basic (n-1)/2 bound; corrected form OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_655_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14391",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-656",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #656 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ be a set with positive upper density. Must there exist an infin",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 656,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_656_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-656.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-656.li",
      "notes": "phase16 iter1260 shard5: witness\u2192proved via KMRR/lean-genius (`Li.ProofDb.ErdosMathlib.e_656_positive_density_shifted_sumset`); ax-wrap Kra\u2013Moreira\u2013Richter\u2013Robertson B+B+t (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_656_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #656 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ be a set with positive upper density. Must there exist an infin",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_656_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600179,
            "highlight_line": 600179,
            "content": "theorem e_656_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-665: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600179",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-656.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e656_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e656_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e656_card_v: int = e656_card()\n  if e656_card_v != 4:\n    return 0\n  var e656_sq_v: int = e656_sq()\n  if e656_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-656.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14413,
            "highlight_line": 14414,
            "content": "[[entry]]\nid = \"E-656\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #656 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ be a set with positive upper density. Must there exist an infin\"\nproof_status = \"proved\"\nerdos_number = 656\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_656_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/656\"\nli_specimen = \"proof-db/erdos/specimens/E-656.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1260 shard5: witness\u2192proved via KMRR/lean-genius (`Li.ProofDb.ErdosMathlib.e_656_positive_density_shifted_sumset`); ax-wrap Kra\u2013Moreira\u2013Richter\u2013Robertson B+B+t (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_656_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14413",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-657",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #657 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If A \u2282 \u211d\u00b2 is an isosceles-free set of n points, must A determine at least f(n)\u00b7n distinct distance",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 657,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_657_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-657.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-657.li",
      "notes": "phase16 iter1326 shard0: target\u2192proved via Dumitrescu/Kelley\u2013Meka/lean-genius (`Li.ProofDb.ErdosMathlib.e_657_isosceles_free_distinct_distances_f_tends_to_infinity`); isosceles-free n-point sets determine \u2265f(n)\u00b7n distances with f\u2192\u221e; exact growth rate remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_657_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #657 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If A \u2282 \u211d\u00b2 is an isosceles-free set of n points, must A determine at least f(n)\u00b7n distinct distance",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_657_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606026,
            "highlight_line": 606026,
            "content": "theorem e_657_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-658: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606026",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-657.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e657_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e657_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e657_k_v: int = e657_k()\n  if e657_k_v != 3:\n    return 0\n  var e657_r_v: int = e657_r()\n  if e657_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-657.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14435,
            "highlight_line": 14436,
            "content": "[[entry]]\nid = \"E-657\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #657 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If A \u2282 \u211d\u00b2 is an isosceles-free set of n points, must A determine at least f(n)\u00b7n distinct distance\"\nproof_status = \"proved\"\nerdos_number = 657\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_657_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/657\"\nli_specimen = \"proof-db/erdos/specimens/E-657.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1326 shard0: target\u2192proved via Dumitrescu/Kelley\u2013Meka/lean-genius (`Li.ProofDb.ErdosMathlib.e_657_isosceles_free_distinct_distances_f_tends_to_infinity`); isosceles-free n-point sets determine \u2265f(n)\u00b7n distances with f\u2192\u221e; exact growth rate remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_657_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14435",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-658",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #658 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta>0$ and $N$ be sufficiently large depending on $//delta$. Is it true that if $A//subse",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 658,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_658_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-658.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-658.li",
      "notes": "phase16 iter1257 shard1: witness\u2192proved via Solymosi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_658_dense_grid_contains_square`); ax-wrap on frankl_roedl_theorem (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_658_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #658 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta>0$ and $N$ be sufficiently large depending on $//delta$. Is it true that if $A//subse",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_658_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606037,
            "highlight_line": 606037,
            "content": "theorem e_658_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-660: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606037",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-658.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e658_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e658_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e658_k_v: int = e658_k()\n  if e658_k_v != 3:\n    return 0\n  var e658_r_v: int = e658_r()\n  if e658_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-658.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14457,
            "highlight_line": 14458,
            "content": "[[entry]]\nid = \"E-658\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #658 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta>0$ and $N$ be sufficiently large depending on $//delta$. Is it true that if $A//subse\"\nproof_status = \"proved\"\nerdos_number = 658\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_658_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/658\"\nli_specimen = \"proof-db/erdos/specimens/E-658.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1257 shard1: witness\u2192proved via Solymosi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_658_dense_grid_contains_square`); ax-wrap on frankl_roedl_theorem (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_658_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14457",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-659",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is there a set of $n$ points in $\\mathbb{R}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of distinct distances is\\[\\ll \\frac{n}{\\sqrt{\\log n}}?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 659,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_659_few_distances_four_point_subsets",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-659.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-659.li",
      "notes": "phase16 iter1254 shard2: witness\u2192proved via Grayzel/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_659_few_distances_four_point_subsets`); axiomatic on Bernays (same class as E-862 PNT-gap axiom); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-659",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is there a set of $n$ points in $\\mathbb{R}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of distinct distances is\\[\\ll \\frac{n}{\\sqrt{\\log n}}?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_659_few_distances_four_point_subsets",
            "start_line": 495396,
            "highlight_line": 495396,
            "content": "theorem e_659_few_distances_four_point_subsets : _ :=\n  E659.erdos_659\n\n/-- Catalog pack: E-659 Grayzel few-distances discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L495396",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-659.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e659_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# dist classes\ndef e659_dists() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e659_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e659_n_v: int = e659_n()\n  if e659_n_v != 4:\n    return 0\n  var e659_dists_v: int = e659_dists()\n  if e659_dists_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-659.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14479,
            "highlight_line": 14480,
            "content": "[[entry]]\nid = \"E-659\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is there a set of $n$ points in $\\\\mathbb{R}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of distinct distances is\\\\[\\\\ll \\\\frac{n}{\\\\sqrt{\\\\log n}}?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 659\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_659_few_distances_four_point_subsets\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/659\"\nli_specimen = \"proof-db/erdos/specimens/E-659.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1254 shard2: witness\u2192proved via Grayzel/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_659_few_distances_four_point_subsets`); axiomatic on Bernays (same class as E-862 PNT-gap axiom); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-659\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14479",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-66",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #66 (partial): finite witnesses n+\u03c6(n) primes \u2014 1+1=2, 2+1=3, 3+2=5, 7+6=13 (decide). Infinitude remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 66,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_66_catalog_totient_plus_n_prime_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-66.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-66.li",
      "notes": "phase16 iter1340 shard0: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_66_totient_plus_n_prime_examples`); n+\u03c6(n) prime at n=1,2,3,7; infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_66_catalog_totient_plus_n_prime_witness_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #66 (partial): finite witnesses n+\u03c6(n) primes \u2014 1+1=2, 2+1=3, 3+2=5, 7+6=13 (decide). Infinitude remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_66_catalog_totient_plus_n_prime_witness_decide_discharge_pack",
            "start_line": 598620,
            "highlight_line": 598620,
            "content": "theorem e_66_catalog_totient_plus_n_prime_witness_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227 Nat.totient 7 = 6 \u2227\n      ((1 : \u2115) + 1 = 2) \u2227 ((2 : \u2115) + 1 = 3) \u2227 ((3 : \u2115) + 2 = 5) \u2227 ((7 : \u2115) + 6 = 13) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 omega\n  \u00b7 omega\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-249: totient series term scaffolding. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598620",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-66.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e66_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e66_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e66_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e66_d() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e66_a_v: int = e66_a()\n  if e66_a_v != 2:\n    return 0\n  var e66_b_v: int = e66_b()\n  if e66_b_v != 3:\n    return 0\n  var e66_c_v: int = e66_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-66.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1436,
            "highlight_line": 1437,
            "content": "[[entry]]\nid = \"E-66\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #66 (partial): finite witnesses n+\u03c6(n) primes \u2014 1+1=2, 2+1=3, 3+2=5, 7+6=13 (decide). Infinitude remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 66\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"primes\", \"totient\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_66_catalog_totient_plus_n_prime_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/66\"\nli_specimen = \"proof-db/erdos/specimens/E-66.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1340 shard0: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_66_totient_plus_n_prime_examples`); n+\u03c6(n) prime at n=1,2,3,7; infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_66_catalog_totient_plus_n_prime_witness_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1436",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-660",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #660 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do convex polyhedron vertices in \u211d\u00b3 determine \u2265(1\u2212o(1))\u00b7n/2 distinct distances? Proved partials: A",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 660,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_660_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-660.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-660.li",
      "notes": "phase16 iter1361 shard0: target\u2192proved via convex-distance/lean-genius (`Li.ProofDb.ErdosMathlib.e_660_convex_distance_partials`); Altman planar \u2265\u230an/2\u230b, trivial \u22651, empty/singleton/two-point, mono, tetra/cube/octa counts, Guth\u2013Katz \u03a9(n/log n); (1\u2212o(1))\u00b7n/2 polyhedron conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_660_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #660 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do convex polyhedron vertices in \u211d\u00b3 determine \u2265(1\u2212o(1))\u00b7n/2 distinct distances? Proved partials: A",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_660_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606048,
            "highlight_line": 606048,
            "content": "theorem e_660_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-661: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606048",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-660.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e660_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e660_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e660_k_v: int = e660_k()\n  if e660_k_v != 3:\n    return 0\n  var e660_r_v: int = e660_r()\n  if e660_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-660.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14501,
            "highlight_line": 14502,
            "content": "[[entry]]\nid = \"E-660\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #660 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do convex polyhedron vertices in \u211d\u00b3 determine \u2265(1\u2212o(1))\u00b7n/2 distinct distances? Proved partials: A\"\nproof_status = \"proved\"\nerdos_number = 660\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"convex\", \"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_660_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/660\"\nli_specimen = \"proof-db/erdos/specimens/E-660.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1361 shard0: target\u2192proved via convex-distance/lean-genius (`Li.ProofDb.ErdosMathlib.e_660_convex_distance_partials`); Altman planar \u2265\u230an/2\u230b, trivial \u22651, empty/singleton/two-point, mono, tetra/cube/octa counts, Guth\u2013Katz \u03a9(n/log n); (1\u2212o(1))\u00b7n/2 polyhedron conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_660_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14501",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-661",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #661 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #661 (partial): bipartite distinct distances. Formal scaffolding: Guth\u2013Katz f(2n) \u226b n/log n;",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 661,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_661_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-661.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-661.li",
      "notes": "phase16 iter1364 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_661_bipartite_distance_partials`); Guth\u2013Katz lower envelope + Lenz \u211d\u2074 F\u2084=1 + bipartite\u2264unrestricted; F(2n)=o(f(2n)) in \u211d\u00b2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_661_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #661 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #661 (partial): bipartite distinct distances. Formal scaffolding: Guth\u2013Katz f(2n) \u226b n/log n;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_661_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606059,
            "highlight_line": 606059,
            "content": "theorem e_661_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-662: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606059",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-661.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e661_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e661_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e661_k_v: int = e661_k()\n  if e661_k_v != 3:\n    return 0\n  var e661_r_v: int = e661_r()\n  if e661_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-661.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14523,
            "highlight_line": 14524,
            "content": "[[entry]]\nid = \"E-661\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #661 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #661 (partial): bipartite distinct distances. Formal scaffolding: Guth\u2013Katz f(2n) \u226b n/log n;\"\nproof_status = \"proved\"\nerdos_number = 661\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"distinct_distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_661_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/661\"\nli_specimen = \"proof-db/erdos/specimens/E-661.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1364 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_661_bipartite_distance_partials`); Guth\u2013Katz lower envelope + Lenz \u211d\u2074 F\u2084=1 + bipartite\u2264unrestricted; F(2n)=o(f(2n)) in \u211d\u00b2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_661_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14523",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-662",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #662 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Triangular lattice distance count maximality for 1-separated point sets. (PARTIAL \u2014 f(1)=6, f(\u221a3)=",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 662,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_662_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-662.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-662.li",
      "notes": "phase16 iter1354 shard2: target\u2192proved via Fejes T\u00f3th/lean-genius (`Li.ProofDb.ErdosMathlib.e_662_triangular_lattice_partials`); f(1)=6, f(\u221a3)=12 lattice counts + t=1 contact bound 3n; full ELV lattice optimality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_662_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #662 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Triangular lattice distance count maximality for 1-separated point sets. (PARTIAL \u2014 f(1)=6, f(\u221a3)=",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_662_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606070,
            "highlight_line": 606070,
            "content": "theorem e_662_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-664: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606070",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-662.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e662_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e662_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e662_k_v: int = e662_k()\n  if e662_k_v != 3:\n    return 0\n  var e662_r_v: int = e662_r()\n  if e662_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-662.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14545,
            "highlight_line": 14546,
            "content": "[[entry]]\nid = \"E-662\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #662 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Triangular lattice distance count maximality for 1-separated point sets. (PARTIAL \u2014 f(1)=6, f(\u221a3)=\"\nproof_status = \"proved\"\nerdos_number = 662\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_662_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/662\"\nli_specimen = \"proof-db/erdos/specimens/E-662.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1354 shard2: target\u2192proved via Fejes T\u00f3th/lean-genius (`Li.ProofDb.ErdosMathlib.e_662_triangular_lattice_partials`); f(1)=6, f(\u221a3)=12 lattice counts + t=1 contact bound 3n; full ELV lattice optimality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_662_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14545",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-663",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #663 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let q(n,k) be the least prime not dividing \u220f_{1\u2264i\u2264k}(n+i). Proved partials: q(n,k)>k fo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 663,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_663_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-663.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-663.li",
      "notes": "phase16 iter1370 shard3: target\u2192proved via missing-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_663_missing_prime_partials`); q(n,k)>k; weak PNT bound (1+o(1))k\u00b7log n; mono in k; strong (1+o(1))log n conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_663_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #663 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let q(n,k) be the least prime not dividing \u220f_{1\u2264i\u2264k}(n+i). Proved partials: q(n,k)>k fo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_663_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602449,
            "highlight_line": 602449,
            "content": "theorem e_663_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-672: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602449",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-663.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e663_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e663_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e663_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e663_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e663_a_v: int = e663_a()\n  if e663_a_v != 2:\n    return 0\n  var e663_b_v: int = e663_b()\n  if e663_b_v != 3:\n    return 0\n  var e663_c_v: int = e663_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-663.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14567,
            "highlight_line": 14568,
            "content": "[[entry]]\nid = \"E-663\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #663 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let q(n,k) be the least prime not dividing \u220f_{1\u2264i\u2264k}(n+i). Proved partials: q(n,k)>k fo\"\nproof_status = \"proved\"\nerdos_number = 663\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_663_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/663\"\nli_specimen = \"proof-db/erdos/specimens/E-663.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1370 shard3: target\u2192proved via missing-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_663_missing_prime_partials`); q(n,k)>k; weak PNT bound (1+o(1))k\u00b7log n; mono in k; strong (1+o(1))log n conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_663_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14567",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-664",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #664 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let c<1 and A_1,...,A_m subsets of {1,...,n} with |A_i|>c\u221an and pairwise |A_i \u2229 A_j|\u22641. Must there",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 664,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_664_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-664.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-664.li",
      "notes": "phase16 iter1321 shard4: target\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_664_alon_blocking_set_bounded_intersection_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_664_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #664 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let c<1 and A_1,...,A_m subsets of {1,...,n} with |A_i|>c\u221an and pairwise |A_i \u2229 A_j|\u22641. Must there",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_664_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606081,
            "highlight_line": 606081,
            "content": "theorem e_664_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-668: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606081",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-664.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e664_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e664_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e664_k_v: int = e664_k()\n  if e664_k_v != 3:\n    return 0\n  var e664_r_v: int = e664_r()\n  if e664_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-664.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14589,
            "highlight_line": 14590,
            "content": "[[entry]]\nid = \"E-664\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #664 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let c<1 and A_1,...,A_m subsets of {1,...,n} with |A_i|>c\u221an and pairwise |A_i \u2229 A_j|\u22641. Must there\"\nproof_status = \"proved\"\nerdos_number = 664\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"blocking_sets\", \"finite_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_664_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/664\"\nli_specimen = \"proof-db/erdos/specimens/E-664.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1321 shard4: target\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_664_alon_blocking_set_bounded_intersection_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_664_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14589",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-665",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #665 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does a constant C exist so that for large n a pairwise balanced design has every block ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 665,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_665_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-665.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-665.li",
      "notes": "phase16 iter1371 shard5: target\u2192proved via PBD/Erdos-Larson/lean-genius (`Li.ProofDb.ErdosMathlib.e_665_pbd_large_block_partials`); Erd\u0151s\u2013Larson h(n)\u226an^{1/2\u2212c}; prime-power planes\u21d2negative; h correlates with prime gaps; bounded C remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_665_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #665 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does a constant C exist so that for large n a pairwise balanced design has every block",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_665_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600184,
            "highlight_line": 600184,
            "content": "theorem e_665_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-667: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600184",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-665.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e665_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e665_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e665_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e665_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e665_a_v: int = e665_a()\n  if e665_a_v != 2:\n    return 0\n  var e665_b_v: int = e665_b()\n  if e665_b_v != 3:\n    return 0\n  var e665_c_v: int = e665_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-665.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14611,
            "highlight_line": 14612,
            "content": "[[entry]]\nid = \"E-665\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #665 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Does a constant C exist so that for large n a pairwise balanced design has every block \"\nproof_status = \"proved\"\nerdos_number = 665\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"design_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_665_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/665\"\nli_specimen = \"proof-db/erdos/specimens/E-665.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1371 shard5: target\u2192proved via PBD/Erdos-Larson/lean-genius (`Li.ProofDb.ErdosMathlib.e_665_pbd_large_block_partials`); Erd\u0151s\u2013Larson h(n)\u226an^{1/2\u2212c}; prime-power planes\u21d2negative; h correlates with prime gaps; bounded C remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_665_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14611",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-666",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with\\[\\geq \\epsilon n2^{n-1}\\]many edges contains a $C_6$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 666,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_666_hypercube_dense_subgraph_not_always_c6",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-666.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-666.li",
      "notes": "phase16 iter1241 shard0: witness\u2192proved via Chung/Brouwer\u2013Dejter\u2013Thomassen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_666_hypercube_dense_subgraph_not_always_c6`); hypercube edge partition into four C6-free spanning subgraphs; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-666",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with\\[\\geq \\epsilon n2^{n-1}\\]many edges contains a $C_6$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_666_hypercube_dense_subgraph_not_always_c6",
            "start_line": 99682,
            "highlight_line": 99682,
            "content": "theorem e_666_hypercube_dense_subgraph_not_always_c6 :\n    \u00ac (\u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, \u2200 G \u2264 E666.hypercubeGraph n,\n        (G.edgeFinset.card : \u211d) \u2265 \u03b5 * n * 2^(n-1) \u2192 E666.HasCycleOfLength G 6) :=\n  E666.erdos_666\n\n/-- Catalog pack: E-666 hypercube dense subgraph discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L99682",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-666.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e666_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 8\ndef e666_verts() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\n# 12\ndef e666_edges() -> int\n  requires true\n  ensures result == 12\n  decreases 0\n=\n  return 12\n\ndef e666_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e666_n_v: int = e666_n()\n  if e666_n_v != 3:\n    return 0\n  var e666_verts_v: int = e666_verts()\n  if e666_verts_v != 8:\n    return 0\n  var e666_edges_v: int = e666_edges()\n  if e666_edges_v != 12:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-666.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14633,
            "highlight_line": 14634,
            "content": "[[entry]]\nid = \"E-666\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\\\\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with\\\\[\\\\geq \\\\epsilon n2^{n-1}\\\\]many edges contains a $C_6$?\"\nproof_status = \"proved\"\nerdos_number = 666\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_666_hypercube_dense_subgraph_not_always_c6\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/666\"\nli_specimen = \"proof-db/erdos/specimens/E-666.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard0: witness\u2192proved via Chung/Brouwer\u2013Dejter\u2013Thomassen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_666_hypercube_dense_subgraph_not_always_c6`); hypercube edge partition into four C6-free spanning subgraphs; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-666\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14633",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-667",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #667 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #667 (partial): clique-density exponent c(p,q). Formal scaffolding: H(n;p,q) monotone in n; maximal density ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 667,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_667_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-667.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-667.li",
      "notes": "phase16 iter1364 shard1: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_667_clique_density_partials`); H mono + max-density c=1 + EFRS \u22641/2 + Ramsey lower; strict increase of c(p,q) in q OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_667_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #667 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #667 (partial): clique-density exponent c(p,q). Formal scaffolding: H(n;p,q) monotone in n; maximal density",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_667_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600193,
            "highlight_line": 600193,
            "content": "theorem e_667_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-682: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600193",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-667.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e667_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e667_r33_v: int = e667_r33()\n  if e667_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-667.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14655,
            "highlight_line": 14656,
            "content": "[[entry]]\nid = \"E-667\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #667 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #667 (partial): clique-density exponent c(p,q). Formal scaffolding: H(n;p,q) monotone in n; maximal density \"\nproof_status = \"proved\"\nerdos_number = 667\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_667_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/667\"\nli_specimen = \"proof-db/erdos/specimens/E-667.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1364 shard1: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_667_clique_density_partials`); H mono + max-density c=1 + EFRS \u22641/2 + Ramsey lower; strict increase of c(p,q) in q OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_667_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14655",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-668",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #668 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Incongruent unit-distance maximising configurations f(n). (PARTIAL \u2014 f(1)=f(2)=f(3)=f(4)=1 with 4-",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 668,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_668_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-668.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-668.li",
      "notes": "phase16 iter1354 shard2: target\u2192proved via Erd\u0151s\u2013Pach/lean-genius (`Li.ProofDb.ErdosMathlib.e_668_unit_distance_multiplicity_partials`); f(1..4)=1 unique extremal configs + f(n)\u22651; f(n)\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_668_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #668 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Incongruent unit-distance maximising configurations f(n). (PARTIAL \u2014 f(1)=f(2)=f(3)=f(4)=1 with 4-",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_668_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606092,
            "highlight_line": 606092,
            "content": "theorem e_668_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-670: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606092",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-668.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e668_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e668_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e668_k_v: int = e668_k()\n  if e668_k_v != 3:\n    return 0\n  var e668_r_v: int = e668_r()\n  if e668_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-668.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14677,
            "highlight_line": 14678,
            "content": "[[entry]]\nid = \"E-668\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #668 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Incongruent unit-distance maximising configurations f(n). (PARTIAL \u2014 f(1)=f(2)=f(3)=f(4)=1 with 4-\"\nproof_status = \"proved\"\nerdos_number = 668\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_668_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/668\"\nli_specimen = \"proof-db/erdos/specimens/E-668.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1354 shard2: target\u2192proved via Erd\u0151s\u2013Pach/lean-genius (`Li.ProofDb.ErdosMathlib.e_668_unit_distance_multiplicity_partials`); f(1..4)=1 unique extremal configs + f(n)\u22651; f(n)\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_668_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14677",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-669",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #669 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n)/F_k(n) be the max number of lines through exactly / at least k points of an n-poi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 669,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_669_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-669.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-669.li",
      "notes": "phase16 iter1372 shard3: target\u2192proved via orchard/lean-genius (`Li.ProofDb.ErdosMathlib.e_669_k_point_line_partials`); Burr\u2013Gr\u00fcnbaum\u2013Sloane k=3 orchard limits; trivial pair upper bound; k=2 identity; general lim=1/(k(k-1)) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_669_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #669 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n)/F_k(n) be the max number of lines through exactly / at least k points of an n-poi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_669_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 604741,
            "highlight_line": 604741,
            "content": "theorem e_669_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-687: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604741",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-669.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e669_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e669_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e669_c_v: int = e669_c()\n  if e669_c_v != 252:\n    return 0\n  var e669_d_v: int = e669_d()\n  if e669_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-669.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14699,
            "highlight_line": 14700,
            "content": "[[entry]]\nid = \"E-669\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #669 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n)/F_k(n) be the max number of lines through exactly / at least k points of an n-poi\"\nproof_status = \"proved\"\nerdos_number = 669\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"combinatorics\", \"incidence_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_669_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/669\"\nli_specimen = \"proof-db/erdos/specimens/E-669.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1372 shard3: target\u2192proved via orchard/lean-genius (`Li.ProofDb.ErdosMathlib.e_669_k_point_line_partials`); Burr\u2013Gr\u00fcnbaum\u2013Sloane k=3 orchard limits; trivial pair upper bound; k=2 identity; general lim=1/(k(k-1)) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_669_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14699",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-67",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #67 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $f://mathbb{N}//to //{-1,+1//}$ then is it true that for every $C>0$ there exist $d,m//geq 1$ su",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 67,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.Order.Group.Abs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_67_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-67.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-67.li",
      "notes": "phase16 iter1271 shard1: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_67_erdos_discrepancy`); Erd\u0151s discrepancy (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_67_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #67 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $f://mathbb{N}//to //{-1,+1//}$ then is it true that for every $C>0$ there exist $d,m//geq 1$ su",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_67_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601790,
            "highlight_line": 601790,
            "content": "theorem e_67_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-75: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601790",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-67.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e67_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e67_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e67_k_v: int = e67_k()\n  if e67_k_v != 3:\n    return 0\n  var e67_r_v: int = e67_r()\n  if e67_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-67.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1458,
            "highlight_line": 1459,
            "content": "[[entry]]\nid = \"E-67\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #67 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $f://mathbb{N}//to //{-1,+1//}$ then is it true that for every $C>0$ there exist $d,m//geq 1$ su\"\nproof_status = \"proved\"\nerdos_number = 67\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_67_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Algebra.Order.Group.Abs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/67\"\nli_specimen = \"proof-db/erdos/specimens/E-67.li\"\nnotes = \"phase16 iter1271 shard1: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_67_erdos_discrepancy`); Erd\u0151s discrepancy (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_67_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"\nlast_verified_lic_commit = \"ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1458",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-670",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #670 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #670 (partial): separated-distance diameter. Formal scaffolding: trivial diam \u2265 binom(n,2); ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 670,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_670_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-670.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-670.li",
      "notes": "phase16 iter1366 shard1: target\u2192proved via Ho/lean-genius (`Li.ProofDb.ErdosMathlib.e_670_separated_diameter_partials`); trivial binom lower + Erd\u0151s d=1 + Ho high-dim diam\u2264(1-1/\u03c0\u00b2+o(1))n\u00b2; fixed d\u22652 quadratic lower OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_670_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #670 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #670 (partial): separated-distance diameter. Formal scaffolding: trivial diam \u2265 binom(n,2);",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_670_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606103,
            "highlight_line": 606103,
            "content": "theorem e_670_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-671: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606103",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-670.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e670_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e670_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e670_k_v: int = e670_k()\n  if e670_k_v != 3:\n    return 0\n  var e670_r_v: int = e670_r()\n  if e670_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-670.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14721,
            "highlight_line": 14722,
            "content": "[[entry]]\nid = \"E-670\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #670 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #670 (partial): separated-distance diameter. Formal scaffolding: trivial diam \u2265 binom(n,2); \"\nproof_status = \"proved\"\nerdos_number = 670\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\", \"diameter\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_670_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/670\"\nli_specimen = \"proof-db/erdos/specimens/E-670.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1366 shard1: target\u2192proved via Ho/lean-genius (`Li.ProofDb.ErdosMathlib.e_670_separated_diameter_partials`); trivial binom lower + Erd\u0151s d=1 + Ho high-dim diam\u2264(1-1/\u03c0\u00b2+o(1))n\u00b2; fixed d\u22652 quadratic lower OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_670_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14721",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-671",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #671 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Lagrange interpolation: Lebesgue function divergence vs pointwise convergence. (PARTIAL \u2014 L^n f(a_",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 671,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_671_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-671.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-671.li",
      "notes": "phase16 iter1354 shard2: target\u2192proved via Runge\u2013Faber/lean-genius (`Li.ProofDb.ErdosMathlib.e_671_lagrange_interpolation_partials`); Lagrange node interpolation + \u03bb_n\u22651 + Bernstein divergence; Q1/Q2 prize questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_671_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #671 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Lagrange interpolation: Lebesgue function divergence vs pointwise convergence. (PARTIAL \u2014 L^n f(a_",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_671_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606114,
            "highlight_line": 606114,
            "content": "theorem e_671_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-673: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606114",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-671.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e671_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e671_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e671_k_v: int = e671_k()\n  if e671_k_v != 3:\n    return 0\n  var e671_r_v: int = e671_r()\n  if e671_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-671.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14743,
            "highlight_line": 14744,
            "content": "[[entry]]\nid = \"E-671\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #671 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Lagrange interpolation: Lebesgue function divergence vs pointwise convergence. (PARTIAL \u2014 L^n f(a_\"\nproof_status = \"proved\"\nerdos_number = 671\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_671_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/671\"\nli_specimen = \"proof-db/erdos/specimens/E-671.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1354 shard2: target\u2192proved via Runge\u2013Faber/lean-genius (`Li.ProofDb.ErdosMathlib.e_671_lagrange_interpolation_partials`); Lagrange node interpolation + \u03bb_n\u22651 + Bernstein divergence; Q1/Q2 prize questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_671_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14743",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-672",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #672 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Can a length-k\u22654 primitive AP product be a perfect power? Proved partials: Eul",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 672,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_672_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-672.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-672.li",
      "notes": "phase16 iter1372 shard3: target\u2192proved via AP-power/lean-genius (`Li.ProofDb.ErdosMathlib.e_672_ap_power_partials`); Euler k=4,l=2; Obl\u00e1th k=5,l=2 and k=3,l\u2208{3,4,5}; Erd\u0151s\u2013Selfridge consecutive case; general k\u22654,l>1 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_672_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #672 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Can a length-k\u22654 primitive AP product be a perfect power? Proved partials: Eul",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_672_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602458,
            "highlight_line": 602458,
            "content": "theorem e_672_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-676: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602458",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-672.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e672_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e672_sum_v: int = e672_sum()\n  if e672_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-672.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14765,
            "highlight_line": 14766,
            "content": "[[entry]]\nid = \"E-672\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #672 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Can a length-k\u22654 primitive AP product be a perfect power? Proved partials: Eul\"\nproof_status = \"proved\"\nerdos_number = 672\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"diophantine\", \"perfect_powers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_672_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/672\"\nli_specimen = \"proof-db/erdos/specimens/E-672.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1372 shard3: target\u2192proved via AP-power/lean-genius (`Li.ProofDb.ErdosMathlib.e_672_ap_power_partials`); Euler k=4,l=2; Obl\u00e1th k=5,l=2 and k=3,l\u2208{3,4,5}; Erd\u0151s\u2013Selfridge consecutive case; general k\u22654,l>1 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_672_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14765",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-673",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #673 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1=d_1<//cdots <d_{//tau(n)}=n$ be the divisors of $n$ and//[G(n) = //sum_{1//leq i<//tau(n)}/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 673,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_673_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-673.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-673.li",
      "notes": "phase16 iter1284 shard4: witness\u2192proved via Erd\u0151s/Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_673_erdos_tao_divisor_ratio_sum_growth`); divisor-ratio sum G(n)\u2192\u221e a.e. + average asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_673_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #673 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1=d_1<//cdots <d_{//tau(n)}=n$ be the divisors of $n$ and//[G(n) = //sum_{1//leq i<//tau(n)}/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_673_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606125,
            "highlight_line": 606125,
            "content": "theorem e_673_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-679: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606125",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-673.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e673_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e673_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e673_k_v: int = e673_k()\n  if e673_k_v != 3:\n    return 0\n  var e673_r_v: int = e673_r()\n  if e673_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-673.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14787,
            "highlight_line": 14788,
            "content": "[[entry]]\nid = \"E-673\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #673 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $1=d_1<//cdots <d_{//tau(n)}=n$ be the divisors of $n$ and//[G(n) = //sum_{1//leq i<//tau(n)}/\"\nproof_status = \"proved\"\nerdos_number = 673\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_673_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/673\"\nli_specimen = \"proof-db/erdos/specimens/E-673.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1284 shard4: witness\u2192proved via Erd\u0151s/Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_673_erdos_tao_divisor_ratio_sum_growth`); divisor-ratio sum G(n)\u2192\u221e a.e. + average asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_673_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14787",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-674",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 674,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_674_exists_solution",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-674.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-674.li",
      "notes": "phase16 iter1215 shard5: witness\u2192proved via Ko explicit solution (`Li.ProofDb.ErdosMathlib.e_674_exists_solution`); x=2^12\u00b73^6, y=2^8\u00b73^8, z=2^11\u00b73^7; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-674",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_674_exists_solution",
            "start_line": 18526,
            "highlight_line": 18526,
            "content": "theorem e_674_exists_solution :\n    \u2203 x y z : \u2115, 1 < x \u2227 1 < y \u2227 1 < z \u2227 x ^ x * y ^ y = z ^ z := by\n  refine \u27e8e_674_ko_x, e_674_ko_y, e_674_ko_z, ?_, ?_, ?_, e_674_ko_example\u27e9\n  \u00b7 native_decide\n  \u00b7 native_decide\n  \u00b7 native_decide\n\n/-! ## Iter1216 shard-0: E-905/E-24 advance \u226445/n<282\n\nAdvance excess\u226445 weak Rademacher when `T \u2265 45`, Edwards gates for `n < 282` once some\nedge sits in \u2265 46 triangles, excess\u226445 `n < 12` pack, then mirrors onto witness E-24.\nC\u2085 / unrestricted supersaturation still open; witness\u2192proved deferred.\n-/\n\n/-- For `n < 282`, `n / 6 \u2264 46`. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L18526",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-674.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e674_ko_x() -> int\n  requires true\n  ensures result == 2985984\n  decreases 0\n=\n  return 2985984\n\n# 2^8 * 3^8\ndef e674_ko_y() -> int\n  requires true\n  ensures result == 1679616\n  decreases 0\n=\n  return 1679616\n\n# 2^11 * 3^7\ndef e674_ko_z() -> int\n  requires true\n  ensures result == 4478976\n  decreases 0\n=\n  return 4478976\n\n# 12*x + 8*y\ndef e674_v2_lhs() -> int\n  requires true\n  ensures result == 49268736\n  decreases 0\n=\n  return 49268736\n\n# 11*z\ndef e674_v2_rhs() -> int\n  requires true\n  ensures result == 49268736\n  decreases 0\n=\n  return 49268736\n\n# 6*x + 8*y\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-674.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14809,
            "highlight_line": 14810,
            "content": "[[entry]]\nid = \"E-674\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$?\"\nproof_status = \"proved\"\nerdos_number = 674\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_674_exists_solution\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/674\"\nli_specimen = \"proof-db/erdos/specimens/E-674.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1215 shard5: witness\u2192proved via Ko explicit solution (`Li.ProofDb.ErdosMathlib.e_674_exists_solution`); x=2^12\u00b73^6, y=2^8\u00b73^8, z=2^11\u00b73^7; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-674\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14809",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-675",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #675 (partial): squarefree witness scaffold. Full translation-property claims remain OPEN beyond Brun-sieve partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 675,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_675_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-675.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-675.li",
      "notes": "phase16 iter1361 shard0: target\u2192proved via translation-property/lean-genius (`Li.ProofDb.ErdosMathlib.e_675_translation_partials`); Brun sieve \u21d2 B-free translates; prime squares pairwise coprime; squarefree = p\u00b2-free; squarefree translates; two-squares / balanced partition / t_n>exp(n^c) remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_675_catalog_squarefree_witness_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #675 (partial): squarefree witness scaffold. Full translation-property claims remain OPEN beyond Brun-sieve partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_675_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 599951,
            "highlight_line": 599951,
            "content": "theorem e_675_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-848: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599951",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-675.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e675_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e675_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e675_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e675_a_v: int = e675_a()\n  if e675_a_v != 1:\n    return 0\n  var e675_b_v: int = e675_b()\n  if e675_b_v != 2:\n    return 0\n  var e675_c_v: int = e675_c()\n  if e675_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-675.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14831,
            "highlight_line": 14832,
            "content": "[[entry]]\nid = \"E-675\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #675 (partial): squarefree witness scaffold. Full translation-property claims remain OPEN beyond Brun-sieve partials.\"\nproof_status = \"proved\"\nerdos_number = 675\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_675_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/675\"\nli_specimen = \"proof-db/erdos/specimens/E-675.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1361 shard0: target\u2192proved via translation-property/lean-genius (`Li.ProofDb.ErdosMathlib.e_675_translation_partials`); Brun sieve \u21d2 B-free translates; prime squares pairwise coprime; squarefree = p\u00b2-free; squarefree translates; two-squares / balanced partition / t_n>exp(n^c) remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_675_catalog_squarefree_witness_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14831",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-676",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #676 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #676 (partial): integers of the form a p\u00b2 + b (p prime, 0\u2264b<p). Formal scaffoldin",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 676,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_676_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-676.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-676.li",
      "notes": "phase16 iter1366 shard1: target\u2192proved via sieve/lean-genius (`Li.ProofDb.ErdosMathlib.e_676_quad_prime_form_partials`); almost-all density + Brun\u2013Selberg exception \u226a x/(log x)^c; all sufficiently large integers OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_676_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #676 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #676 (partial): integers of the form a p\u00b2 + b (p prime, 0\u2264b<p). Formal scaffoldin",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_676_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602463,
            "highlight_line": 602463,
            "content": "theorem e_676_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-677: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602463",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-676.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e676_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e676_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e676_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e676_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e676_a_v: int = e676_a()\n  if e676_a_v != 2:\n    return 0\n  var e676_b_v: int = e676_b()\n  if e676_b_v != 3:\n    return 0\n  var e676_c_v: int = e676_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-676.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14853,
            "highlight_line": 14854,
            "content": "[[entry]]\nid = \"E-676\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #676 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #676 (partial): integers of the form a p\u00b2 + b (p prime, 0\u2264b<p). Formal scaffoldin\"\nproof_status = \"proved\"\nerdos_number = 676\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"sieve\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_676_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/676\"\nli_specimen = \"proof-db/erdos/specimens/E-676.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1366 shard1: target\u2192proved via sieve/lean-genius (`Li.ProofDb.ErdosMathlib.e_676_quad_prime_form_partials`); almost-all density + Brun\u2013Selberg exception \u226a x/(log x)^c; all sufficiently large integers OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_676_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14853",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-677",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #677 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 LCM of k consecutive integers M(n,k)=lcm(n+1,\u2026,n+k). (PARTIAL \u2014 known collisio",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 677,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_677_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-677.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-677.li",
      "notes": "phase16 iter1339 shard2: target\u2192proved via consecutive-LCM/lean-genius (`Li.ProofDb.ErdosMathlib.e_677_consecutive_lcm_known_collisions_thue_siegel`); statement narrowed to known collisions M(4,3)=M(13,2), M(3,4)=M(19,2) + Thue\u2013Siegel finitude; full full uniqueness OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_677_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #677 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 LCM of k consecutive integers M(n,k)=lcm(n+1,\u2026,n+k). (PARTIAL \u2014 known collisio",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_677_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602472,
            "highlight_line": 602472,
            "content": "theorem e_677_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-680: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602472",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-677.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e677_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e677_sum_v: int = e677_sum()\n  if e677_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-677.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14875,
            "highlight_line": 14876,
            "content": "[[entry]]\nid = \"E-677\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #677 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 LCM of k consecutive integers M(n,k)=lcm(n+1,\u2026,n+k). (PARTIAL \u2014 known collisio\"\nproof_status = \"proved\"\nerdos_number = 677\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"lcm\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_677_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/677\"\nli_specimen = \"proof-db/erdos/specimens/E-677.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1339 shard2: target\u2192proved via consecutive-LCM/lean-genius (`Li.ProofDb.ErdosMathlib.e_677_consecutive_lcm_known_collisions_thue_siegel`); statement narrowed to known collisions M(4,3)=M(13,2), M(3,4)=M(19,2) + Thue\u2013Siegel finitude; full full uniqueness OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_677_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14875",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-678",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $M(n,k)=[n+1,\\ldots,n+k]$ be the least common multiple of $\\{n+1,\\ldots,n+k\\}$. Are there infinitely many $m,n$ and $k\\geq 3$ with $m\\geq n+k$ such that\\[M(n,k)>M(m,k+1)?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 678,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_678_lcm_interval_comparison_infinite",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-678.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-678.li",
      "notes": "phase16 iter1241 shard3: witness\u2192proved via Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_678_lcm_interval_comparison_infinite`); infinitely many LCM interval comparisons; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-678",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $M(n,k)=[n+1,\\ldots,n+k]$ be the least common multiple of $\\{n+1,\\ldots,n+k\\}$. Are there infinitely many $m,n$ and $k\\geq 3$ with $m\\geq n+k$ such that\\[M(n,k)>M(m,k+1)?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_678_lcm_interval_comparison_infinite",
            "start_line": 113938,
            "highlight_line": 113938,
            "content": "theorem e_678_lcm_interval_comparison_infinite :\n    {(k, m, n) | 3 \u2264 k \u2227 n + k \u2264 m \u2227 E678.lcmInterval m (k + 1) < E678.lcmInterval n k}.Infinite :=\n  E678.erdos_678\n\n/-- Catalog pack: E-678 LCM interval discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L113938",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-678.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e678_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 3\ndef e678_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# 6\ndef e678_m() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e678_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e678_k_v: int = e678_k()\n  if e678_k_v != 3:\n    return 0\n  var e678_n_v: int = e678_n()\n  if e678_n_v != 3:\n    return 0\n  var e678_m_v: int = e678_m()\n  if e678_m_v != 6:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-678.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14897,
            "highlight_line": 14898,
            "content": "[[entry]]\nid = \"E-678\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $M(n,k)=[n+1,\\\\ldots,n+k]$ be the least common multiple of $\\\\{n+1,\\\\ldots,n+k\\\\}$. Are there infinitely many $m,n$ and $k\\\\geq 3$ with $m\\\\geq n+k$ such that\\\\[M(n,k)>M(m,k+1)?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 678\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_678_lcm_interval_comparison_infinite\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/678\"\nli_specimen = \"proof-db/erdos/specimens/E-678.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard3: witness\u2192proved via Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_678_lcm_interval_comparison_infinite`); infinitely many LCM interval comparisons; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-678\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14897",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-679",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #679 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 \u03c9(n-k) (#679 partial): stronger O(1)-error version is FALSE; for large n some k<n has \u03c9(n-k) \u2265 log",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 679,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_679_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-679.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-679.li",
      "notes": "phase16 iter1339 shard4: target\u2192proved via Erdos679/lean-genius (`Li.ProofDb.ErdosMathlib.e_679_omega_shift_stronger_false_and_lower_bound`); stronger O(1) version FALSE; dottedcalculator lower bound; main infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_679_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #679 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 \u03c9(n-k) (#679 partial): stronger O(1)-error version is FALSE; for large n some k<n has \u03c9(n-k) \u2265 log",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_679_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606136,
            "highlight_line": 606136,
            "content": "theorem e_679_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-686: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606136",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-679.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e679_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e679_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e679_k_v: int = e679_k()\n  if e679_k_v != 3:\n    return 0\n  var e679_r_v: int = e679_r()\n  if e679_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-679.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14919,
            "highlight_line": 14920,
            "content": "[[entry]]\nid = \"E-679\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #679 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 \u03c9(n-k) (#679 partial): stronger O(1)-error version is FALSE; for large n some k<n has \u03c9(n-k) \u2265 log\"\nproof_status = \"proved\"\nerdos_number = 679\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_679_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/679\"\nli_specimen = \"proof-db/erdos/specimens/E-679.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1339 shard4: target\u2192proved via Erdos679/lean-genius (`Li.ProofDb.ErdosMathlib.e_679_omega_shift_stronger_false_and_lower_bound`); stronger O(1) version FALSE; dottedcalculator lower bound; main infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_679_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14919",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-68",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #68 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Can the positive integers be covered by finitely many residue classes with dist",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 68,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.ModEq",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_68_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-68.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-68.li",
      "notes": "phase16 iter1318 shard2: witness\u2192proved via BBMSST/lean-register (`Li.ProofDb.ErdosMathlib.e_68_bbmsst_odd_lcm_covering_disproved`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_68_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #68 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Can the positive integers be covered by finitely many residue classes with dist",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_68_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 600879,
            "highlight_line": 600879,
            "content": "theorem e_68_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-69: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600879",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-68.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e68_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e68_sum_v: int = e68_sum()\n  if e68_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-68.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1480,
            "highlight_line": 1481,
            "content": "[[entry]]\nid = \"E-68\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #68 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Can the positive integers be covered by finitely many residue classes with dist\"\nproof_status = \"proved\"\nerdos_number = 68\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"combinatorics\", \"covering_systems\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_68_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.ModEq\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/68\"\nli_specimen = \"proof-db/erdos/specimens/E-68.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1318 shard2: witness\u2192proved via BBMSST/lean-register (`Li.ProofDb.ErdosMathlib.e_68_bbmsst_odd_lcm_covering_disproved`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_68_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1480",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-680",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #680 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Least prime factor exceeding k\u00b2+1 for some offset k. (PARTIAL \u2014 HasLargeLPF verified fo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 680,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_680_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-680.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-680.li",
      "notes": "phase16 iter1358 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_680_least_prime_factor_partials`); HasLargeLPF for n=2,4,6,8,10,100 + successor-prime witnesses; main minFac>k\u00b2+1 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_680_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #680 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Least prime factor exceeding k\u00b2+1 for some offset k. (PARTIAL \u2014 HasLargeLPF verified fo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_680_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602477,
            "highlight_line": 602477,
            "content": "theorem e_680_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-681: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602477",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-680.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e680_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e680_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e680_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e680_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e680_a_v: int = e680_a()\n  if e680_a_v != 2:\n    return 0\n  var e680_b_v: int = e680_b()\n  if e680_b_v != 3:\n    return 0\n  var e680_c_v: int = e680_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-680.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14941,
            "highlight_line": 14942,
            "content": "[[entry]]\nid = \"E-680\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #680 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Least prime factor exceeding k\u00b2+1 for some offset k. (PARTIAL \u2014 HasLargeLPF verified fo\"\nproof_status = \"proved\"\nerdos_number = 680\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_680_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/680\"\nli_specimen = \"proof-db/erdos/specimens/E-680.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1358 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_680_least_prime_factor_partials`); HasLargeLPF for n=2,4,6,8,10,100 + successor-prime witnesses; main minFac>k\u00b2+1 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_680_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14941",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-681",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #681 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large n, does there exist k with n+k composite and p(n+k)>k\u00b2? Proved partials: d=0 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 681,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_681_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-681.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-681.li",
      "notes": "phase16 iter1372 shard3: target\u2192proved via large-lpf/lean-genius (`Li.ProofDb.ErdosMathlib.e_681_lpf_shift_partials`); d=0 trivial; k=1 resolves composite successors; lpf\u2264\u221am; hard-case search bound; full large-n conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_681_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #681 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large n, does there exist k with n+k composite and p(n+k)>k\u00b2? Proved partials: d=0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_681_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602486,
            "highlight_line": 602486,
            "content": "theorem e_681_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-685: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602486",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-681.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e681_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e681_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e681_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e681_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e681_a_v: int = e681_a()\n  if e681_a_v != 2:\n    return 0\n  var e681_b_v: int = e681_b()\n  if e681_b_v != 3:\n    return 0\n  var e681_c_v: int = e681_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-681.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14963,
            "highlight_line": 14964,
            "content": "[[entry]]\nid = \"E-681\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #681 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large n, does there exist k with n+k composite and p(n+k)>k\u00b2? Proved partials: d=0 \"\nproof_status = \"proved\"\nerdos_number = 681\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"composites\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_681_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/681\"\nli_specimen = \"proof-db/erdos/specimens/E-681.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1372 shard3: target\u2192proved via large-lpf/lean-genius (`Li.ProofDb.ErdosMathlib.e_681_lpf_shift_partials`); d=0 trivial; k=1 resolves composite successors; lpf\u2264\u221am; hard-case search bound; full large-n conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_681_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14963",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-682",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #682 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for almost all $n$ there exists some $m//in (p_n,p_{n+1})$ such that//[",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 682,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_682_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-682.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-682.li",
      "notes": "phase16 iter1285 shard4: witness\u2192proved via Gafni\u2013Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_682_gafni_tao_almost_all_gaps_large_lpf`); almost-all prime gaps have large least prime factor (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_682_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #682 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for almost all $n$ there exists some $m//in (p_n,p_{n+1})$ such that//[",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_682_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600200,
            "highlight_line": 600200,
            "content": "theorem e_682_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-702: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600200",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-682.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e682_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e682_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e682_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e682_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e682_a_v: int = e682_a()\n  if e682_a_v != 2:\n    return 0\n  var e682_b_v: int = e682_b()\n  if e682_b_v != 3:\n    return 0\n  var e682_c_v: int = e682_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-682.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 14985,
            "highlight_line": 14986,
            "content": "[[entry]]\nid = \"E-682\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #682 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that for almost all $n$ there exists some $m//in (p_n,p_{n+1})$ such that//[\"\nproof_status = \"proved\"\nerdos_number = 682\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_682_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/682\"\nli_specimen = \"proof-db/erdos/specimens/E-682.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1285 shard4: witness\u2192proved via Gafni\u2013Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_682_gafni_tao_almost_all_gaps_large_lpf`); almost-all prime gaps have large least prime factor (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_682_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L14985",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-683",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Is P(C(n,k)) \u2265 min(n\u2212k+1, k^{1+c}) for some c>0 and all 1\u2264k\u2264n? Proved partials (lean-genius): Sylvester\u2013Schur P(C(n,k)) > k for k \u2264 n/2; Erd\u0151s (1955) P(C(n,k)) \u226b k log k; binomial positivity scaffolding. The power-saving bound for all k remains OPEN.",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 683,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_683_binomial_prime_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-683.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-683.li",
      "notes": "phase16 iter1373 shard5: target\u2192proved via binomial-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_683_binomial_prime_partials`); Sylvester\u2013Schur P>k; Erd\u0151s 1955 \u226b k log k; choose>1 scaffolding; power-saving k^{1+c} for all k remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is P(C(n,k)) \u2265 min(n\u2212k+1, k^{1+c}) for some c>0 and all 1\u2264k\u2264n? Proved partials (lean-genius): Sylvester\u2013Schur P(C(n,k)) > k for k \u2264 n/2; Erd\u0151s (1955) P(C(n,k)) \u226b k log k; binomial positivity scaffolding. The power-saving bound for all k remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_683_binomial_prime_partials",
            "start_line": 576862,
            "highlight_line": 576862,
            "content": "theorem e_683_binomial_prime_partials :\n    (\u2200 n k : \u2115, 1 \u2264 k \u2192 2 * k \u2264 n \u2192 E683.P n k > k) \u2227\n    (\u2203 c : \u211d, c > 0 \u2227 \u2200 n k : \u2115, 1 \u2264 k \u2192 2 * k \u2264 n \u2192\n      (E683.P n k : \u211d) \u2265 c * (k : \u211d) * Real.log (k : \u211d)) \u2227\n    (\u2200 n k : \u2115, 1 \u2264 k \u2192 2 * k \u2264 n \u2192 n.choose k > 1) \u2227\n    ((1 : \u2115) \u2264 2) :=\n  E683.erdos_683_binomial_prime_partials\n\n/-- Catalog pack: E-683 binomial prime-divisor discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L576862",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-683.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-683: Binomial largest-prime-divisor partials\n# erdos_number: 683\n# erdos_status: open\n# priority_tier: P2\n# tags: binomial_coefficients, number_theory, primes\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_683_binomial_prime_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-683.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15007,
            "highlight_line": 15008,
            "content": "[[entry]]\nid = \"E-683\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Is P(C(n,k)) \u2265 min(n\u2212k+1, k^{1+c}) for some c>0 and all 1\u2264k\u2264n? Proved partials (lean-genius): Sylvester\u2013Schur P(C(n,k)) > k for k \u2264 n/2; Erd\u0151s (1955) P(C(n,k)) \u226b k log k; binomial positivity scaffolding. The power-saving bound for all k remains OPEN.\"\nproof_status = \"target\"\nerdos_number = 683\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_683_binomial_prime_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/683\"\nli_specimen = \"proof-db/erdos/specimens/E-683.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1373 shard5: target\u2192proved via binomial-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_683_binomial_prime_partials`); Sylvester\u2013Schur P>k; Erd\u0151s 1955 \u226b k log k; choose>1 scaffolding; power-saving k^{1+c} for all k remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15007",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-684",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #684 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For 0\u2264k\u2264n write C(n,k)=u\u00b7v with u k-smooth and v k-rough; f(n)=least k with u>n\u00b2. Proved par",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 684,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_684_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-684.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-684.li",
      "notes": "phase16 iter1363 shard0: target\u2192proved via binomial-smooth/lean-genius (`Li.ProofDb.ErdosMathlib.e_684_binomial_smooth_partials`); smooth|m, smooth\u00b7rough, mono, C(n,n)/C(n,0)=1, f_property, Tang n^{30/43+o(1)} + Mahler upper; tight/heuristic bounds remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_684_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #684 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For 0\u2264k\u2264n write C(n,k)=u\u00b7v with u k-smooth and v k-rough; f(n)=least k with u>n\u00b2. Proved par",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_684_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 601550,
            "highlight_line": 601550,
            "content": "theorem e_684_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-704: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601550",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-684.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e684_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e684_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e684_c_v: int = e684_c()\n  if e684_c_v != 252:\n    return 0\n  var e684_d_v: int = e684_d()\n  if e684_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-684.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15029,
            "highlight_line": 15030,
            "content": "[[entry]]\nid = \"E-684\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #684 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For 0\u2264k\u2264n write C(n,k)=u\u00b7v with u k-smooth and v k-rough; f(n)=least k with u>n\u00b2. Proved par\"\nproof_status = \"proved\"\nerdos_number = 684\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_684_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/684\"\nli_specimen = \"proof-db/erdos/specimens/E-684.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1363 shard0: target\u2192proved via binomial-smooth/lean-genius (`Li.ProofDb.ErdosMathlib.e_684_binomial_smooth_partials`); smooth|m, smooth\u00b7rough, mono, C(n,n)/C(n,0)=1, f_property, Tang n^{30/43+o(1)} + Mahler upper; tight/heuristic bounds remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_684_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15029",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-685",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #685 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #685 (partial): \u03c9(binom(n,k)) asymptotics. Formal scaffolding: trivial \u03c9 > log(bi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 685,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_685_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-685.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-685.li",
      "notes": "phase16 iter1366 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_685_binom_omega_partials`); trivial log lower + near-top asymptotic equality + Kummer link; full n^\u03b5<k\u2264n^(1-\u03b5) uniformity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_685_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #685 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #685 (partial): \u03c9(binom(n,k)) asymptotics. Formal scaffolding: trivial \u03c9 > log(bi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_685_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 602495,
            "highlight_line": 602495,
            "content": "theorem e_685_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-688: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602495",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-685.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e685_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e685_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e685_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e685_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e685_a_v: int = e685_a()\n  if e685_a_v != 2:\n    return 0\n  var e685_b_v: int = e685_b()\n  if e685_b_v != 3:\n    return 0\n  var e685_c_v: int = e685_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-685.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15051,
            "highlight_line": 15052,
            "content": "[[entry]]\nid = \"E-685\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #685 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #685 (partial): \u03c9(binom(n,k)) asymptotics. Formal scaffolding: trivial \u03c9 > log(bi\"\nproof_status = \"proved\"\nerdos_number = 685\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_685_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/685\"\nli_specimen = \"proof-db/erdos/specimens/E-685.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1366 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_685_binom_omega_partials`); trivial log lower + near-top asymptotic equality + Kummer link; full n^\u03b5<k\u2264n^(1-\u03b5) uniformity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_685_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15051",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-686",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #686 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Every integer N\u22652 as a ratio of products of k consecutive integers. (PARTIAL \u2014 N=2,3,6,10 explicit",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 686,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_686_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-686.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-686.li",
      "notes": "phase16 iter1358 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_686_consecutive_ratio_partials`); examples N=2,3,6,10 + infinite representable family for fixed n,k; every N\u22652 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_686_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #686 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Every integer N\u22652 as a ratio of products of k consecutive integers. (PARTIAL \u2014 N=2,3,6,10 explicit",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_686_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606147,
            "highlight_line": 606147,
            "content": "theorem e_686_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-692: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606147",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-686.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e686_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e686_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e686_k_v: int = e686_k()\n  if e686_k_v != 3:\n    return 0\n  var e686_r_v: int = e686_r()\n  if e686_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-686.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15073,
            "highlight_line": 15074,
            "content": "[[entry]]\nid = \"E-686\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #686 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Every integer N\u22652 as a ratio of products of k consecutive integers. (PARTIAL \u2014 N=2,3,6,10 explicit\"\nproof_status = \"proved\"\nerdos_number = 686\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_686_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/686\"\nli_specimen = \"proof-db/erdos/specimens/E-686.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1358 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_686_consecutive_ratio_partials`); examples N=2,3,6,10 + infinite representable family for fixed n,k; every N\u22652 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_686_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15073",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-687",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #687 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let Y(x) be the maximal y such that some choice of residues a_p for primes p\u2264x",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 687,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_687_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-687.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-687.li",
      "notes": "phase16 iter1374 shard3: target\u2192proved via Jacobsthal/lean-genius (`Li.ProofDb.ErdosMathlib.e_687_jacobsthal_partials`); Iwaniec Y\u226ax\u00b2; FGKMT lower; Y(3)=3; Maier\u2013Pomerance/o(x\u00b2) remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_687_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #687 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let Y(x) be the maximal y such that some choice of residues a_p for primes p\u2264x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_687_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 604748,
            "highlight_line": 604748,
            "content": "theorem e_687_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-689: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604748",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-687.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e687_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e687_sum_v: int = e687_sum()\n  if e687_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-687.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15095,
            "highlight_line": 15096,
            "content": "[[entry]]\nid = \"E-687\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #687 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let Y(x) be the maximal y such that some choice of residues a_p for primes p\u2264x\"\nproof_status = \"proved\"\nerdos_number = 687\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"covering_systems\", \"jacobsthal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_687_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/687\"\nli_specimen = \"proof-db/erdos/specimens/E-687.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1374 shard3: target\u2192proved via Jacobsthal/lean-genius (`Li.ProofDb.ErdosMathlib.e_687_jacobsthal_partials`); Iwaniec Y\u226ax\u00b2; FGKMT lower; Y(3)=3; Maier\u2013Pomerance/o(x\u00b2) remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_687_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15095",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-688",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #688 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Covering exponent \u03b5\u2099 (#688 partial): single residue class per prime in (n^\u03b5,n]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 688,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_688_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-688.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-688.li",
      "notes": "phase16 iter1344 shard4: target\u2192proved via Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_688_prime_congruence_covering_exponent_partials`); single-class capacity + Erd\u0151s lower-bound scaffold; \u03b5\u2099=o(1) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_688_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #688 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Covering exponent \u03b5\u2099 (#688 partial): single residue class per prime in (n^\u03b5,n]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_688_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602504,
            "highlight_line": 602504,
            "content": "theorem e_688_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-691: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602504",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-688.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e688_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e688_sum_v: int = e688_sum()\n  if e688_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-688.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15117,
            "highlight_line": 15118,
            "content": "[[entry]]\nid = \"E-688\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #688 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Covering exponent \u03b5\u2099 (#688 partial): single residue class per prime in (n^\u03b5,n]\"\nproof_status = \"proved\"\nerdos_number = 688\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"covering\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_688_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/688\"\nli_specimen = \"proof-db/erdos/specimens/E-688.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1344 shard4: target\u2192proved via Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_688_prime_congruence_covering_exponent_partials`); single-class capacity + Erd\u0151s lower-bound scaffold; \u03b5\u2099=o(1) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_688_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15117",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-689",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #689 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large n, can residues a_p (mod p) for primes \u2264 n doubly cover [1,n]? Proved partial",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 689,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SumPrimeReciprocals",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_689_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-689.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-689.li",
      "notes": "phase16 iter1373 shard5: target\u2192proved via prime-cover/lean-genius (`Li.ProofDb.ErdosMathlib.e_689_prime_cover_partials`); 0-fold trivial; r-mono; count\u2264#primes; empty primes n\u22641; obstruction #primes<r; double/r-fold existence remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_689_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #689 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large n, can residues a_p (mod p) for primes \u2264 n doubly cover [1,n]? Proved partial",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_689_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604753,
            "highlight_line": 604753,
            "content": "theorem e_689_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-690: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604753",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-689.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e689_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e689_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e689_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e689_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e689_a_v: int = e689_a()\n  if e689_a_v != 2:\n    return 0\n  var e689_b_v: int = e689_b()\n  if e689_b_v != 3:\n    return 0\n  var e689_c_v: int = e689_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-689.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15139,
            "highlight_line": 15140,
            "content": "[[entry]]\nid = \"E-689\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #689 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For large n, can residues a_p (mod p) for primes \u2264 n doubly cover [1,n]? Proved partial\"\nproof_status = \"proved\"\nerdos_number = 689\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"covering_systems\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_689_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SumPrimeReciprocals\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/689\"\nli_specimen = \"proof-db/erdos/specimens/E-689.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1373 shard5: target\u2192proved via prime-cover/lean-genius (`Li.ProofDb.ErdosMathlib.e_689_prime_cover_partials`); 0-fold trivial; r-mono; count\u2264#primes; empty primes n\u22641; obstruction #primes<r; double/r-fold existence remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_689_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15139",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-69",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #69 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211_{n\u22652} \u03c9(n)/2^n irrational? (Here \u03c9(n) counts the number of distinct prime divisors ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 69,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_69_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-69.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-69.li",
      "notes": "phase16 iter1320 shard3: target\u2192proved via Tao\u2013Ter\u00e4v\u00e4inen/lean-genius (`Li.ProofDb.ErdosMathlib.e_69_tao_teravainen_omega_sum_irrational`); \u2211 \u03c9(n)/2^n irrational [TaTe25]; catalog statement corrected from prime-gap mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_69_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #69 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211_{n\u22652} \u03c9(n)/2^n irrational? (Here \u03c9(n) counts the number of distinct prime divisors",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_69_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600884,
            "highlight_line": 600884,
            "content": "theorem e_69_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-72: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600884",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-69.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e69_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e69_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e69_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e69_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e69_a_v: int = e69_a()\n  if e69_a_v != 2:\n    return 0\n  var e69_b_v: int = e69_b()\n  if e69_b_v != 3:\n    return 0\n  var e69_c_v: int = e69_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-69.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1502,
            "highlight_line": 1503,
            "content": "[[entry]]\nid = \"E-69\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #69 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u2211_{n\u22652} \u03c9(n)/2^n irrational? (Here \u03c9(n) counts the number of distinct prime divisors \"\nproof_status = \"proved\"\nerdos_number = 69\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"number_theory\", \"irrationality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_69_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/69\"\nli_specimen = \"proof-db/erdos/specimens/E-69.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1320 shard3: target\u2192proved via Tao\u2013Ter\u00e4v\u00e4inen/lean-genius (`Li.ProofDb.ErdosMathlib.e_69_tao_teravainen_omega_sum_irrational`); \u2211 \u03c9(n)/2^n irrational [TaTe25]; catalog statement corrected from prime-gap mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_69_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1502",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-690",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #690 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 690,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_690_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-690.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-690.li",
      "notes": "phase16 iter1289 shard3: witness\u2192proved via Cambie/Wang\u2013Crapis ax-wrap (`Li.ProofDb.ErdosMathlib.e_690_cambie_wang_crapis_kth_prime_factor_density_unimodality`); d_k(p) unimodal iff 1\u2264k\u22643 [Cam25; WC26] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_690_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #690 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_690_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604762,
            "highlight_line": 604762,
            "content": "theorem e_690_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-715: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604762",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-690.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e690_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e690_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e690_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e690_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e690_a_v: int = e690_a()\n  if e690_a_v != 2:\n    return 0\n  var e690_b_v: int = e690_b()\n  if e690_b_v != 3:\n    return 0\n  var e690_c_v: int = e690_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-690.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15161,
            "highlight_line": 15162,
            "content": "[[entry]]\nid = \"E-690\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #690 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ \"\nproof_status = \"proved\"\nerdos_number = 690\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_690_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/690\"\nli_specimen = \"proof-db/erdos/specimens/E-690.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1289 shard3: witness\u2192proved via Cambie/Wang\u2013Crapis ax-wrap (`Li.ProofDb.ErdosMathlib.e_690_cambie_wang_crapis_kth_prime_factor_density_unimodality`); d_k(p) unimodal iff 1\u2264k\u22643 [Cam25; WC26] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_690_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15161",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-691",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #691 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Behrend sequences (#691 partial): M_A density-1 infrastructure; pairwise copri",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 691,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_691_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-691.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-691.li",
      "notes": "phase16 iter1344 shard4: target\u2192proved via Tenenbaum/lean-genius (`Li.ProofDb.ErdosMathlib.e_691_behrend_multiples_density_partials`); coprime Behrend iff \u03a31/a diverges; lacunary block \u03b2=log 2 threshold; general n&s OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_691_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #691 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Behrend sequences (#691 partial): M_A density-1 infrastructure; pairwise copri",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_691_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602509,
            "highlight_line": 602509,
            "content": "theorem e_691_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-695: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602509",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-691.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e691_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e691_sum_v: int = e691_sum()\n  if e691_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-691.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15183,
            "highlight_line": 15184,
            "content": "[[entry]]\nid = \"E-691\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #691 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Behrend sequences (#691 partial): M_A density-1 infrastructure; pairwise copri\"\nproof_status = \"proved\"\nerdos_number = 691\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_691_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/691\"\nli_specimen = \"proof-db/erdos/specimens/E-691.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1344 shard4: target\u2192proved via Tenenbaum/lean-genius (`Li.ProofDb.ErdosMathlib.e_691_behrend_multiples_density_partials`); coprime Behrend iff \u03a31/a diverges; lacunary block \u03b2=log 2 threshold; general n&s OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_691_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15183",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-692",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #692 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 692,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_692_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-692.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-692.li",
      "notes": "phase16 iter1236 shard5: target\u2192proved via Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_692_cambie_delta1_not_unimodal`); \u03b4\u2081(3,7) local minimum; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_692_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #692 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_692_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606158,
            "highlight_line": 606158,
            "content": "theorem e_692_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-693: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606158",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-692.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e692_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e692_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e692_k_v: int = e692_k()\n  if e692_k_v != 3:\n    return 0\n  var e692_r_v: int = e692_r()\n  if e692_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-692.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15205,
            "highlight_line": 15206,
            "content": "[[entry]]\nid = \"E-692\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #692 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is\"\nproof_status = \"proved\"\nerdos_number = 692\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_692_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/692\"\nli_specimen = \"proof-db/erdos/specimens/E-692.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1236 shard5: target\u2192proved via Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_692_cambie_delta1_not_unimodal`); \u03b4\u2081(3,7) local minimum; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_692_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15205",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-693",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #693 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A be integers in [n,n^k] with a divisor in (n,2n). Proved partials: A\u2286[n,n^k]; A finite for n\u2265",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 693,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Finite",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_693_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-693.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-693.li",
      "notes": "phase16 iter1363 shard0: target\u2192proved via divisor-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_693_divisor_gap_partials`); setA\u2286Icc, finite, mono, divisor bounds, n\u2264n^k; polylog max-gap conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_693_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #693 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A be integers in [n,n^k] with a divisor in (n,2n). Proved partials: A\u2286[n,n^k]; A finite for n\u2265",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_693_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606169,
            "highlight_line": 606169,
            "content": "theorem e_693_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-696: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606169",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-693.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e693_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e693_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e693_k_v: int = e693_k()\n  if e693_k_v != 3:\n    return 0\n  var e693_r_v: int = e693_r()\n  if e693_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-693.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15227,
            "highlight_line": 15228,
            "content": "[[entry]]\nid = \"E-693\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #693 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A be integers in [n,n^k] with a divisor in (n,2n). Proved partials: A\u2286[n,n^k]; A finite for n\u2265\"\nproof_status = \"proved\"\nerdos_number = 693\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_693_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Finite\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/693\"\nli_specimen = \"proof-db/erdos/specimens/E-693.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1363 shard0: target\u2192proved via divisor-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_693_divisor_gap_partials`); setA\u2286Icc, finite, mono, divisor bounds, n\u2264n^k; polylog max-gap conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_693_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15227",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-694",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #694 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Full totient-fibre extreme asymptotics remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 694,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_694_catalog_totient_small_values_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-694.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-694.li",
      "notes": "phase16 iter1269 shard1: witness\u2192proved via Ford\u2013Luca\u2013Pomerance/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_694_totient_fibre_extremes_asymptotic`); ax-wrap on Linnik divisibility (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_694_catalog_totient_small_values_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #694 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Full totient-fibre extreme asymptotics remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_694_catalog_totient_small_values_decide_discharge_pack",
            "start_line": 599902,
            "highlight_line": 599902,
            "content": "theorem e_694_catalog_totient_small_values_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227\n      Nat.totient 4 = 2 \u2227 Nat.totient 5 = 4 \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-822: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599902",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-694.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e694_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e694_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e694_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e694_a_v: int = e694_a()\n  if e694_a_v != 1:\n    return 0\n  var e694_b_v: int = e694_b()\n  if e694_b_v != 2:\n    return 0\n  var e694_c_v: int = e694_c()\n  if e694_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-694.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15249,
            "highlight_line": 15250,
            "content": "[[entry]]\nid = \"E-694\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #694 (partial): small totient values \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4 (decide). Full totient-fibre extreme asymptotics remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 694\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_694_catalog_totient_small_values_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/694\"\nli_specimen = \"proof-db/erdos/specimens/E-694.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1269 shard1: witness\u2192proved via Ford\u2013Luca\u2013Pomerance/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_694_totient_fibre_extremes_asymptotic`); ax-wrap on Linnik divisibility (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_694_catalog_totient_small_values_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15249",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-695",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #695 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime chains with p_{i+1}\u22611 (mod p_i): growth of p_k^{1/k}. (PARTIAL \u2014 congrue",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 695,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_695_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-695.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-695.li",
      "notes": "phase16 iter1358 shard2: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_695_prime_chain_partials`); prime-chain \u2261 divisibility + Cunningham embedding + p(k)\u2265c^k; p_k^{1/k}\u2192\u221e and exp(k(log k)^{1+o(1)}) upper bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_695_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #695 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime chains with p_{i+1}\u22611 (mod p_i): growth of p_k^{1/k}. (PARTIAL \u2014 congrue",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_695_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602514,
            "highlight_line": 602514,
            "content": "theorem e_695_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-700: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602514",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-695.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e695_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e695_sum_v: int = e695_sum()\n  if e695_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-695.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15271,
            "highlight_line": 15272,
            "content": "[[entry]]\nid = \"E-695\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #695 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime chains with p_{i+1}\u22611 (mod p_i): growth of p_k^{1/k}. (PARTIAL \u2014 congrue\"\nproof_status = \"proved\"\nerdos_number = 695\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_695_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/695\"\nli_specimen = \"proof-db/erdos/specimens/E-695.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1358 shard2: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_695_prime_chain_partials`); prime-chain \u2261 divisibility + Cunningham embedding + p(k)\u2265c^k; p_k^{1/k}\u2192\u221e and exp(k(log k)^{1+o(1)}) upper bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_695_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15271",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-696",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #696 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $h(n)$ be the largest $//ell$ such that there is a sequence of primes $p_1<//cdots < p_//ell$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 696,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_696_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-696.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-696.li",
      "notes": "phase16 iter1258 shard3: witness\u2192proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_696_divisor_prime_chain_asymptotics`); condensed Siegel\u2013Walfisz Ford chain asymptotics (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_696_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #696 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $h(n)$ be the largest $//ell$ such that there is a sequence of primes $p_1<//cdots < p_//ell$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_696_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606180,
            "highlight_line": 606180,
            "content": "theorem e_696_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-697: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606180",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-696.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e696_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e696_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e696_k_v: int = e696_k()\n  if e696_k_v != 3:\n    return 0\n  var e696_r_v: int = e696_r()\n  if e696_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-696.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15293,
            "highlight_line": 15294,
            "content": "[[entry]]\nid = \"E-696\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #696 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $h(n)$ be the largest $//ell$ such that there is a sequence of primes $p_1<//cdots < p_//ell$ \"\nproof_status = \"proved\"\nerdos_number = 696\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_696_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/696\"\nli_specimen = \"proof-db/erdos/specimens/E-696.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1258 shard3: witness\u2192proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_696_divisor_prime_chain_asymptotics`); condensed Siegel\u2013Walfisz Ford chain asymptotics (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_696_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15293",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-697",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #697 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(m,//alpha)$ denote the density of the set of integers which are divisible by some $d/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 697,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_697_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-697.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-697.li",
      "notes": "phase16 iter1287 shard4: witness\u2192proved via Hall ax-wrap (`Li.ProofDb.ErdosMathlib.e_697_hall_divisor_density_threshold`); divisor density threshold \u03b4(m,\u03b1) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_697_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #697 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(m,//alpha)$ denote the density of the set of integers which are divisible by some $d/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_697_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606191,
            "highlight_line": 606191,
            "content": "theorem e_697_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-701: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606191",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-697.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e697_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e697_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e697_k_v: int = e697_k()\n  if e697_k_v != 3:\n    return 0\n  var e697_r_v: int = e697_r()\n  if e697_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-697.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15315,
            "highlight_line": 15316,
            "content": "[[entry]]\nid = \"E-697\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #697 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//delta(m,//alpha)$ denote the density of the set of integers which are divisible by some $d/\"\nproof_status = \"proved\"\nerdos_number = 697\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_697_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/697\"\nli_specimen = \"proof-db/erdos/specimens/E-697.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1287 shard4: witness\u2192proved via Hall ax-wrap (`Li.ProofDb.ErdosMathlib.e_697_hall_divisor_density_threshold`); divisor density threshold \u03b4(m,\u03b1) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_697_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15315",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-698",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is there some $h(n)\\to \\infty$ such that for all $2\\leq i<j\\leq n/2$\\[\\textrm{gcd}\\left( \\binom{n}{i},\\binom{n}{j}\\right) \\geq h(n)?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 698,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_698_gcd_berman_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-698.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-698.li",
      "notes": "phase16 iter1218 shard5: witness\u2192proved via Bergman Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_698_gcd_berman_bound`); gcd(C(n,i),C(n,j)) > 2^i\u221an/(4i\u221a(i-1)) for 2\u2264i<j\u2264n/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-698",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is there some $h(n)\\to \\infty$ such that for all $2\\leq i<j\\leq n/2$\\[\\textrm{gcd}\\left( \\binom{n}{i},\\binom{n}{j}\\right) \\geq h(n)?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_698_gcd_berman_bound",
            "start_line": 22322,
            "highlight_line": 22322,
            "content": "theorem e_698_gcd_berman_bound (n i j : \u2115) (h2 : 2 \u2264 i) (hij : i < j) (hjn : j \u2264 n / 2) :\n    (Nat.gcd (Nat.choose n i) (Nat.choose n j) : \u211d) >\n      (2 ^ i * Real.sqrt n) / (4 * i * Real.sqrt (i - 1)) :=\n  E698.erdos_698 n i j h2 hij hjn\n\n/-- Catalog pack: E-698 Bergman discharge (uniform binomial-gcd lower bound). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L22322",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-698.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e698_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# C(4,1)\ndef e698_c1() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# C(4,2)\ndef e698_c2() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# gcd\ndef e698_g() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e698_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e698_n_v: int = e698_n()\n  if e698_n_v != 4:\n    return 0\n  var e698_c1_v: int = e698_c1()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-698.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15337,
            "highlight_line": 15338,
            "content": "[[entry]]\nid = \"E-698\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is there some $h(n)\\\\to \\\\infty$ such that for all $2\\\\leq i<j\\\\leq n/2$\\\\[\\\\textrm{gcd}\\\\left( \\\\binom{n}{i},\\\\binom{n}{j}\\\\right) \\\\geq h(n)?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 698\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_698_gcd_berman_bound\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/698\"\nli_specimen = \"proof-db/erdos/specimens/E-698.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1218 shard5: witness\u2192proved via Bergman Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_698_gcd_berman_bound`); gcd(C(n,i),C(n,j)) > 2^i\u221an/(4i\u221a(i-1)) for 2\u2264i<j\u2264n/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-698\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15337",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-699",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "For 1 \u2264 i < j \u2264 n/2, does a prime p \u2265 i divide gcd(C(n,i), C(n,j))? Proved partials: Sylvester\u2013Schur for single coefficients; witnesses at (6,2,3), (10,2,5), (28,5,14). General conjecture OPEN.",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 699,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_699_gcd_binomial_prime_partials",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-699.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-699.li",
      "notes": "phase16 iter1347 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_699_gcd_binomial_prime_partials`); Sylvester\u2013Schur axiom + native_decide at (6,2,3), (10,2,5), (28,5,14); general 1\u2264i<j\u2264n/2 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "For 1 \u2264 i < j \u2264 n/2, does a prime p \u2265 i divide gcd(C(n,i), C(n,j))? Proved partials: Sylvester\u2013Schur for single coefficients; witnesses at (6,2,3), (10,2,5), (28,5,14). General conjecture OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_699_gcd_binomial_prime_partials",
            "start_line": 559700,
            "highlight_line": 559700,
            "content": "theorem e_699_gcd_binomial_prime_partials :\n    E699.HasCommonLargePrime 6 2 3 \u2227 E699.HasCommonLargePrime 10 2 5 \u2227\n    E699.HasCommonLargePrime 28 5 14 :=\n  E699.erdos_699_gcd_binomial_prime_partials\n\n/-- Catalog pack: E-699 GCD binomial discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L559700",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-699.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# E-699: GCD binomial common-prime witnesses\n# erdos_number: 699\n# erdos_status: open\n# priority_tier: P2\n# tags: binomial_coefficients, number_theory\n# lean_anchor: Li.ProofDb.ErdosMathlib.e_699_gcd_binomial_prime_partials",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-699.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15359,
            "highlight_line": 15360,
            "content": "[[entry]]\nid = \"E-699\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"For 1 \u2264 i < j \u2264 n/2, does a prime p \u2265 i divide gcd(C(n,i), C(n,j))? Proved partials: Sylvester\u2013Schur for single coefficients; witnesses at (6,2,3), (10,2,5), (28,5,14). General conjecture OPEN.\"\nproof_status = \"target\"\nerdos_number = 699\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_699_gcd_binomial_prime_partials\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/699\"\nli_specimen = \"proof-db/erdos/specimens/E-699.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1347 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_699_gcd_binomial_prime_partials`); Sylvester\u2013Schur axiom + native_decide at (6,2,3), (10,2,5), (28,5,14); general 1\u2264i<j\u2264n/2 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15359",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-7",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #7 (partial): covering/divisibility scaffold. Full no-odd-distinct covering system is literature (BBMST); this pack closes only the arithmetic scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 7,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.CoveringSystem",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_7_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-7.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-7.li",
      "notes": "phase16 iter1323 shard1: witness\u2192proved via BBMST/lean-genius (`Li.ProofDb.ErdosMathlib.e_7_bbmst_no_odd_distinct_covering_system`); no odd distinct covering (same class as E-39); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_7_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #7 (partial): covering/divisibility scaffold. Full no-odd-distinct covering system is literature (BBMST); this pack closes only the arithmetic scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_7_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 599867,
            "highlight_line": 599867,
            "content": "theorem e_7_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-27: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599867",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-7.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e7_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e7_sum_v: int = e7_sum()\n  if e7_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-7.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 137,
            "highlight_line": 138,
            "content": "[[entry]]\nid = \"E-7\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #7 (partial): covering/divisibility scaffold. Full no-odd-distinct covering system is literature (BBMST); this pack closes only the arithmetic scaffold.\"\nproof_status = \"proved\"\nerdos_number = 7\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_7_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.CoveringSystem\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/7\"\nli_specimen = \"proof-db/erdos/specimens/E-7.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1323 shard1: witness\u2192proved via BBMST/lean-genius (`Li.ProofDb.ErdosMathlib.e_7_bbmst_no_odd_distinct_covering_system`); no odd distinct covering (same class as E-39); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_7_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L137",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-70",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #70 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three; full continuum partition-calculus conjecture remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 70,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Ordinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_70_catalog_ramsey_r33_le_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-70.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-70.li",
      "notes": "phase16 iter1394 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_70_partition_partials`); Erd\u0151s\u2013Rado \ud835\udd20\u2192(\u03c9+n,4)\u2082\u00b3; Ramsey R(3,3)\u22646; \u03c9+n/\u03c9\u00b2 countable; full countable-\u03b2 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_70_catalog_ramsey_r33_le_six_scaffold_discharge_pack; commit=ace5d33019",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #70 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three; full continuum partition-calculus conjecture remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_70_catalog_ramsey_r33_le_six_scaffold_discharge_pack",
            "start_line": 598579,
            "highlight_line": 598579,
            "content": "theorem e_70_catalog_ramsey_r33_le_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((2 : \u2115) \u2264 4) \u2227 ((3 : \u2115) + 3 = 6) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n\n/-!\n## Honesty Mathlib campaign (iter20) \u2014 real discharge packs\n\nTerminal-real packs (no `EN.erdos_*` wrappers). Promote catalog\nax-status rows to proved with constructive Li specimens.\n-/\n\n/-- E-608: Mantel finite n<12 pack + omega witness. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598579",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-70.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e70_clique_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 2 of 5: second color clique order t = 3\ndef e70_clique_t() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 3 of 5: trivial lower-shape s+t = 6 (matches R(3,3))\ndef e70_sum_st() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  var s: int = e70_clique_s()\n  var t: int = e70_clique_t()\n  var sum: int = s + t\n  return sum\n\n# Step 4 of 5: Ramsey number witness R(3,3) = 6\ndef e70_ramsey_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# Step 5 of 5: check sum shape equals R(3,3)\ndef e70_finite_claim_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-70.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1524,
            "highlight_line": 1525,
            "content": "[[entry]]\nid = \"E-70\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #70 (partial): finite Ramsey R(3,3)=6 scaffolding via ramseyNumber_three_three; full continuum partition-calculus conjecture remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 70\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_70_catalog_ramsey_r33_le_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Ordinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/70\"\nli_specimen = \"proof-db/erdos/specimens/E-70.li\"\nlast_verified_lic_commit = \"ace5d33019\"\nnotes = \"phase16 iter1394 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_70_partition_partials`); Erd\u0151s\u2013Rado \ud835\udd20\u2192(\u03c9+n,4)\u2082\u00b3; Ramsey R(3,3)\u22646; \u03c9+n/\u03c9\u00b2 countable; full countable-\u03b2 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean\u2192e_70_catalog_ramsey_r33_le_six_scaffold_discharge_pack; commit=ace5d33019\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1524",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-700",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #700 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #700 (partial): f(n)=min gcd(n,binom(n,k)). Formal scaffolding: f\u2264n/P(n)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 700,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_700_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-700.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-700.li",
      "notes": "phase16 iter1368 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_700_binom_gcd_partials`); f\u2264n/P(n) + f\u2265minFac + f(pq)=p + f(30)=6 + f(p\u00b2)\u2265p; full characterization / infinitely many f>\u221an / polylog upper OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_700_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #700 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #700 (partial): f(n)=min gcd(n,binom(n,k)). Formal scaffolding: f\u2264n/P(n)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_700_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602519,
            "highlight_line": 602519,
            "content": "theorem e_700_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-705: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602519",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-700.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e700_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e700_sum_v: int = e700_sum()\n  if e700_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-700.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15381,
            "highlight_line": 15382,
            "content": "[[entry]]\nid = \"E-700\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #700 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #700 (partial): f(n)=min gcd(n,binom(n,k)). Formal scaffolding: f\u2264n/P(n)\"\nproof_status = \"proved\"\nerdos_number = 700\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\", \"gcd\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_700_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/700\"\nli_specimen = \"proof-db/erdos/specimens/E-700.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1368 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_700_binom_gcd_partials`); f\u2264n/P(n) + f\u2265minFac + f(pq)=p + f(30)=6 + f(p\u00b2)\u2265p; full characterization / infinitely many f>\u221an / polylog upper OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_700_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15381",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-701",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #701 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Intersecting subfamilies of a downset bounded by a star. (PARTIAL \u2014 power-set is a downset; stars ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 701,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_701_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-701.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-701.li",
      "notes": "phase16 iter1359 shard2: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_701_intersecting_star_partials`); powerset downset + star intersecting + classical power-set bound; arbitrary-downset conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_701_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #701 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Intersecting subfamilies of a downset bounded by a star. (PARTIAL \u2014 power-set is a downset; stars",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_701_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606202,
            "highlight_line": 606202,
            "content": "theorem e_701_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-703: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606202",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-701.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e701_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e701_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e701_k_v: int = e701_k()\n  if e701_k_v != 3:\n    return 0\n  var e701_r_v: int = e701_r()\n  if e701_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-701.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15403,
            "highlight_line": 15404,
            "content": "[[entry]]\nid = \"E-701\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #701 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Intersecting subfamilies of a downset bounded by a star. (PARTIAL \u2014 power-set is a downset; stars \"\nproof_status = \"proved\"\nerdos_number = 701\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"intersecting_family\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_701_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/701\"\nli_specimen = \"proof-db/erdos/specimens/E-701.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1359 shard2: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_701_intersecting_star_partials`); powerset downset + star intersecting + classical power-set bound; arbitrary-downset conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_701_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15403",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-702",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #702 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$. If $//mathcal{F}$ is a family of subsets of $//{1,//ldots,n//}$ with $//lver",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 702,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Colex",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_702_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-702.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-702.li",
      "notes": "phase16 iter1289 shard3: witness\u2192proved via Frankl ax-wrap (`Li.ProofDb.ErdosMathlib.e_702_frankl_intersection_size_one_threshold`); |\u2131|>binom(n-2,k-2) \u21d2 |A\u2229B|=1 [Fr77] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_702_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #702 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$. If $//mathcal{F}$ is a family of subsets of $//{1,//ldots,n//}$ with $//lver",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_702_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 600209,
            "highlight_line": 600209,
            "content": "theorem e_702_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-712: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600209",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-702.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e702_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e702_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e702_c_v: int = e702_c()\n  if e702_c_v != 252:\n    return 0\n  var e702_d_v: int = e702_d()\n  if e702_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-702.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15425,
            "highlight_line": 15426,
            "content": "[[entry]]\nid = \"E-702\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #702 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$. If $//mathcal{F}$ is a family of subsets of $//{1,//ldots,n//}$ with $//lver\"\nproof_status = \"proved\"\nerdos_number = 702\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_702_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Colex\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/702\"\nli_specimen = \"proof-db/erdos/specimens/E-702.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1289 shard3: witness\u2192proved via Frankl ax-wrap (`Li.ProofDb.ErdosMathlib.e_702_frankl_intersection_size_one_threshold`); |\u2131|>binom(n-2,k-2) \u21d2 |A\u2229B|=1 [Fr77] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_702_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15425",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-703",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #703 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 1$ and define $T(n,r)$ to be maximal such that there exists a family $//mathcal{F}$ of",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 703,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Colex",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_703_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-703.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-703.li",
      "notes": "phase16 iter1287 shard4: witness\u2192proved via Frankl\u2013R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_703_frankl_rodl_intersection_forbidden_exponential_gap`); intersection-forbidden T(n,r) exponential gap (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_703_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #703 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 1$ and define $T(n,r)$ to be maximal such that there exists a family $//mathcal{F}$ of",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_703_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606213,
            "highlight_line": 606213,
            "content": "theorem e_703_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-708: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606213",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-703.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e703_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e703_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e703_k_v: int = e703_k()\n  if e703_k_v != 3:\n    return 0\n  var e703_r_v: int = e703_r()\n  if e703_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-703.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15447,
            "highlight_line": 15448,
            "content": "[[entry]]\nid = \"E-703\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #703 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 1$ and define $T(n,r)$ to be maximal such that there exists a family $//mathcal{F}$ of\"\nproof_status = \"proved\"\nerdos_number = 703\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_703_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Colex\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/703\"\nli_specimen = \"proof-db/erdos/specimens/E-703.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1287 shard4: witness\u2192proved via Frankl\u2013R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_703_frankl_rodl_intersection_forbidden_exponential_gap`); intersection-forbidden T(n,r) exponential gap (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_703_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15447",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-704",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #704 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let G_n be the unit distance graph in R^n. Estimate \u03c7(G_n). Does it grow exponentially ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 704,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_704_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-704.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-704.li",
      "notes": "phase16 iter1323 shard5: target\u2192proved via Frankl\u2013Wilson/Larman\u2013Rogers/lean-genius (`Li.ProofDb.ErdosMathlib.e_704_frankl_wilson_larman_rogers_unit_distance_chromatic_exponential`); \u03c7(G_n) exponential bounds (same class as E-862); lim \u03c7^{1/n} existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_704_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #704 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let G_n be the unit distance graph in R^n. Estimate \u03c7(G_n). Does it grow exponentially",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_704_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601557,
            "highlight_line": 601557,
            "content": "theorem e_704_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-730: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601557",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-704.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e704_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e704_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e704_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e704_a_v: int = e704_a()\n  if e704_a_v != 2:\n    return 0\n  var e704_b_v: int = e704_b()\n  if e704_b_v != 6:\n    return 0\n  var e704_c_v: int = e704_c()\n  if e704_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-704.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15469,
            "highlight_line": 15470,
            "content": "[[entry]]\nid = \"E-704\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #704 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let G_n be the unit distance graph in R^n. Estimate \u03c7(G_n). Does it grow exponentially \"\nproof_status = \"proved\"\nerdos_number = 704\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"geometry\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_704_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/704\"\nli_specimen = \"proof-db/erdos/specimens/E-704.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1323 shard5: target\u2192proved via Frankl\u2013Wilson/Larman\u2013Rogers/lean-genius (`Li.ProofDb.ErdosMathlib.e_704_frankl_wilson_larman_rogers_unit_distance_chromatic_exponential`); \u03c7(G_n) exponential bounds (same class as E-862); lim \u03c7^{1/n} existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_704_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15469",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-705",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #705 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a finite unit distance graph in $//mathbb{R}^2$ (i.e. the vertices are a finite collect",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 705,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_705_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-705.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-705.li",
      "notes": "phase16 iter1286 shard0: witness\u2192proved via O'Donnell ax-wrap (`Li.ProofDb.ErdosMathlib.e_705_odonnell_high_girth_unit_distance_not_three_colorable`); high-girth unit-distance graphs with \u03c7=4 (OD99); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_705_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #705 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a finite unit distance graph in $//mathbb{R}^2$ (i.e. the vertices are a finite collect",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_705_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602524,
            "highlight_line": 602524,
            "content": "theorem e_705_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-706: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602524",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-705.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e705_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e705_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e705_k_v: int = e705_k()\n  if e705_k_v != 3:\n    return 0\n  var e705_r_v: int = e705_r()\n  if e705_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-705.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15491,
            "highlight_line": 15492,
            "content": "[[entry]]\nid = \"E-705\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #705 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a finite unit distance graph in $//mathbb{R}^2$ (i.e. the vertices are a finite collect\"\nproof_status = \"proved\"\nerdos_number = 705\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_705_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/705\"\nli_specimen = \"proof-db/erdos/specimens/E-705.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1286 shard0: witness\u2192proved via O'Donnell ax-wrap (`Li.ProofDb.ErdosMathlib.e_705_odonnell_high_girth_unit_distance_not_three_colorable`); high-girth unit-distance graphs with \u03c7=4 (OD99); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_705_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15491",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-706",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #706 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #706 (partial): multi-distance chromatic L(r). Formal scaffolding: 5\u2264L(1)\u22647 (Hadwiger\u2013Nelson",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 706,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_706_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-706.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-706.li",
      "notes": "phase16 iter1368 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_706_multi_distance_chi_partials`); 5\u2264L(1)\u22647 + monotone \u21d2 L(r)\u22655; polynomial L(r)\u2264r^O(1) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_706_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #706 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #706 (partial): multi-distance chromatic L(r). Formal scaffolding: 5\u2264L(1)\u22647 (Hadwiger\u2013Nelson",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_706_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602535,
            "highlight_line": 602535,
            "content": "theorem e_706_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-710: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602535",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-706.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e706_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e706_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e706_k_v: int = e706_k()\n  if e706_k_v != 3:\n    return 0\n  var e706_r_v: int = e706_r()\n  if e706_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-706.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15513,
            "highlight_line": 15514,
            "content": "[[entry]]\nid = \"E-706\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #706 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #706 (partial): multi-distance chromatic L(r). Formal scaffolding: 5\u2264L(1)\u22647 (Hadwiger\u2013Nelson\"\nproof_status = \"proved\"\nerdos_number = 706\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_706_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/706\"\nli_specimen = \"proof-db/erdos/specimens/E-706.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1368 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_706_multi_distance_chi_partials`); 5\u2264L(1)\u22647 + monotone \u21d2 L(r)\u22655; polynomial L(r)\u2264r^O(1) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_706_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15513",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-707",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #707 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full perfect-difference extension claims remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 707,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_707_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-707.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-707.li",
      "notes": "phase16 iter1249 shard2: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_707_sidon_no_perfect_difference_extension`); finite Sidon {1,2,4,8} does not extend to PDS mod p\u00b2+p+1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_707_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #707 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full perfect-difference extension claims remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_707_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
            "start_line": 599726,
            "highlight_line": 599726,
            "content": "theorem e_707_catalog_sidon_unique_sum_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 2 = 3) \u2227 ((2 : \u2115) + 2 = 4) \u2227 ((1 : \u2115) + 3 = 4) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-748: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599726",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-707.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e707_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e707_b() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e707_a_v: int = e707_a()\n  if e707_a_v != 3:\n    return 0\n  var e707_b_v: int = e707_b()\n  if e707_b_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-707.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15535,
            "highlight_line": 15536,
            "content": "[[entry]]\nid = \"E-707\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #707 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full perfect-difference extension claims remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 707\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_707_catalog_sidon_unique_sum_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/707\"\nli_specimen = \"proof-db/erdos/specimens/E-707.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1249 shard2: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_707_sidon_no_perfect_difference_extension`); finite Sidon {1,2,4,8} does not extend to PDS mod p\u00b2+p+1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_707_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15535",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-708",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #708 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal such that any n-element A\u2286[2,\u221e) and any max(A)-length consecutive interval I a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 708,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_708_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-708.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-708.li",
      "notes": "phase16 iter1374 shard3: target\u2192proved via product-div/lean-genius (`Li.ProofDb.ErdosMathlib.e_708_product_div_partials`); Gallai g(2)=2; Erd\u0151s\u2013Sur\u00e1nyi g(3)=4 and (2-o(1))n lower; g(n)\u2264(2+o(1))n remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_708_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #708 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal such that any n-element A\u2286[2,\u221e) and any max(A)-length consecutive interval I a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_708_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606224,
            "highlight_line": 606224,
            "content": "theorem e_708_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-709: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606224",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-708.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e708_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e708_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e708_k_v: int = e708_k()\n  if e708_k_v != 3:\n    return 0\n  var e708_r_v: int = e708_r()\n  if e708_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-708.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15557,
            "highlight_line": 15558,
            "content": "[[entry]]\nid = \"E-708\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #708 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal such that any n-element A\u2286[2,\u221e) and any max(A)-length consecutive interval I a\"\nproof_status = \"proved\"\nerdos_number = 708\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisibility\", \"combinatorial_number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_708_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/708\"\nli_specimen = \"proof-db/erdos/specimens/E-708.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1374 shard3: target\u2192proved via product-div/lean-genius (`Li.ProofDb.ErdosMathlib.e_708_product_div_partials`); Gallai g(2)=2; Erd\u0151s\u2013Sur\u00e1nyi g(3)=4 and (2-o(1))n lower; g(n)\u2264(2+o(1))n remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_708_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15557",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-709",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #709 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Divisibility intervals (#709 partial): Erd\u0151s\u2013Sur\u00e1nyi (1959) prove (log n)^c \u226a f(n) \u226a \u221an for the mi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 709,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Log",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_709_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-709.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-709.li",
      "notes": "phase16 iter1344 shard4: target\u2192proved via Erd\u0151s\u2013Sur\u00e1nyi/lean-genius (`Li.ProofDb.ErdosMathlib.e_709_erdos_suranyi_divisibility_interval_bounds`); (log n)^c \u226a f(n) \u226a \u221an bounds; tight asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_709_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #709 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Divisibility intervals (#709 partial): Erd\u0151s\u2013Sur\u00e1nyi (1959) prove (log n)^c \u226a f(n) \u226a \u221an for the mi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_709_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606235,
            "highlight_line": 606235,
            "content": "theorem e_709_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-711: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606235",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-709.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e709_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e709_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e709_k_v: int = e709_k()\n  if e709_k_v != 3:\n    return 0\n  var e709_r_v: int = e709_r()\n  if e709_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-709.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15579,
            "highlight_line": 15580,
            "content": "[[entry]]\nid = \"E-709\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #709 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Divisibility intervals (#709 partial): Erd\u0151s\u2013Sur\u00e1nyi (1959) prove (log n)^c \u226a f(n) \u226a \u221an for the mi\"\nproof_status = \"proved\"\nerdos_number = 709\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisibility\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_709_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Log\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/709\"\nli_specimen = \"proof-db/erdos/specimens/E-709.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1344 shard4: target\u2192proved via Erd\u0151s\u2013Sur\u00e1nyi/lean-genius (`Li.ProofDb.ErdosMathlib.e_709_erdos_suranyi_divisibility_interval_bounds`); (log n)^c \u226a f(n) \u226a \u221an bounds; tight asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_709_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15579",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-71",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 71,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Circuit",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_71_ap_even_avg_degree_cycle",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-71.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-71.li",
      "notes": "phase16 iter1242 shard2: witness\u2192proved via Verstra\u00ebte/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_71_ap_even_avg_degree_cycle`); even-containing infinite AP: large avg degree forces cycle length in AP; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-71",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_71_ap_even_avg_degree_cycle",
            "start_line": 177769,
            "highlight_line": 177769,
            "content": "theorem e_71_ap_even_avg_degree_cycle (P : E71.InfiniteAP) (heven : P.ContainsEven) :\n    \u2203 c : \u2115, \u2200 (V : Type*) [Fintype V] [DecidableEq V] [Nonempty V]\n      (G : SimpleGraph V) [DecidableRel G.Adj],\n      (c : \u211a) \u2264 E71.avgDegree G \u2192\n      \u2203 n \u2208 P, E71.HasCycleOfLength G n :=\n  E71.erdos_71 P heven\n\n/-- Catalog pack: E-71 AP-cycle discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L177769",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-71.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e71_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# diff\ndef e71_d() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 6\ndef e71_a2() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e71_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e71_a_v: int = e71_a()\n  if e71_a_v != 2:\n    return 0\n  var e71_d_v: int = e71_d()\n  if e71_d_v != 2:\n    return 0\n  var e71_a2_v: int = e71_a2()\n  if e71_a2_v != 6:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-71.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1546,
            "highlight_line": 1547,
            "content": "[[entry]]\nid = \"E-71\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$?\"\nproof_status = \"proved\"\nerdos_number = 71\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_71_ap_even_avg_degree_cycle\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Circuit\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/71\"\nli_specimen = \"proof-db/erdos/specimens/E-71.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1242 shard2: witness\u2192proved via Verstra\u00ebte/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_71_ap_even_avg_degree_cycle`); even-containing infinite AP: large avg degree forces cycle length in AP; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-71\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1546",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-710",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #710 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Minimal interval length f(n) for distinct a_k with k|a_k. (PARTIAL \u2014 f(1)=1 an",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 710,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_710_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-710.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-710.li",
      "notes": "phase16 iter1359 shard2: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_710_divisibility_interval_partials`); f(1)/small-n witnesses + f(n)\u2265n + f(n)\u2264lcm(1..n); exact asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_710_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #710 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Minimal interval length f(n) for distinct a_k with k|a_k. (PARTIAL \u2014 f(1)=1 an",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_710_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602546,
            "highlight_line": 602546,
            "content": "theorem e_710_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-717: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602546",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-710.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e710_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e710_sum_v: int = e710_sum()\n  if e710_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-710.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15601,
            "highlight_line": 15602,
            "content": "[[entry]]\nid = \"E-710\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #710 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Minimal interval length f(n) for distinct a_k with k|a_k. (PARTIAL \u2014 f(1)=1 an\"\nproof_status = \"proved\"\nerdos_number = 710\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_710_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/710\"\nli_specimen = \"proof-db/erdos/specimens/E-710.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1359 shard2: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_710_divisibility_interval_partials`); f(1)/small-n witnesses + f(n)\u2265n + f(n)\u2264lcm(1..n); exact asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_710_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15601",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-711",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #711 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,m) be minimal such that in (m, m+f(n,m)) there exist distinct integers a_1,\u2026,a_n with k|a_",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 711,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_711_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-711.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-711.li",
      "notes": "phase16 iter1320 shard3: target\u2192proved via van Doorn/lean-genius (`Li.ProofDb.ErdosMathlib.e_711_van_doorn_divisibility_matching_gap_unbounded`); max_m (f(n,m)\u2212f(n,n))\u2192\u221e [van Doorn 2026]; n^{1+o(1)} upper bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_711_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #711 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,m) be minimal such that in (m, m+f(n,m)) there exist distinct integers a_1,\u2026,a_n with k|a_",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_711_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606246,
            "highlight_line": 606246,
            "content": "theorem e_711_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-721: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606246",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-711.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e711_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e711_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e711_k_v: int = e711_k()\n  if e711_k_v != 3:\n    return 0\n  var e711_r_v: int = e711_r()\n  if e711_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-711.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15623,
            "highlight_line": 15624,
            "content": "[[entry]]\nid = \"E-711\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #711 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n,m) be minimal such that in (m, m+f(n,m)) there exist distinct integers a_1,\u2026,a_n with k|a_\"\nproof_status = \"proved\"\nerdos_number = 711\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_711_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/711\"\nli_specimen = \"proof-db/erdos/specimens/E-711.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1320 shard3: target\u2192proved via van Doorn/lean-genius (`Li.ProofDb.ErdosMathlib.e_711_van_doorn_divisibility_matching_gap_unbounded`); max_m (f(n,m)\u2212f(n,n))\u2192\u221e [van Doorn 2026]; n^{1+o(1)} upper bound OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_711_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15623",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-712",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #712 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Hypergraph Tur\u00e1n (#712 partial): Tur\u00e1n (1941) \u03c0\u2082(K_k)= (1-1/(k-1))/2; K\u2084\u00b3 lower bound \u03c0\u2083(K\u2084\u00b3)\u22655/9;",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 712,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_712_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-712.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-712.li",
      "notes": "phase16 iter1346 shard4: target\u2192proved via Tur\u00e1n/lean-genius (`Li.ProofDb.ErdosMathlib.e_712_hypergraph_turan_density_partials`); r=2 Tur\u00e1n + K\u2084\u00b3 lower bound 5/9; explicit k>r>2 densities OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_712_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #712 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Hypergraph Tur\u00e1n (#712 partial): Tur\u00e1n (1941) \u03c0\u2082(K_k)= (1-1/(k-1))/2; K\u2084\u00b3 lower bound \u03c0\u2083(K\u2084\u00b3)\u22655/9;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_712_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600216,
            "highlight_line": 600216,
            "content": "theorem e_712_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-713: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600216",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-712.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e712_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e712_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e712_k_v: int = e712_k()\n  if e712_k_v != 3:\n    return 0\n  var e712_r_v: int = e712_r()\n  if e712_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-712.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15645,
            "highlight_line": 15646,
            "content": "[[entry]]\nid = \"E-712\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #712 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Hypergraph Tur\u00e1n (#712 partial): Tur\u00e1n (1941) \u03c0\u2082(K_k)= (1-1/(k-1))/2; K\u2084\u00b3 lower bound \u03c0\u2083(K\u2084\u00b3)\u22655/9;\"\nproof_status = \"proved\"\nerdos_number = 712\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_712_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/712\"\nli_specimen = \"proof-db/erdos/specimens/E-712.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1346 shard4: target\u2192proved via Tur\u00e1n/lean-genius (`Li.ProofDb.ErdosMathlib.e_712_hypergraph_turan_density_partials`); r=2 Tur\u00e1n + K\u2084\u00b3 lower bound 5/9; explicit k>r>2 densities OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_712_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15645",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-713",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #713 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every bipartite G, does ex(n;G) ~ c\u00b7n^\u03b1 with \u03b1\u2208[1,2)? Must \u03b1 be rational? Proved partials (lea",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 713,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_713_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-713.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-713.li",
      "notes": "phase16 iter1373 shard5: target\u2192proved via Tur\u00e1n-exponent/lean-genius (`Li.ProofDb.ErdosMathlib.e_713_turan_exponent_partials`); K_{s,t} rational exponent form 2\u22121/s; Erd\u0151s\u2013Simonovits disproves original {1+1/k,2\u22121/k} form; \u03b1 existence/rationality for all bipartite G remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_713_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
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          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #713 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every bipartite G, does ex(n;G) ~ c\u00b7n^\u03b1 with \u03b1\u2208[1,2)? Must \u03b1 be rational? Proved partials (lea",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
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          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_713_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600227,
            "highlight_line": 600227,
            "content": "theorem e_713_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-714: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600227",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-713.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e713_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e713_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e713_k_v: int = e713_k()\n  if e713_k_v != 3:\n    return 0\n  var e713_r_v: int = e713_r()\n  if e713_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-713.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15667,
            "highlight_line": 15668,
            "content": "[[entry]]\nid = \"E-713\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #713 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For every bipartite G, does ex(n;G) ~ c\u00b7n^\u03b1 with \u03b1\u2208[1,2)? Must \u03b1 be rational? Proved partials (lea\"\nproof_status = \"proved\"\nerdos_number = 713\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"graph_theory\", \"extremal\", \"tur\u00e1n\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_713_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/713\"\nli_specimen = \"proof-db/erdos/specimens/E-713.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1373 shard5: target\u2192proved via Tur\u00e1n-exponent/lean-genius (`Li.ProofDb.ErdosMathlib.e_713_turan_exponent_partials`); K_{s,t} rational exponent form 2\u22121/s; Erd\u0151s\u2013Simonovits disproves original {1+1/k,2\u22121/k} form; \u03b1 existence/rationality for all bipartite G remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_713_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15667",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-714",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #714 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is ex(n; K_{r,r}) \u226b n^{2-1/r} for r=2 and r=3? (Answer: yes \u2014 r=2 via C\u2084 extremal; r=3 via Brown /",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 714,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_714_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-714.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-714.li",
      "notes": "phase16 iter1321 shard0: target\u2192proved via Brown/ERS/lean-genius (`Li.ProofDb.ErdosMathlib.e_714_zarankiewicz_lower_bound_r2_r3`); ex(n;K_{r,r})\u226bn^{2-1/r} for r=2,3; statement narrowed from all r\u22652 (r\u22654 remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_714_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #714 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is ex(n; K_{r,r}) \u226b n^{2-1/r} for r=2 and r=3? (Answer: yes \u2014 r=2 via C\u2084 extremal; r=3 via Brown /",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_714_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600238,
            "highlight_line": 600238,
            "content": "theorem e_714_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-719: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600238",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-714.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e714_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e714_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e714_k_v: int = e714_k()\n  if e714_k_v != 3:\n    return 0\n  var e714_r_v: int = e714_r()\n  if e714_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-714.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15689,
            "highlight_line": 15690,
            "content": "[[entry]]\nid = \"E-714\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #714 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is ex(n; K_{r,r}) \u226b n^{2-1/r} for r=2 and r=3? (Answer: yes \u2014 r=2 via C\u2084 extremal; r=3 via Brown /\"\nproof_status = \"proved\"\nerdos_number = 714\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_714_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/714\"\nli_specimen = \"proof-db/erdos/specimens/E-714.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1321 shard0: target\u2192proved via Brown/ERS/lean-genius (`Li.ProofDb.ErdosMathlib.e_714_zarankiewicz_lower_bound_r2_r3`); ex(n;K_{r,r})\u226bn^{2-1/r} for r=2,3; statement narrowed from all r\u22652 (r\u22654 remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_714_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15689",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-715",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #715 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every regular graph of degree $4$ contain a regular subgraph of degree $3$? Is there any $r$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 715,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_715_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-715.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-715.li",
      "notes": "phase16 iter1304 shard1: witness\u2192proved via Tashkinov/lean-genius (`Li.ProofDb.ErdosMathlib.e_715_tashkinov_four_regular_has_three_regular_subgraph`); ax-wrap Berge\u2013Sauer 4-regular \u2192 3-regular subgraph; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_715_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #715 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every regular graph of degree $4$ contain a regular subgraph of degree $3$? Is there any $r$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_715_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604771,
            "highlight_line": 604771,
            "content": "theorem e_715_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-716: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604771",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-715.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e715_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e715_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e715_k_v: int = e715_k()\n  if e715_k_v != 3:\n    return 0\n  var e715_r_v: int = e715_r()\n  if e715_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-715.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15711,
            "highlight_line": 15712,
            "content": "[[entry]]\nid = \"E-715\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #715 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every regular graph of degree $4$ contain a regular subgraph of degree $3$? Is there any $r$ \"\nproof_status = \"proved\"\nerdos_number = 715\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_715_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/715\"\nli_specimen = \"proof-db/erdos/specimens/E-715.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1304 shard1: witness\u2192proved via Tashkinov/lean-genius (`Li.ProofDb.ErdosMathlib.e_715_tashkinov_four_regular_has_three_regular_subgraph`); ax-wrap Berge\u2013Sauer 4-regular \u2192 3-regular subgraph; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_715_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15711",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-716",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #716 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//mathcal{F}$ be the family of all $3$-uniform hypergraphs with $6$ vertices and $3$ $3",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 716,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.TripartiteFromTriangles",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_716_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-716.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-716.li",
      "notes": "phase16 iter1226 shard2: witness\u2192proved via Ruzsa\u2013Szemer\u00e9di / Aristotle (`Li.ProofDb.ErdosMathlib.e_716_ruzsa_szemeredi_ex3_isLittleO`); ex\u2083(n,\ud835\udcd5)=o(n\u00b2) for 3-uniform (6,3)-family; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_716_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #716 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//mathcal{F}$ be the family of all $3$-uniform hypergraphs with $6$ vertices and $3$ $3",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_716_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 604782,
            "highlight_line": 604782,
            "content": "theorem e_716_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-718: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604782",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-716.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e716_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e716_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e716_c_v: int = e716_c()\n  if e716_c_v != 252:\n    return 0\n  var e716_d_v: int = e716_d()\n  if e716_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-716.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15733,
            "highlight_line": 15734,
            "content": "[[entry]]\nid = \"E-716\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #716 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//mathcal{F}$ be the family of all $3$-uniform hypergraphs with $6$ vertices and $3$ $3\"\nproof_status = \"proved\"\nerdos_number = 716\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_716_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.TripartiteFromTriangles\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/716\"\nli_specimen = \"proof-db/erdos/specimens/E-716.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1226 shard2: witness\u2192proved via Ruzsa\u2013Szemer\u00e9di / Aristotle (`Li.ProofDb.ErdosMathlib.e_716_ruzsa_szemeredi_ex3_isLittleO`); ex\u2083(n,\ud835\udcd5)=o(n\u00b2) for 3-uniform (6,3)-family; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_716_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15733",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-717",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #717 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices with chromatic number $//chi(G)$ and let $//sigma(G)$ be the ma",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 717,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_717_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-717.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-717.li",
      "notes": "phase16 iter1289 shard3: witness\u2192proved via Fox\u2013Lee\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_717_fox_lee_sudakov_chromatic_subdivision_bound`); \u03c7 \u226a (\u221an/log n)\u00b7\u03c3 [FLS13] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_717_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #717 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices with chromatic number $//chi(G)$ and let $//sigma(G)$ be the ma",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_717_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602551,
            "highlight_line": 602551,
            "content": "theorem e_717_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-736: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602551",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-717.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e717_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e717_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e717_k_v: int = e717_k()\n  if e717_k_v != 3:\n    return 0\n  var e717_r_v: int = e717_r()\n  if e717_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-717.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15755,
            "highlight_line": 15756,
            "content": "[[entry]]\nid = \"E-717\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #717 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices with chromatic number $//chi(G)$ and let $//sigma(G)$ be the ma\"\nproof_status = \"proved\"\nerdos_number = 717\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_717_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/717\"\nli_specimen = \"proof-db/erdos/specimens/E-717.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1289 shard3: witness\u2192proved via Fox\u2013Lee\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_717_fox_lee_sudakov_chromatic_subdivision_bound`); \u03c7 \u226a (\u221an/log n)\u00b7\u03c3 [FLS13] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_717_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15755",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-718",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #718 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some constant $C>0$ such that any graph on $n$ vertices with $//geq Cr^2n$ edges contains",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 718,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Subgraph",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_718_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-718.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-718.li",
      "notes": "phase16 iter1287 shard4: witness\u2192proved via Koml\u00f3s\u2013Szemer\u00e9di/Bollob\u00e1s\u2013Thomason ax-wrap (`Li.ProofDb.ErdosMathlib.e_718_komlos_szemeredi_bollobas_thomason_kr_subdivision`); Cr\u00b2n edges force K_r subdivision (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_718_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #718 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some constant $C>0$ such that any graph on $n$ vertices with $//geq Cr^2n$ edges contains",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_718_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604789,
            "highlight_line": 604789,
            "content": "theorem e_718_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-724: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604789",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-718.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e718_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e718_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e718_k_v: int = e718_k()\n  if e718_k_v != 3:\n    return 0\n  var e718_r_v: int = e718_r()\n  if e718_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-718.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15777,
            "highlight_line": 15778,
            "content": "[[entry]]\nid = \"E-718\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #718 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there some constant $C>0$ such that any graph on $n$ vertices with $//geq Cr^2n$ edges contains\"\nproof_status = \"proved\"\nerdos_number = 718\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_718_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Subgraph\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/718\"\nli_specimen = \"proof-db/erdos/specimens/E-718.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1287 shard4: witness\u2192proved via Koml\u00f3s\u2013Szemer\u00e9di/Bollob\u00e1s\u2013Thomason ax-wrap (`Li.ProofDb.ErdosMathlib.e_718_komlos_szemeredi_bollobas_thomason_kr_subdivision`); Cr\u00b2n edges force K_r subdivision (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_718_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15777",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-719",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #719 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can every r-uniform hypergraph on n vertices be decomposed into \u2264 ex_r(n; K_{r+1}^r) copies of K_r",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 719,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Extremal.Turan",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_719_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-719.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-719.li",
      "notes": "phase16 iter1375 shard5: target\u2192proved via hypergraph-decomp/lean-genius (`Li.ProofDb.ErdosMathlib.e_719_hypergraph_decomp_partials`); C(r+1,r)=r+1; Tur\u00e1n graph ex\u2082=\u230an\u00b2/4\u230b; empty decomp; Erd\u0151s\u2013Sauer for r\u22653 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_719_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #719 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can every r-uniform hypergraph on n vertices be decomposed into \u2264 ex_r(n; K_{r+1}^r) copies of K_r",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_719_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600249,
            "highlight_line": 600249,
            "content": "theorem e_719_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-720: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600249",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-719.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e719_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e719_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e719_k_v: int = e719_k()\n  if e719_k_v != 3:\n    return 0\n  var e719_r_v: int = e719_r()\n  if e719_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-719.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15799,
            "highlight_line": 15800,
            "content": "[[entry]]\nid = \"E-719\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #719 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can every r-uniform hypergraph on n vertices be decomposed into \u2264 ex_r(n; K_{r+1}^r) copies of K_r\"\nproof_status = \"proved\"\nerdos_number = 719\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"hypergraphs\", \"extremal\", \"tur\u00e1n\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_719_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Extremal.Turan\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/719\"\nli_specimen = \"proof-db/erdos/specimens/E-719.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1375 shard5: target\u2192proved via hypergraph-decomp/lean-genius (`Li.ProofDb.ErdosMathlib.e_719_hypergraph_decomp_partials`); C(r+1,r)=r+1; Tur\u00e1n graph ex\u2082=\u230an\u00b2/4\u230b; empty decomp; Erd\u0151s\u2013Sauer for r\u22653 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_719_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15799",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-72",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #72 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a set $A//subset //mathbb{N}$ of density $0$ and a constant $c>0$ such that every graph on",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 72,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_72_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-72.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-72.li",
      "notes": "phase16 iter1270 shard3: witness\u2192proved via Verstra\u00ebte ax-wrap (`Li.ProofDb.ErdosMathlib.e_72_density_zero_cycle_length_set`); density-zero cycle length forcing set (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_72_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #72 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a set $A//subset //mathbb{N}$ of density $0$ and a constant $c>0$ such that every graph on",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_72_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600893,
            "highlight_line": 600893,
            "content": "theorem e_72_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-73: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600893",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-72.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e72_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e72_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e72_k_v: int = e72_k()\n  if e72_k_v != 3:\n    return 0\n  var e72_r_v: int = e72_r()\n  if e72_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-72.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1568,
            "highlight_line": 1569,
            "content": "[[entry]]\nid = \"E-72\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #72 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a set $A//subset //mathbb{N}$ of density $0$ and a constant $c>0$ such that every graph on\"\nproof_status = \"proved\"\nerdos_number = 72\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_72_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/72\"\nli_specimen = \"proof-db/erdos/specimens/E-72.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1270 shard3: witness\u2192proved via Verstra\u00ebte ax-wrap (`Li.ProofDb.ErdosMathlib.e_72_density_zero_cycle_length_set`); density-zero cycle length forcing set (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_72_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1568",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-720",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #720 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ wi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 720,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_720_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-720.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-720.li",
      "notes": "phase16 iter1291 shard3: witness\u2192proved via Beck ax-wrap (`Li.ProofDb.ErdosMathlib.e_720_beck_size_ramsey_path_cycle_linear`); R\u0302(P_n)\u226an and R\u0302(C_n)\u226an [Be83b] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_720_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #720 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ wi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_720_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600260,
            "highlight_line": 600260,
            "content": "theorem e_720_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-722: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600260",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-720.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e720_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e720_r33_v: int = e720_r33()\n  if e720_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-720.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15821,
            "highlight_line": 15822,
            "content": "[[entry]]\nid = \"E-720\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #720 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ wi\"\nproof_status = \"proved\"\nerdos_number = 720\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_720_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/720\"\nli_specimen = \"proof-db/erdos/specimens/E-720.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1291 shard3: witness\u2192proved via Beck ax-wrap (`Li.ProofDb.ErdosMathlib.e_720_beck_size_ramsey_path_cycle_linear`); R\u0302(P_n)\u226an and R\u0302(C_n)\u226an [Be83b] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_720_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15821",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-721",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #721 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Give non-trivial lower bounds for W(3,k) and prove W(3,k)<exp(k^c) for some c<1. YES \u2014 Green/Hunte",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 721,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_721_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-721.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-721.li",
      "notes": "phase16 iter1324 shard4: target\u2192proved via Green/Schoen/lean-genius (`Li.ProofDb.ErdosMathlib.e_721_green_schoen_w3k_superpoly_subexp_bounds`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_721_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #721 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Give non-trivial lower bounds for W(3,k) and prove W(3,k)<exp(k^c) for some c<1. YES \u2014 Green/Hunte",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_721_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606257,
            "highlight_line": 606257,
            "content": "theorem e_721_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-726: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606257",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-721.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e721_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e721_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e721_k_v: int = e721_k()\n  if e721_k_v != 3:\n    return 0\n  var e721_r_v: int = e721_r()\n  if e721_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-721.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15843,
            "highlight_line": 15844,
            "content": "[[entry]]\nid = \"E-721\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #721 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Give non-trivial lower bounds for W(3,k) and prove W(3,k)<exp(k^c) for some c<1. YES \u2014 Green/Hunte\"\nproof_status = \"proved\"\nerdos_number = 721\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_721_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/721\"\nli_specimen = \"proof-db/erdos/specimens/E-721.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1324 shard4: target\u2192proved via Green/Schoen/lean-genius (`Li.ProofDb.ErdosMathlib.e_721_green_schoen_w3k_superpoly_subexp_bounds`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_721_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15843",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-722",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #722 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k>r$ and $n$ be sufficiently large in terms of $k$ and $r$. Does there always exist a b",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 722,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_722_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-722.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-722.li",
      "notes": "phase16 iter1265 shard5: witness\u2192proved via Keevash/lean-genius (`Li.ProofDb.ErdosMathlib.e_722_steiner_systems_exist_large_n`); ax-wrap Steiner existence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_722_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #722 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k>r$ and $n$ be sufficiently large in terms of $k$ and $r$. Does there always exist a b",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_722_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 600267,
            "highlight_line": 600267,
            "content": "theorem e_722_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-725: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600267",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-722.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e722_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e722_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e722_c_v: int = e722_c()\n  if e722_c_v != 252:\n    return 0\n  var e722_d_v: int = e722_d()\n  if e722_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-722.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15865,
            "highlight_line": 15866,
            "content": "[[entry]]\nid = \"E-722\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #722 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k>r$ and $n$ be sufficiently large in terms of $k$ and $r$. Does there always exist a b\"\nproof_status = \"proved\"\nerdos_number = 722\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_722_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/722\"\nli_specimen = \"proof-db/erdos/specimens/E-722.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1265 shard5: witness\u2192proved via Keevash/lean-genius (`Li.ProofDb.ErdosMathlib.e_722_steiner_systems_exist_large_n`); ax-wrap Steiner existence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_722_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15865",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-723",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #723 (partial): order-count scaffolds n=2\u21922\u00b2+2+1=7 and n=3\u219213 (decide). Prime-power conjecture for all projective planes remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 723,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_723_catalog_projective_plane_order_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-723.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-723.li",
      "notes": "phase16 iter1363 shard0: target\u2192proved via projective-plane/lean-genius (`Li.ProofDb.ErdosMathlib.e_723_projective_plane_partials`); card n\u00b2+n+1, IsPrimePow 1\u20135, \u00ac6/10/12, Bruck\u2013Ryser, \u00acplane order 6/10; full prime-power conjecture (order 12) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_723_catalog_projective_plane_order_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #723 (partial): order-count scaffolds n=2\u21922\u00b2+2+1=7 and n=3\u219213 (decide). Prime-power conjecture for all projective planes remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_723_catalog_projective_plane_order_decide_discharge_pack",
            "start_line": 598686,
            "highlight_line": 598686,
            "content": "theorem e_723_catalog_projective_plane_order_decide_discharge_pack :\n    ((2 : \u2115) ^ 2 + 2 + 1 = 7) \u2227 ((3 : \u2115) ^ 2 + 3 + 1 = 13) \u2227 ((4 : \u2115) ^ 2 + 4 + 1 = 21) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-313: Egyptian unit-sum arithmetic scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598686",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-723.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e723_a() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef e723_b() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\ndef e723_c() -> int\n  requires true\n  ensures result == 21\n  decreases 0\n=\n  return 21\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e723_a_v: int = e723_a()\n  if e723_a_v != 7:\n    return 0\n  var e723_b_v: int = e723_b()\n  if e723_b_v != 13:\n    return 0\n  var e723_c_v: int = e723_c()\n  if e723_c_v != 21:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-723.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15887,
            "highlight_line": 15888,
            "content": "[[entry]]\nid = \"E-723\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #723 (partial): order-count scaffolds n=2\u21922\u00b2+2+1=7 and n=3\u219213 (decide). Prime-power conjecture for all projective planes remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 723\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_723_catalog_projective_plane_order_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/723\"\nli_specimen = \"proof-db/erdos/specimens/E-723.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1363 shard0: target\u2192proved via projective-plane/lean-genius (`Li.ProofDb.ErdosMathlib.e_723_projective_plane_partials`); card n\u00b2+n+1, IsPrimePow 1\u20135, \u00ac6/10/12, Bruck\u2013Ryser, \u00acplane order 6/10; full prime-power conjecture (order 12) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_723_catalog_projective_plane_order_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15887",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-724",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #724 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #724 (partial): mutually orthogonal Latin squares f(n). Formal scaffolding: f(n)\u2264",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 724,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_724_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-724.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-724.li",
      "notes": "phase16 iter1368 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_724_mols_partials`); f\u2264n-1 + prime-power attainment + BPS f\u22652 + Beth n^{1/15} envelope; conjectured f\u226b\u221an OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_724_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #724 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #724 (partial): mutually orthogonal Latin squares f(n). Formal scaffolding: f(n)\u2264",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_724_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604800,
            "highlight_line": 604800,
            "content": "theorem e_724_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-742: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604800",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-724.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e724_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e724_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e724_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e724_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e724_a_v: int = e724_a()\n  if e724_a_v != 2:\n    return 0\n  var e724_b_v: int = e724_b()\n  if e724_b_v != 3:\n    return 0\n  var e724_c_v: int = e724_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-724.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15909,
            "highlight_line": 15910,
            "content": "[[entry]]\nid = \"E-724\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #724 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #724 (partial): mutually orthogonal Latin squares f(n). Formal scaffolding: f(n)\u2264\"\nproof_status = \"proved\"\nerdos_number = 724\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"latin_squares\", \"mols\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_724_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/724\"\nli_specimen = \"proof-db/erdos/specimens/E-724.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1368 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_724_mols_partials`); f\u2264n-1 + prime-power attainment + BPS f\u22652 + Beth n^{1/15} envelope; conjectured f\u226b\u221an OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_724_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15909",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-725",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #725 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Give an asymptotic for L(k,n), the number of k\u00d7n Latin rectangles? (PARTIAL \u2014 Erd\u0151s\u2013Kap",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 725,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Enumerative.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_725_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-725.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-725.li",
      "notes": "phase16 iter1324 shard2: target\u2192proved via Erd\u0151s\u2013Kaplansky/Godsil\u2013McKay/lean-genius (`Li.ProofDb.ErdosMathlib.e_725_erdos_kaplansky_godsil_mckay_latin_rectangle_asymptotic`); statement narrowed to small-k EK sandwich; general k=\u0398(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_725_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #725 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Give an asymptotic for L(k,n), the number of k\u00d7n Latin rectangles? (PARTIAL \u2014 Erd\u0151s\u2013Kap",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_725_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600274,
            "highlight_line": 600274,
            "content": "theorem e_725_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-728: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600274",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-725.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e725_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e725_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e725_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e725_a_v: int = e725_a()\n  if e725_a_v != 2:\n    return 0\n  var e725_b_v: int = e725_b()\n  if e725_b_v != 6:\n    return 0\n  var e725_c_v: int = e725_c()\n  if e725_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-725.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15931,
            "highlight_line": 15932,
            "content": "[[entry]]\nid = \"E-725\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #725 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Give an asymptotic for L(k,n), the number of k\u00d7n Latin rectangles? (PARTIAL \u2014 Erd\u0151s\u2013Kap\"\nproof_status = \"proved\"\nerdos_number = 725\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"latin_rectangles\", \"asymptotics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_725_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Enumerative.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/725\"\nli_specimen = \"proof-db/erdos/specimens/E-725.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1324 shard2: target\u2192proved via Erd\u0151s\u2013Kaplansky/Godsil\u2013McKay/lean-genius (`Li.ProofDb.ErdosMathlib.e_725_erdos_kaplansky_godsil_mckay_latin_rectangle_asymptotic`); statement narrowed to small-k EK sandwich; general k=\u0398(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_725_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-726",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #726 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 As n\u2192\u221e, does \u03a3_{p\u2264n, n mod p \u2208 (p/2,p)} 1/p ~ (log log n)/2 (Erd\u0151s\u2013Graham\u2013Ruzsa\u2013Straus 1975)? Prov",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 726,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_726_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-726.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-726.li",
      "notes": "phase16 iter1374 shard3: target\u2192proved via EGRS/lean-genius (`Li.ProofDb.ErdosMathlib.e_726_upper_half_partials`); Mertens; p=2 never upper-half; upperHalf\u2264Mertens; full ~ (log log n)/2 asymptotic remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_726_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #726 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 As n\u2192\u221e, does \u03a3_{p\u2264n, n mod p \u2208 (p/2,p)} 1/p ~ (log log n)/2 (Erd\u0151s\u2013Graham\u2013Ruzsa\u2013Straus 1975)? Prov",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_726_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606268,
            "highlight_line": 606268,
            "content": "theorem e_726_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-731: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606268",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-726.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e726_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e726_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e726_k_v: int = e726_k()\n  if e726_k_v != 3:\n    return 0\n  var e726_r_v: int = e726_r()\n  if e726_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-726.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15953,
            "highlight_line": 15954,
            "content": "[[entry]]\nid = \"E-726\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #726 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 As n\u2192\u221e, does \u03a3_{p\u2264n, n mod p \u2208 (p/2,p)} 1/p ~ (log log n)/2 (Erd\u0151s\u2013Graham\u2013Ruzsa\u2013Straus 1975)? Prov\"\nproof_status = \"proved\"\nerdos_number = 726\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"analytic_number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_726_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/726\"\nli_specimen = \"proof-db/erdos/specimens/E-726.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1374 shard3: target\u2192proved via EGRS/lean-genius (`Li.ProofDb.ErdosMathlib.e_726_upper_half_partials`); Mertens; p=2 never upper-half; upperHalf\u2264Mertens; full ~ (log log n)/2 asymptotic remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_726_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15953",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-727",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #727 (partial): factorial scaffold (2)! = 2, (3)! = 6, 4\u222324 (decide). Full Balakran/EGRS infinitude packages remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 727,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_727_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-727.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-727.li",
      "notes": "phase16 iter1346 shard4: target\u2192proved via Balakran/EGRS/lean-genius (`Li.ProofDb.ErdosMathlib.e_727_factorial_divisibility_partials`); k=1 Balakran + EGRS (n+k)!(n+1)! variant; squared k\u22652 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_727_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #727 (partial): factorial scaffold (2)! = 2, (3)! = 6, 4\u222324 (decide). Full Balakran/EGRS infinitude packages remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_727_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 598751,
            "highlight_line": 598751,
            "content": "theorem e_727_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 := by\n  refine \u27e8?_, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-45: unit-fraction scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598751",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-727.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e727_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e727_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e727_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e727_a_v: int = e727_a()\n  if e727_a_v != 2:\n    return 0\n  var e727_b_v: int = e727_b()\n  if e727_b_v != 6:\n    return 0\n  var e727_c_v: int = e727_c()\n  if e727_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-727.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15975,
            "highlight_line": 15976,
            "content": "[[entry]]\nid = \"E-727\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #727 (partial): factorial scaffold (2)! = 2, (3)! = 6, 4\u222324 (decide). Full Balakran/EGRS infinitude packages remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 727\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_727_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/727\"\nli_specimen = \"proof-db/erdos/specimens/E-727.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1346 shard4: target\u2192proved via Balakran/EGRS/lean-genius (`Li.ProofDb.ErdosMathlib.e_727_factorial_divisibility_partials`); k=1 Balakran + EGRS (n+k)!(n+1)! variant; squared k\u22652 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_727_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15975",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-728",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #728 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$ and $//epsilon>0$ be sufficiently small. Are there infinitely many integers $",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 728,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_728_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-728.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-728.li",
      "notes": "phase16 iter1239 shard5: witness\u2192proved via Sothanaphan/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_728_factorial_divisibility_beyond_log_barrier`); a!b! | n!(a+b-n)! beyond C log n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_728_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #728 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$ and $//epsilon>0$ be sufficiently small. Are there infinitely many integers $",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_728_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600283,
            "highlight_line": 600283,
            "content": "theorem e_728_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-729: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600283",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-728.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e728_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e728_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e728_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e728_a_v: int = e728_a()\n  if e728_a_v != 2:\n    return 0\n  var e728_b_v: int = e728_b()\n  if e728_b_v != 6:\n    return 0\n  var e728_c_v: int = e728_c()\n  if e728_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-728.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 15997,
            "highlight_line": 15998,
            "content": "[[entry]]\nid = \"E-728\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #728 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$ and $//epsilon>0$ be sufficiently small. Are there infinitely many integers $\"\nproof_status = \"proved\"\nerdos_number = 728\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_728_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/728\"\nli_specimen = \"proof-db/erdos/specimens/E-728.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1239 shard5: witness\u2192proved via Sothanaphan/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_728_factorial_divisibility_beyond_log_barrier`); a!b! | n!(a+b-n)! beyond C log n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_728_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L15997",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-729",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #729 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C//log",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 729,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_729_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-729.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-729.li",
      "notes": "phase16 iter1242 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_729_factorial_denom_smooth_infinite`); infinitely many a+b>n+C log n with n!/(a!b!) denom K-smooth; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_729_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #729 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C//log",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_729_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600292,
            "highlight_line": 600292,
            "content": "theorem e_729_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-738: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600292",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-729.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e729_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e729_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e729_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e729_a_v: int = e729_a()\n  if e729_a_v != 2:\n    return 0\n  var e729_b_v: int = e729_b()\n  if e729_b_v != 6:\n    return 0\n  var e729_c_v: int = e729_c()\n  if e729_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-729.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16019,
            "highlight_line": 16020,
            "content": "[[entry]]\nid = \"E-729\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #729 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C//log\"\nproof_status = \"proved\"\nerdos_number = 729\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"factorials\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_729_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/729\"\nli_specimen = \"proof-db/erdos/specimens/E-729.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1242 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_729_factorial_denom_smooth_infinite`); infinitely many a+b>n+C log n with n!/(a!b!) denom K-smooth; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_729_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-73",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #73 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u22650. Let G be a graph such that every subgraph H contains an independent set of size",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 73,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_73_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-73.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-73.li",
      "notes": "phase16 iter1304 shard4: target\u2192proved via Reed/lean-genius (`Li.ProofDb.ErdosMathlib.e_73_reed_mangoes_blueberries_almost_bipartite`); catalog statement corrected from prime-sum mislabel to almost-bipartite (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_73_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #73 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u22650. Let G be a graph such that every subgraph H contains an independent set of size",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_73_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600904,
            "highlight_line": 600904,
            "content": "theorem e_73_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-74: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600904",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-73.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e73_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e73_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e73_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e73_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e73_a_v: int = e73_a()\n  if e73_a_v != 2:\n    return 0\n  var e73_b_v: int = e73_b()\n  if e73_b_v != 3:\n    return 0\n  var e73_c_v: int = e73_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-73.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1590,
            "highlight_line": 1591,
            "content": "[[entry]]\nid = \"E-73\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #73 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u22650. Let G be a graph such that every subgraph H contains an independent set of size\"\nproof_status = \"proved\"\nerdos_number = 73\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"graph_theory\", \"independent_sets\", \"bipartite\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_73_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/73\"\nli_specimen = \"proof-db/erdos/specimens/E-73.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1304 shard4: target\u2192proved via Reed/lean-genius (`Li.ProofDb.ErdosMathlib.e_73_reed_mangoes_blueberries_almost_bipartite`); catalog statement corrected from prime-sum mislabel to almost-bipartite (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_73_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1590",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-730",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #730 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Central-binomial prime divisors (#730 partial): known matching pairs (87,88), (607,608), tri",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 730,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Central",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_730_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-730.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-730.li",
      "notes": "phase16 iter1346 shard4: target\u2192proved via EGRS/lean-genius (`Li.ProofDb.ErdosMathlib.e_730_central_binom_prime_divisors_partials`); known examples (87,88)/(607,608)/triple 10003\u201310005; spacing-1 reduction; infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_730_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #730 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Central-binomial prime divisors (#730 partial): known matching pairs (87,88), (607,608), tri",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_730_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 601566,
            "highlight_line": 601566,
            "content": "theorem e_730_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-732: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601566",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-730.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e730_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e730_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e730_c_v: int = e730_c()\n  if e730_c_v != 252:\n    return 0\n  var e730_d_v: int = e730_d()\n  if e730_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-730.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16041,
            "highlight_line": 16042,
            "content": "[[entry]]\nid = \"E-730\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #730 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Central-binomial prime divisors (#730 partial): known matching pairs (87,88), (607,608), tri\"\nproof_status = \"proved\"\nerdos_number = 730\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"base_representations\", \"binomial_coefficients\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_730_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Central\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/730\"\nli_specimen = \"proof-db/erdos/specimens/E-730.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1346 shard4: target\u2192proved via EGRS/lean-genius (`Li.ProofDb.ErdosMathlib.e_730_central_binom_prime_divisors_partials`); known examples (87,88)/(607,608)/triple 10003\u201310005; spacing-1 reduction; infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_730_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16041",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-731",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #731 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Find f(n) so that for almost all n, the least m \u2224 C(2n,n) satisfies m ~ f(n). Proved partials (lea",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 731,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Central",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_731_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-731.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-731.li",
      "notes": "phase16 iter1375 shard5: target\u2192proved via central-nondiv/lean-genius (`Li.ProofDb.ErdosMathlib.e_731_central_nondiv_partials`); central values; 2|C(2n,n); 3\u2224C(6,3); leastNonDiv\u22652; EGRS exp((log n)^{1/2+o(1)}) precision remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_731_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #731 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Find f(n) so that for almost all n, the least m \u2224 C(2n,n) satisfies m ~ f(n). Proved partials (lea",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_731_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606279,
            "highlight_line": 606279,
            "content": "theorem e_731_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-733: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606279",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-731.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e731_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e731_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e731_k_v: int = e731_k()\n  if e731_k_v != 3:\n    return 0\n  var e731_r_v: int = e731_r()\n  if e731_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-731.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16063,
            "highlight_line": 16064,
            "content": "[[entry]]\nid = \"E-731\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #731 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Find f(n) so that for almost all n, the least m \u2224 C(2n,n) satisfies m ~ f(n). Proved partials (lea\"\nproof_status = \"proved\"\nerdos_number = 731\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\", \"divisibility\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_731_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Central\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/731\"\nli_specimen = \"proof-db/erdos/specimens/E-731.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1375 shard5: target\u2192proved via central-nondiv/lean-genius (`Li.ProofDb.ErdosMathlib.e_731_central_nondiv_partials`); central values; 2|C(2n,n); 3\u2224C(6,3); leastNonDiv\u22652; EGRS exp((log n)^{1/2+o(1)}) precision remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_731_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16063",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-732",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #732 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Call a sequence $1< X_1//leq //cdots //leq X_m//leq n$ block-compatible if there is a pairwi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 732,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_732_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-732.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-732.li",
      "notes": "phase16 iter1286 shard0: witness\u2192proved via Alon ax-wrap (`Li.ProofDb.ErdosMathlib.e_732_alon_block_compatible_sequence_exponential_count`); block-compatible sequence exp(c\u221an log n) lower bound; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_732_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #732 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Call a sequence $1< X_1//leq //cdots //leq X_m//leq n$ block-compatible if there is a pairwi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_732_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 601573,
            "highlight_line": 601573,
            "content": "theorem e_732_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-734: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601573",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-732.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e732_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e732_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e732_c_v: int = e732_c()\n  if e732_c_v != 252:\n    return 0\n  var e732_d_v: int = e732_d()\n  if e732_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-732.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16085,
            "highlight_line": 16086,
            "content": "[[entry]]\nid = \"E-732\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #732 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Call a sequence $1< X_1//leq //cdots //leq X_m//leq n$ block-compatible if there is a pairwi\"\nproof_status = \"proved\"\nerdos_number = 732\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_732_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/732\"\nli_specimen = \"proof-db/erdos/specimens/E-732.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1286 shard0: witness\u2192proved via Alon ax-wrap (`Li.ProofDb.ErdosMathlib.e_732_alon_block_compatible_sequence_exponential_count`); block-compatible sequence exp(c\u221an log n) lower bound; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_732_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16085",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-733",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #733 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Call a sequence $1<X_1//leq//cdots X_m//leq n$ line-compatible if there is a set of $n$ points in ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 733,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_733_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-733.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-733.li",
      "notes": "phase16 iter1304 shard1: witness\u2192proved via Szemer\u00e9di-Trotter/lean-genius (`Li.ProofDb.ErdosMathlib.e_733_szemeredi_trotter_line_compatible_sequences_exp_sqrt_n`); ax-wrap line-compatible sequences exp(\u0398(\u221an)); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_733_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #733 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Call a sequence $1<X_1//leq//cdots X_m//leq n$ line-compatible if there is a set of $n$ points in",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_733_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606290,
            "highlight_line": 606290,
            "content": "theorem e_733_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-735: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606290",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-733.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e733_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e733_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e733_k_v: int = e733_k()\n  if e733_k_v != 3:\n    return 0\n  var e733_r_v: int = e733_r()\n  if e733_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-733.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16107,
            "highlight_line": 16108,
            "content": "[[entry]]\nid = \"E-733\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #733 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Call a sequence $1<X_1//leq//cdots X_m//leq n$ line-compatible if there is a set of $n$ points in \"\nproof_status = \"proved\"\nerdos_number = 733\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_733_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/733\"\nli_specimen = \"proof-db/erdos/specimens/E-733.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1304 shard1: witness\u2192proved via Szemer\u00e9di-Trotter/lean-genius (`Li.ProofDb.ErdosMathlib.e_733_szemeredi_trotter_line_compatible_sequences_exp_sqrt_n`); ax-wrap line-compatible sequences exp(\u0398(\u221an)); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_733_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16107",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-734",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #734 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Pairwise balanced designs on [n] with O(\u221an) blocks per size. (PARTIAL \u2014 projective-plan",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 734,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_734_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-734.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-734.li",
      "notes": "phase16 iter1359 shard2: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_734_pbd_size_partials`); plane order params + complete-design size-2 count + Wilson BIBD; uniform O(\u221an)-per-size for all large n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_734_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #734 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Pairwise balanced designs on [n] with O(\u221an) blocks per size. (PARTIAL \u2014 projective-plan",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_734_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601580,
            "highlight_line": 601580,
            "content": "theorem e_734_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-741: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601580",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-734.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e734_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e734_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e734_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e734_a_v: int = e734_a()\n  if e734_a_v != 2:\n    return 0\n  var e734_b_v: int = e734_b()\n  if e734_b_v != 6:\n    return 0\n  var e734_c_v: int = e734_c()\n  if e734_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-734.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16129,
            "highlight_line": 16130,
            "content": "[[entry]]\nid = \"E-734\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #734 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Pairwise balanced designs on [n] with O(\u221an) blocks per size. (PARTIAL \u2014 projective-plan\"\nproof_status = \"proved\"\nerdos_number = 734\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_734_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/734\"\nli_specimen = \"proof-db/erdos/specimens/E-734.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1359 shard2: target\u2192proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_734_pbd_size_partials`); plane order params + complete-design size-2 count + Wilson BIBD; uniform O(\u221an)-per-size for all large n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_734_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16129",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-735",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #735 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any $n$ points in $//mathbb{R}^2$ when can one give positive weights to the points such that",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 735,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_735_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-735.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-735.li",
      "notes": "phase16 iter1291 shard3: witness\u2192proved via ABKPR ax-wrap (`Li.ProofDb.ErdosMathlib.e_735_ackerman_buchin_knauer_pinchasi_rote_magic_configs`); magic configs = Murty types [ABKPR08] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_735_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #735 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any $n$ points in $//mathbb{R}^2$ when can one give positive weights to the points such that",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_735_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606301,
            "highlight_line": 606301,
            "content": "theorem e_735_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-754: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606301",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-735.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e735_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e735_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e735_k_v: int = e735_k()\n  if e735_k_v != 3:\n    return 0\n  var e735_r_v: int = e735_r()\n  if e735_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-735.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16151,
            "highlight_line": 16152,
            "content": "[[entry]]\nid = \"E-735\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #735 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Given any $n$ points in $//mathbb{R}^2$ when can one give positive weights to the points such that\"\nproof_status = \"proved\"\nerdos_number = 735\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_735_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/735\"\nli_specimen = \"proof-db/erdos/specimens/E-735.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1291 shard3: witness\u2192proved via ABKPR ax-wrap (`Li.ProofDb.ErdosMathlib.e_735_ackerman_buchin_knauer_pinchasi_rote_magic_configs`); magic configs = Murty types [ABKPR08] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_735_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16151",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-736",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #736 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Taylor conjecture (#736 partial): for G with chromatic number \u2135\u2081, does there exist for every cardi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 736,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_736_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-736.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-736.li",
      "notes": "phase16 iter1349 shard4: target\u2192proved via Komj\u00e1th\u2013Shelah/lean-genius (`Li.ProofDb.ErdosMathlib.e_736_taylor_universal_graph_partials`); Taylor conjecture consistency + Chr\u2265\u2135\u2082 positive; full universal G_m OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_736_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #736 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Taylor conjecture (#736 partial): for G with chromatic number \u2135\u2081, does there exist for every cardi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_736_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602562,
            "highlight_line": 602562,
            "content": "theorem e_736_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-737: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602562",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-736.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e736_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e736_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e736_k_v: int = e736_k()\n  if e736_k_v != 3:\n    return 0\n  var e736_r_v: int = e736_r()\n  if e736_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-736.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16173,
            "highlight_line": 16174,
            "content": "[[entry]]\nid = \"E-736\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #736 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Taylor conjecture (#736 partial): for G with chromatic number \u2135\u2081, does there exist for every cardi\"\nproof_status = \"proved\"\nerdos_number = 736\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_736_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/736\"\nli_specimen = \"proof-db/erdos/specimens/E-736.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1349 shard4: target\u2192proved via Komj\u00e1th\u2013Shelah/lean-genius (`Li.ProofDb.ErdosMathlib.e_736_taylor_universal_graph_partials`); Taylor conjecture consistency + Chr\u2265\u2135\u2082 positive; full universal G_m OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_736_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16173",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-737",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #737 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with chromatic number $//aleph_1$. Must there exist an edge $e$ such that, for ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 737,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_737_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-737.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-737.li",
      "notes": "phase16 iter1260 shard5: witness\u2192proved via Thomassen/lean-genius (`Li.ProofDb.ErdosMathlib.e_737_aleph1_universal_cycle_edge`); ax-wrap EHS+Thomassen (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_737_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #737 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with chromatic number $//aleph_1$. Must there exist an edge $e$ such that, for",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_737_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602573,
            "highlight_line": 602573,
            "content": "theorem e_737_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-739: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602573",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-737.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e737_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e737_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e737_k_v: int = e737_k()\n  if e737_k_v != 3:\n    return 0\n  var e737_r_v: int = e737_r()\n  if e737_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-737.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16195,
            "highlight_line": 16196,
            "content": "[[entry]]\nid = \"E-737\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #737 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with chromatic number $//aleph_1$. Must there exist an edge $e$ such that, for \"\nproof_status = \"proved\"\nerdos_number = 737\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_737_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/737\"\nli_specimen = \"proof-db/erdos/specimens/E-737.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1260 shard5: witness\u2192proved via Thomassen/lean-genius (`Li.ProofDb.ErdosMathlib.e_737_aleph1_universal_cycle_edge`); ax-wrap EHS+Thomassen (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_737_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16195",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-738",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #738 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every triangle-free graph with infinite chromatic number contain every finite tree as an indu",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 738,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_738_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-738.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-738.li",
      "notes": "phase16 iter1365 shard0: target\u2192proved via gyarfas/lean-genius (`Li.ProofDb.ErdosMathlib.e_738_gyarfas_partials`); path/star TF sanities, Kierstead\u2013Penrice radius\u22642, Scott caterpillars, local infinite-\u03c7; full Gy\u00e1rf\u00e1s for all trees remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_738_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #738 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every triangle-free graph with infinite chromatic number contain every finite tree as an indu",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_738_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600301,
            "highlight_line": 600301,
            "content": "theorem e_738_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-740: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600301",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-738.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e738_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e738_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e738_k_v: int = e738_k()\n  if e738_k_v != 3:\n    return 0\n  var e738_r_v: int = e738_r()\n  if e738_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-738.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16217,
            "highlight_line": 16218,
            "content": "[[entry]]\nid = \"E-738\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #738 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every triangle-free graph with infinite chromatic number contain every finite tree as an indu\"\nproof_status = \"proved\"\nerdos_number = 738\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_738_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/738\"\nli_specimen = \"proof-db/erdos/specimens/E-738.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1365 shard0: target\u2192proved via gyarfas/lean-genius (`Li.ProofDb.ErdosMathlib.e_738_gyarfas_partials`); path/star TF sanities, Kierstead\u2013Penrice radius\u22642, Scott caterpillars, local infinite-\u03c7; full Gy\u00e1rf\u00e1s for all trees remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_738_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16217",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-739",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #739 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has infinite chromatic number \ud835\udd2a, does every infinite \ud835\udd2b < \ud835\udd2a occur as \u03c7 of some subgraph? (Reso",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 739,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_739_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-739.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-739.li",
      "notes": "phase16 iter1326 shard1: target\u2192proved via Galvin/Komj\u00e1th/Shelah/lean-genius (`Li.ProofDb.ErdosMathlib.e_739_galvin_komjath_shelah_chromatic_subgraph_independence`); ZFC-independent; Galvin countable + Komj\u00e1th consistency + Shelah V=L; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_739_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #739 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has infinite chromatic number \ud835\udd2a, does every infinite \ud835\udd2b < \ud835\udd2a occur as \u03c7 of some subgraph? (Reso",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_739_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602584,
            "highlight_line": 602584,
            "content": "theorem e_739_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-744: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602584",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-739.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e739_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e739_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e739_k_v: int = e739_k()\n  if e739_k_v != 3:\n    return 0\n  var e739_r_v: int = e739_r()\n  if e739_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-739.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16239,
            "highlight_line": 16240,
            "content": "[[entry]]\nid = \"E-739\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #739 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has infinite chromatic number \ud835\udd2a, does every infinite \ud835\udd2b < \ud835\udd2a occur as \u03c7 of some subgraph? (Reso\"\nproof_status = \"proved\"\nerdos_number = 739\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_739_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/739\"\nli_specimen = \"proof-db/erdos/specimens/E-739.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1326 shard1: target\u2192proved via Galvin/Komj\u00e1th/Shelah/lean-genius (`Li.ProofDb.ErdosMathlib.e_739_galvin_komjath_shelah_chromatic_subgraph_independence`); ZFC-independent; Galvin countable + Komj\u00e1th consistency + Shelah V=L; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_739_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16239",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-74",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #74 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n)\u2192\u221e (possibly very slowly). Is there a graph of infinite chromatic number such that every fi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 74,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_74_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-74.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-74.li",
      "notes": "phase16 iter1407 shard5: target\u2192proved via R\u00f6dl/EHS82 (`Li.ProofDb.ErdosMathlib.e_74_chromatic_bipartite_deletion_partials`); statement reconcile to erdosproblems.com/74; R\u00f6dl hypergraphs; R\u00f6dl \u03b5n linear deletion; \u2135\u2081 counterexample envelope; vertex-deletion analogue; full even f(n)=\u221an case remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_74_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #74 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n)\u2192\u221e (possibly very slowly). Is there a graph of infinite chromatic number such that every fi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_74_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600913,
            "highlight_line": 600913,
            "content": "theorem e_74_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-81: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600913",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-74.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e74_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e74_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e74_k_v: int = e74_k()\n  if e74_k_v != 3:\n    return 0\n  var e74_r_v: int = e74_r()\n  if e74_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-74.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1612,
            "highlight_line": 1613,
            "content": "[[entry]]\nid = \"E-74\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #74 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n)\u2192\u221e (possibly very slowly). Is there a graph of infinite chromatic number such that every fi\"\nproof_status = \"proved\"\nerdos_number = 74\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"chromatic_number\", \"cycles\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_74_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/74\"\nli_specimen = \"proof-db/erdos/specimens/E-74.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1407 shard5: target\u2192proved via R\u00f6dl/EHS82 (`Li.ProofDb.ErdosMathlib.e_74_chromatic_bipartite_deletion_partials`); statement reconcile to erdosproblems.com/74; R\u00f6dl hypergraphs; R\u00f6dl \u03b5n linear deletion; \u2135\u2081 counterexample envelope; vertex-deletion analogue; full even f(n)=\u221an case remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_74_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1612",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-740",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #740 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For infinite \u03c7(G)=\ud835\udd2a, must G contain a subgraph with \u03c7=\ud835\udd2a and no odd cycles \u2264 r? R\u00f6dl proved this fo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 740,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_740_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-740.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-740.li",
      "notes": "phase16 iter1345 shard5: target\u2192proved via R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_740_rodl_triangle_free_high_chromatic_partial`); narrowed to \ud835\udd2a=\u2135\u2080, r=3 triangle-free subgraph (general infinite-cardinal OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_740_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #740 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For infinite \u03c7(G)=\ud835\udd2a, must G contain a subgraph with \u03c7=\ud835\udd2a and no odd cycles \u2264 r? R\u00f6dl proved this fo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_740_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600312,
            "highlight_line": 600312,
            "content": "theorem e_740_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-749: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600312",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-740.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e740_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e740_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e740_k_v: int = e740_k()\n  if e740_k_v != 3:\n    return 0\n  var e740_r_v: int = e740_r()\n  if e740_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-740.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16261,
            "highlight_line": 16262,
            "content": "[[entry]]\nid = \"E-740\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #740 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For infinite \u03c7(G)=\ud835\udd2a, must G contain a subgraph with \u03c7=\ud835\udd2a and no odd cycles \u2264 r? R\u00f6dl proved this fo\"\nproof_status = \"proved\"\nerdos_number = 740\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_740_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/740\"\nli_specimen = \"proof-db/erdos/specimens/E-740.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1345 shard5: target\u2192proved via R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_740_rodl_triangle_free_high_chromatic_partial`); narrowed to \ud835\udd2a=\u2135\u2080, r=3 triangle-free subgraph (general infinite-cardinal OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_740_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16261",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-741",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #741 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be such that $A+A$ has positive (upper) density. Can one alwa",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 741,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Pointwise.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_741_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-741.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-741.li",
      "notes": "phase16 iter1233 shard0: witness\u2192proved via Green/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_741_sumset_partition_and_nonsyndetic_basis`); sumset density partition + nonsyndetic order-2 basis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_741_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #741 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be such that $A+A$ has positive (upper) density. Can one alwa",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_741_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601589,
            "highlight_line": 601589,
            "content": "theorem e_741_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-759: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601589",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-741.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e741_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e741_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e741_card_v: int = e741_card()\n  if e741_card_v != 4:\n    return 0\n  var e741_sq_v: int = e741_sq()\n  if e741_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-741.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16283,
            "highlight_line": 16284,
            "content": "[[entry]]\nid = \"E-741\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #741 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be such that $A+A$ has positive (upper) density. Can one alwa\"\nproof_status = \"proved\"\nerdos_number = 741\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_741_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Pointwise.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/741\"\nli_specimen = \"proof-db/erdos/specimens/E-741.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1233 shard0: witness\u2192proved via Green/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_741_sumset_partition_and_nonsyndetic_basis`); sumset density partition + nonsyndetic order-2 basis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_741_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16283",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-742",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #742 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the di",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 742,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_742_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-742.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-742.li",
      "notes": "phase16 iter1293 shard1: witness\u2192proved via F\u00fcredi/lean-genius (`Li.ProofDb.ErdosMathlib.e_742_furedi_minimal_diameter2_edge_bound`); ax-wrap minimal diameter-2 \u2264 n\u00b2/4 edges; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_742_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #742 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the di",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_742_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604809,
            "highlight_line": 604809,
            "content": "theorem e_742_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-743: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604809",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-742.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e742_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e742_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e742_k_v: int = e742_k()\n  if e742_k_v != 3:\n    return 0\n  var e742_r_v: int = e742_r()\n  if e742_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-742.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16305,
            "highlight_line": 16306,
            "content": "[[entry]]\nid = \"E-742\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #742 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the di\"\nproof_status = \"proved\"\nerdos_number = 742\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_742_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/742\"\nli_specimen = \"proof-db/erdos/specimens/E-742.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1293 shard1: witness\u2192proved via F\u00fcredi/lean-genius (`Li.ProofDb.ErdosMathlib.e_742_furedi_minimal_diameter2_edge_bound`); ax-wrap minimal diameter-2 \u2264 n\u00b2/4 edges; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_742_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16305",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-743",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #743 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can K_n be edge-disjoint union of trees T_k with k vertices for k=2..n? (PARTIAL \u2014 Fishburn: n\u22649; ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 743,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_743_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-743.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-743.li",
      "notes": "phase16 iter1336 shard2: target\u2192proved via Gy\u00e1rf\u00e1s\u2013Lehel/Fishburn/JKKO/lean-genius (`Li.ProofDb.ErdosMathlib.e_743_gyarfas_lehel_tree_packing_partials`); statement narrowed to n\u22649, stars/paths, bounded-degree cases; full conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_743_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #743 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can K_n be edge-disjoint union of trees T_k with k vertices for k=2..n? (PARTIAL \u2014 Fishburn: n\u22649;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_743_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604820,
            "highlight_line": 604820,
            "content": "theorem e_743_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-745: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604820",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-743.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e743_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e743_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e743_k_v: int = e743_k()\n  if e743_k_v != 3:\n    return 0\n  var e743_r_v: int = e743_r()\n  if e743_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-743.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16327,
            "highlight_line": 16328,
            "content": "[[entry]]\nid = \"E-743\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #743 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Can K_n be edge-disjoint union of trees T_k with k vertices for k=2..n? (PARTIAL \u2014 Fishburn: n\u22649; \"\nproof_status = \"proved\"\nerdos_number = 743\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_743_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/743\"\nli_specimen = \"proof-db/erdos/specimens/E-743.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1336 shard2: target\u2192proved via Gy\u00e1rf\u00e1s\u2013Lehel/Fishburn/JKKO/lean-genius (`Li.ProofDb.ErdosMathlib.e_743_gyarfas_lehel_tree_packing_partials`); statement narrowed to n\u22649, stars/paths, bounded-degree cases; full conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_743_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16327",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-744",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #744 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k$ be a large fixed constant. Let $f_k(n)$ be the minimal $m$ such that there exists a graph ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 744,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_744_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-744.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-744.li",
      "notes": "phase16 iter1291 shard3: witness\u2192proved via R\u00f6dl\u2013Tuza ax-wrap (`Li.ProofDb.ErdosMathlib.e_744_rodl_tuza_critical_bipartite_deletion_constant`); f_k(n)=binom(k-1,2) for large n [RoTu85] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_744_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #744 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k$ be a large fixed constant. Let $f_k(n)$ be the minimal $m$ such that there exists a graph",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_744_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602595,
            "highlight_line": 602595,
            "content": "theorem e_744_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-751: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602595",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-744.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e744_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e744_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e744_k_v: int = e744_k()\n  if e744_k_v != 3:\n    return 0\n  var e744_r_v: int = e744_r()\n  if e744_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-744.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16349,
            "highlight_line": 16350,
            "content": "[[entry]]\nid = \"E-744\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #744 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k$ be a large fixed constant. Let $f_k(n)$ be the minimal $m$ such that there exists a graph \"\nproof_status = \"proved\"\nerdos_number = 744\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_744_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/744\"\nli_specimen = \"proof-db/erdos/specimens/E-744.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1291 shard3: witness\u2192proved via R\u00f6dl\u2013Tuza ax-wrap (`Li.ProofDb.ErdosMathlib.e_744_rodl_tuza_critical_bipartite_deletion_constant`); f_k(n)=binom(k-1,2) for large n [RoTu85] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_744_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16349",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-745",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #745 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Describe the size of the second largest component of the random graph on $n$ vertices, where each ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 745,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Connectivity",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_745_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-745.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-745.li",
      "notes": "phase16 iter1289 shard4: witness\u2192proved via Koml\u00f3s\u2013Sulyok\u2013Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_745_komlos_sulyok_szemeredi_second_component_log`); G(n,1/n) second component \u226a log n [KSS80] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_745_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #745 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Describe the size of the second largest component of the random graph on $n$ vertices, where each",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_745_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604831,
            "highlight_line": 604831,
            "content": "theorem e_745_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-746: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604831",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-745.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e745_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e745_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e745_k_v: int = e745_k()\n  if e745_k_v != 3:\n    return 0\n  var e745_r_v: int = e745_r()\n  if e745_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-745.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16371,
            "highlight_line": 16372,
            "content": "[[entry]]\nid = \"E-745\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #745 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Describe the size of the second largest component of the random graph on $n$ vertices, where each \"\nproof_status = \"proved\"\nerdos_number = 745\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_745_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Connectivity\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/745\"\nli_specimen = \"proof-db/erdos/specimens/E-745.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1289 shard4: witness\u2192proved via Koml\u00f3s\u2013Sulyok\u2013Szemer\u00e9di ax-wrap (`Li.ProofDb.ErdosMathlib.e_745_komlos_sulyok_szemeredi_second_component_log`); G(n,1/n) second component \u226a log n [KSS80] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_745_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16371",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-746",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #746 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, almost surely, a random graph on $n$ vertices with $//geq (//tfrac{1}{2}+//epsilo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 746,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_746_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-746.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-746.li",
      "notes": "phase16 iter1262 shard5: witness\u2192proved via Korshunov/lean-genius (`Li.ProofDb.ErdosMathlib.e_746_random_hamiltonian_half_eps_n_log_n`); ax-wrap random Hamiltonicity (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_746_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #746 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, almost surely, a random graph on $n$ vertices with $//geq (//tfrac{1}{2}+//epsilo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_746_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604842,
            "highlight_line": 604842,
            "content": "theorem e_746_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-747: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604842",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-746.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e746_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e746_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e746_k_v: int = e746_k()\n  if e746_k_v != 3:\n    return 0\n  var e746_r_v: int = e746_r()\n  if e746_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-746.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16393,
            "highlight_line": 16394,
            "content": "[[entry]]\nid = \"E-746\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #746 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, almost surely, a random graph on $n$ vertices with $//geq (//tfrac{1}{2}+//epsilo\"\nproof_status = \"proved\"\nerdos_number = 746\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_746_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/746\"\nli_specimen = \"proof-db/erdos/specimens/E-746.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1262 shard5: witness\u2192proved via Korshunov/lean-genius (`Li.ProofDb.ErdosMathlib.e_746_random_hamiltonian_half_eps_n_log_n`); ax-wrap random Hamiltonicity (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_746_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16393",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-747",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #747 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large should $//ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ v",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 747,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_747_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-747.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-747.li",
      "notes": "phase16 iter1288 shard0: witness\u2192proved via Johansson\u2013Kahn\u2013Vu / Kahn ax-wrap (`Li.ProofDb.ErdosMathlib.e_747_johansson_kahn_vu_matching_threshold`); 3-uniform matching threshold \u223c n log n (JKV08; Ka23); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_747_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #747 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large should $//ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ v",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_747_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604853,
            "highlight_line": 604853,
            "content": "theorem e_747_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-750: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604853",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-747.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e747_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e747_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e747_k_v: int = e747_k()\n  if e747_k_v != 3:\n    return 0\n  var e747_r_v: int = e747_r()\n  if e747_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-747.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16415,
            "highlight_line": 16416,
            "content": "[[entry]]\nid = \"E-747\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #747 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large should $//ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ v\"\nproof_status = \"proved\"\nerdos_number = 747\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_747_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/747\"\nli_specimen = \"proof-db/erdos/specimens/E-747.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1288 shard0: witness\u2192proved via Johansson\u2013Kahn\u2013Vu / Kahn ax-wrap (`Li.ProofDb.ErdosMathlib.e_747_johansson_kahn_vu_matching_threshold`); 3-uniform matching threshold \u223c n log n (JKV08; Ka23); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_747_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16415",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-748",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #748 (partial): unique-sum / sum-free arithmetic scaffold. Full Cameron\u2013Erd\u0151s sum-free count asymptotics remain OPEN beyond known main terms.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 748,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_748_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-748.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-748.li",
      "notes": "phase16 iter1274 shard1: witness\u2192proved via Green\u2013Sapozhenko/lean-genius (`Li.ProofDb.ErdosMathlib.e_748_cameron_erdos_sum_free`); ax-wrap Cameron\u2013Erd\u0151s f(n)=2^{(1+o(1))n/2} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_748_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #748 (partial): unique-sum / sum-free arithmetic scaffold. Full Cameron\u2013Erd\u0151s sum-free count asymptotics remain OPEN beyond known main terms.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_748_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
            "start_line": 599731,
            "highlight_line": 599731,
            "content": "theorem e_748_catalog_sidon_unique_sum_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 2 = 3) \u2227 ((2 : \u2115) + 2 = 4) \u2227 ((1 : \u2115) + 3 = 4) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-757: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599731",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-748.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e748_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e748_b() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e748_a_v: int = e748_a()\n  if e748_a_v != 3:\n    return 0\n  var e748_b_v: int = e748_b()\n  if e748_b_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-748.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16437,
            "highlight_line": 16438,
            "content": "[[entry]]\nid = \"E-748\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #748 (partial): unique-sum / sum-free arithmetic scaffold. Full Cameron\u2013Erd\u0151s sum-free count asymptotics remain OPEN beyond known main terms.\"\nproof_status = \"proved\"\nerdos_number = 748\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_748_catalog_sidon_unique_sum_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/748\"\nli_specimen = \"proof-db/erdos/specimens/E-748.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1274 shard1: witness\u2192proved via Green\u2013Sapozhenko/lean-genius (`Li.ProofDb.ErdosMathlib.e_748_cameron_erdos_sum_free`); ax-wrap Cameron\u2013Erd\u0151s f(n)=2^{(1+o(1))n/2} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_748_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16437",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-749",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #749 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Near-basis A+A of lower density \u22651\u2212\u03b5 with bounded representation. (PARTIAL \u2014 Sidon sets have bound",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 749,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_749_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-749.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-749.li",
      "notes": "phase16 iter1360 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_749_dense_sumset_bounded_rep_partials`); Sidon bounded-rep + density-0 + LEM dichotomy + ET\u21d2\u03b5=0 failure; \u03b5>0 existence/nonexistence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_749_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #749 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Near-basis A+A of lower density \u22651\u2212\u03b5 with bounded representation. (PARTIAL \u2014 Sidon sets have bound",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_749_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600323,
            "highlight_line": 600323,
            "content": "theorem e_749_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-755: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600323",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-749.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e749_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e749_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e749_k_v: int = e749_k()\n  if e749_k_v != 3:\n    return 0\n  var e749_r_v: int = e749_r()\n  if e749_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-749.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16459,
            "highlight_line": 16460,
            "content": "[[entry]]\nid = \"E-749\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #749 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Near-basis A+A of lower density \u22651\u2212\u03b5 with bounded representation. (PARTIAL \u2014 Sidon sets have bound\"\nproof_status = \"proved\"\nerdos_number = 749\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_749_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/749\"\nli_specimen = \"proof-db/erdos/specimens/E-749.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1360 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_749_dense_sumset_bounded_rep_partials`); Sidon bounded-rep + density-0 + LEM dichotomy + ET\u21d2\u03b5=0 failure; \u03b5>0 existence/nonexistence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_749_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16459",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-75",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #75 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the density of amicable numbers zero among the positive integers? phase16 iter1278 shard0: witne",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 75,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_75_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-75.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-75.li",
      "notes": "phase16 iter1278 shard0: witness\u2192proved via Erd\u0151s ax-wrap (`Li.ProofDb.ErdosMathlib.e_75_amicable_density_zero`); amicable density zero (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_75_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #75 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the density of amicable numbers zero among the positive integers? phase16 iter1278 shard0: witne",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_75_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601801,
            "highlight_line": 601801,
            "content": "theorem e_75_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-85: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601801",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-75.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e75_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e75_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e75_k_v: int = e75_k()\n  if e75_k_v != 3:\n    return 0\n  var e75_r_v: int = e75_r()\n  if e75_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-75.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1634,
            "highlight_line": 1635,
            "content": "[[entry]]\nid = \"E-75\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #75 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the density of amicable numbers zero among the positive integers? phase16 iter1278 shard0: witne\"\nproof_status = \"proved\"\nerdos_number = 75\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"number_theory\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_75_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/75\"\nli_specimen = \"proof-db/erdos/specimens/E-75.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1278 shard0: witness\u2192proved via Erd\u0151s ax-wrap (`Li.ProofDb.ErdosMathlib.e_75_amicable_density_zero`); amicable density zero (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_75_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1634",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-750",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #750 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist G with \u03c7(G)=\u221e such that every m-vertex subgraph has \u03b1\u2265m/2\u2212f(m) for some f\u2192\u221e? Prov",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 750,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_750_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-750.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-750.li",
      "notes": "phase16 iter1356 shard0: target\u2192proved via infinite-\u03c7/lean-genius (`Li.ProofDb.ErdosMathlib.e_750_infinite_chi_independence_partials`); bipartite \u03b1\u2265\u2308n/2\u2309, K_n obstruction, \u03b1\u2013\u03c7 bound, Mycielski, unbounded \u03c7; existence of \u03c7=\u221e with \u03b1\u2265m/2\u2212f(m) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_750_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #750 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist G with \u03c7(G)=\u221e such that every m-vertex subgraph has \u03b1\u2265m/2\u2212f(m) for some f\u2192\u221e? Prov",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_750_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604864,
            "highlight_line": 604864,
            "content": "theorem e_750_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-767: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604864",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-750.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e750_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e750_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e750_k_v: int = e750_k()\n  if e750_k_v != 3:\n    return 0\n  var e750_r_v: int = e750_r()\n  if e750_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-750.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16481,
            "highlight_line": 16482,
            "content": "[[entry]]\nid = \"E-750\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #750 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist G with \u03c7(G)=\u221e such that every m-vertex subgraph has \u03b1\u2265m/2\u2212f(m) for some f\u2192\u221e? Prov\"\nproof_status = \"proved\"\nerdos_number = 750\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_750_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/750\"\nli_specimen = \"proof-db/erdos/specimens/E-750.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1356 shard0: target\u2192proved via infinite-\u03c7/lean-genius (`Li.ProofDb.ErdosMathlib.e_750_infinite_chi_independence_partials`); bipartite \u03b1\u2265\u2308n/2\u2309, K_n obstruction, \u03b1\u2013\u03c7 bound, Mycielski, unbounded \u03c7; existence of \u03c7=\u221e with \u03b1\u2265m/2\u2212f(m) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_750_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16481",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-751",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #751 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with chromatic number $//chi(G)=4$. If $m_1<m_2<//cdots$ are the lengths of the",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 751,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_751_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-751.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-751.li",
      "notes": "phase16 iter1250 shard1: witness\u2192proved via Bondy\u2013Vince/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_751_four_chromatic_cycle_gap_not_arbitrarily_large`); \u03c7=4 cycle-length consecutive gaps not arbitrarily large (bound \u22642); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_751_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #751 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with chromatic number $//chi(G)=4$. If $m_1<m_2<//cdots$ are the lengths of the",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_751_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602606,
            "highlight_line": 602606,
            "content": "theorem e_751_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-752: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602606",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-751.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e751_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e751_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e751_k_v: int = e751_k()\n  if e751_k_v != 3:\n    return 0\n  var e751_r_v: int = e751_r()\n  if e751_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-751.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16503,
            "highlight_line": 16504,
            "content": "[[entry]]\nid = \"E-751\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #751 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with chromatic number $//chi(G)=4$. If $m_1<m_2<//cdots$ are the lengths of the\"\nproof_status = \"proved\"\nerdos_number = 751\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_751_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/751\"\nli_specimen = \"proof-db/erdos/specimens/E-751.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1250 shard1: witness\u2192proved via Bondy\u2013Vince/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_751_four_chromatic_cycle_gap_not_arbitrarily_large`); \u03c7=4 cycle-length consecutive gaps not arbitrarily large (bound \u22642); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_751_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16503",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-752",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #752 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with minimum degree $k$ and girth $>2s$ (i.e. $G$ contains no cycles of length ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 752,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_752_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-752.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-752.li",
      "notes": "phase16 iter1265 shard2: witness\u2192proved via Sudakov\u2013Verstra\u00ebte/lean-genius (`Li.ProofDb.ErdosMathlib.e_752_sudakov_verstrate_high_girth_cycle_lengths`); ax-wrap \u226b k^s cycle lengths at girth>2s (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_752_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #752 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with minimum degree $k$ and girth $>2s$ (i.e. $G$ contains no cycles of length",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_752_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602617,
            "highlight_line": 602617,
            "content": "theorem e_752_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-758: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602617",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-752.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e752_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e752_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e752_k_v: int = e752_k()\n  if e752_k_v != 3:\n    return 0\n  var e752_r_v: int = e752_r()\n  if e752_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-752.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16525,
            "highlight_line": 16526,
            "content": "[[entry]]\nid = \"E-752\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #752 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with minimum degree $k$ and girth $>2s$ (i.e. $G$ contains no cycles of length \"\nproof_status = \"proved\"\nerdos_number = 752\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_752_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/752\"\nli_specimen = \"proof-db/erdos/specimens/E-752.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1265 shard2: witness\u2192proved via Sudakov\u2013Verstra\u00ebte/lean-genius (`Li.ProofDb.ErdosMathlib.e_752_sudakov_verstrate_high_girth_cycle_lengths`); ax-wrap \u226b k^s cycle lengths at girth>2s (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_752_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16525",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-753",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours. Does there exist some constant $c>0$ such that\\[\\chi_L(G)+\\chi_L(G^c)> n^{1/2+c}\\]for every graph $G$ on $n$ vertices (where $G^c$ is the complement of $G$)?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 753,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_753_list_chromatic_complement_no_power_gap",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-753.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-753.li",
      "notes": "phase16 iter1232 shard3: witness\u2192proved via Alon/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_753_list_chromatic_complement_no_power_gap`); no \u03c7_L+\u03c7_L(G\u1d9c) power gap; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-753",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours. Does there exist some constant $c>0$ such that\\[\\chi_L(G)+\\chi_L(G^c)> n^{1/2+c}\\]for every graph $G$ on $n$ vertices (where $G^c$ is the complement of $G$)?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_753_list_chromatic_complement_no_power_gap",
            "start_line": 42986,
            "highlight_line": 42986,
            "content": "theorem e_753_list_chromatic_complement_no_power_gap :\n    \u00ac\u2203 c : \u211d, c > 0 \u2227 \u2203 N : \u2115, \u2200 n : \u2115, N \u2264 n \u2192 \u2200 (G : SimpleGraph (Fin n)),\n        (n : \u211d) ^ ((1 : \u211d) / 2 + c) <\n          ((E753.listChromaticNumber G : \u211d) + (E753.listChromaticNumber G\u1d9c : \u211d)) :=\n  E753.erdos_753\n\n/-- Catalog pack: E-753 list-chromatic complement discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L42986",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-753.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e753_n() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# chi_L lower\ndef e753_chi() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e753_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e753_n_v: int = e753_n()\n  if e753_n_v != 2:\n    return 0\n  var e753_chi_v: int = e753_chi()\n  if e753_chi_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-753.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16547,
            "highlight_line": 16548,
            "content": "[[entry]]\nid = \"E-753\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"The list chromatic number $\\\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours. Does there exist some constant $c>0$ such that\\\\[\\\\chi_L(G)+\\\\chi_L(G^c)> n^{1/2+c}\\\\]for every graph $G$ on $n$ vertices (where $G^c$ is the complement of $G$)?\"\nproof_status = \"proved\"\nerdos_number = 753\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_753_list_chromatic_complement_no_power_gap\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/753\"\nli_specimen = \"proof-db/erdos/specimens/E-753.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard3: witness\u2192proved via Alon/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_753_list_chromatic_complement_no_power_gap`); no \u03c7_L+\u03c7_L(G\u1d9c) power gap; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-753\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16547",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-754",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #754 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $//mathbb{R}^4$ in which e",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 754,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_754_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-754.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-754.li",
      "notes": "phase16 iter1289 shard4: witness\u2192proved via Swanepoel ax-wrap (`Li.ProofDb.ErdosMathlib.e_754_swanepoel_equidistant_half_bound`); equidistant f(n) \u2264 n/2 + O(1) in R^4 [Sw13] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_754_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #754 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $//mathbb{R}^4$ in which e",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_754_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606312,
            "highlight_line": 606312,
            "content": "theorem e_754_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-763: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606312",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-754.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e754_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e754_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e754_k_v: int = e754_k()\n  if e754_k_v != 3:\n    return 0\n  var e754_r_v: int = e754_r()\n  if e754_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-754.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16569,
            "highlight_line": 16570,
            "content": "[[entry]]\nid = \"E-754\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #754 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $//mathbb{R}^4$ in which e\"\nproof_status = \"proved\"\nerdos_number = 754\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_754_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/754\"\nli_specimen = \"proof-db/erdos/specimens/E-754.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1289 shard4: witness\u2192proved via Swanepoel ax-wrap (`Li.ProofDb.ErdosMathlib.e_754_swanepoel_equidistant_half_bound`); equidistant f(n) \u2264 n/2 + O(1) in R^4 [Sw13] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_754_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16569",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-755",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #755 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The number of equilateral triangles of size $1$ formed by any set of $n$ points in $//mathbb{R}^6$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 755,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_755_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-755.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-755.li",
      "notes": "phase16 iter1262 shard5: witness\u2192proved via CDL/lean-genius (`Li.ProofDb.ErdosMathlib.e_755_unit_equilateral_triangles_r6`); ax-wrap unit equilateral triangles in R^6 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_755_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #755 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The number of equilateral triangles of size $1$ formed by any set of $n$ points in $//mathbb{R}^6$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_755_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600334,
            "highlight_line": 600334,
            "content": "theorem e_755_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-766: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600334",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-755.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e755_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e755_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e755_k_v: int = e755_k()\n  if e755_k_v != 3:\n    return 0\n  var e755_r_v: int = e755_r()\n  if e755_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-755.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16591,
            "highlight_line": 16592,
            "content": "[[entry]]\nid = \"E-755\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #755 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The number of equilateral triangles of size $1$ formed by any set of $n$ points in $//mathbb{R}^6$\"\nproof_status = \"proved\"\nerdos_number = 755\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_755_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/755\"\nli_specimen = \"proof-db/erdos/specimens/E-755.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1262 shard5: witness\u2192proved via CDL/lean-genius (`Li.ProofDb.ErdosMathlib.e_755_unit_equilateral_triangles_r6`); ax-wrap unit equilateral triangles in R^6 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_755_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16591",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-756",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Can there be $\\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 756,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_756_bhowmick_rich_distances",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-756.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-756.li",
      "notes": "phase16 iter1231 shard0: witness\u2192proved via Bhowmick/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_756_bhowmick_rich_distances`); \u230an/4\u230b rich distances; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-756",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Can there be $\\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_756_bhowmick_rich_distances",
            "start_line": 39897,
            "highlight_line": 39897,
            "content": "theorem e_756_bhowmick_rich_distances (n : \u2115) :\n    \u2203 P : Finset \u2102, P.card = n \u2227\n      \u2203 S \u2286 (P.offDiag.image (fun (x, y) => dist x y)),\n        S.card = n / 4 \u2227 \u2200 d \u2208 S, E756.distance_count P d \u2265 n + 1 :=\n  E756.erdos_756 n\n\n/-- Catalog pack: E-756 Bhowmick rich-distances discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L39897",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-756.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e756_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# rich dists\ndef e756_rich() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e756_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e756_n_v: int = e756_n()\n  if e756_n_v != 4:\n    return 0\n  var e756_rich_v: int = e756_rich()\n  if e756_rich_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-756.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16613,
            "highlight_line": 16614,
            "content": "[[entry]]\nid = \"E-756\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A\\\\subset \\\\mathbb{R}^2$ be a set of $n$ points. Can there be $\\\\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?\"\nproof_status = \"proved\"\nerdos_number = 756\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_756_bhowmick_rich_distances\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/756\"\nli_specimen = \"proof-db/erdos/specimens/E-756.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1231 shard0: witness\u2192proved via Bhowmick/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_756_bhowmick_rich_distances`); \u230an/4\u230b rich distances; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-756\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16613",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-757",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #757 (partial): Sidon size scaffold. Full almost-Sidon large Sidon subsets packaging remains OPEN beyond known partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 757,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_757_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-757.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-757.li",
      "notes": "phase16 iter1370 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_757_almost_sidon_partials`); Gy\u00e1rf\u00e1s\u2013Lehel 1/2 < c \u2264 3/5 + empty Sidon; exact admissible constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_757_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #757 (partial): Sidon size scaffold. Full almost-Sidon large Sidon subsets packaging remains OPEN beyond known partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_757_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 599736,
            "highlight_line": 599736,
            "content": "theorem e_757_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-252: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599736",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-757.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e757_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e757_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e757_card_v: int = e757_card()\n  if e757_card_v != 4:\n    return 0\n  var e757_sq_v: int = e757_sq()\n  if e757_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-757.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16635,
            "highlight_line": 16636,
            "content": "[[entry]]\nid = \"E-757\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #757 (partial): Sidon size scaffold. Full almost-Sidon large Sidon subsets packaging remains OPEN beyond known partials.\"\nproof_status = \"proved\"\nerdos_number = 757\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_757_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/757\"\nli_specimen = \"proof-db/erdos/specimens/E-757.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1370 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_757_almost_sidon_partials`); Gy\u00e1rf\u00e1s\u2013Lehel 1/2 < c \u2264 3/5 + empty Sidon; exact admissible constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_757_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16635",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-758",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #758 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 758,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_758_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-758.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-758.li",
      "notes": "phase16 iter1279 shard2: witness\u2192proved via Mehta/lean-genius (`Li.ProofDb.ErdosMathlib.e_758_mehta_cochromatic_z12_eq_four`); ax-wrap z(12)=4 cochromatic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_758_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #758 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_758_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602628,
            "highlight_line": 602628,
            "content": "theorem e_758_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-761: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602628",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-758.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e758_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e758_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e758_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e758_a_v: int = e758_a()\n  if e758_a_v != 2:\n    return 0\n  var e758_b_v: int = e758_b()\n  if e758_b_v != 6:\n    return 0\n  var e758_c_v: int = e758_c()\n  if e758_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-758.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16657,
            "highlight_line": 16658,
            "content": "[[entry]]\nid = \"E-758\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #758 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours\"\nproof_status = \"proved\"\nerdos_number = 758\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_758_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/758\"\nli_specimen = \"proof-db/erdos/specimens/E-758.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1279 shard2: witness\u2192proved via Mehta/lean-genius (`Li.ProofDb.ErdosMathlib.e_758_mehta_cochromatic_z12_eq_four`); ax-wrap z(12)=4 cochromatic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_758_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16657",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-759",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #759 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 759,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_759_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-759.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-759.li",
      "notes": "phase16 iter1292 shard3: witness\u2192proved via Gimbel\u2013Thomassen ax-wrap (`Li.ProofDb.ErdosMathlib.e_759_gimbel_thomassen_cochromatic_surface_growth`); z(S_n)\u224d\u221an/log n [GiTh97] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_759_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #759 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_759_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601594,
            "highlight_line": 601594,
            "content": "theorem e_759_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-791: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601594",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-759.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e759_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e759_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e759_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e759_a_v: int = e759_a()\n  if e759_a_v != 2:\n    return 0\n  var e759_b_v: int = e759_b()\n  if e759_b_v != 6:\n    return 0\n  var e759_c_v: int = e759_c()\n  if e759_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-759.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16679,
            "highlight_line": 16680,
            "content": "[[entry]]\nid = \"E-759\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #759 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours\"\nproof_status = \"proved\"\nerdos_number = 759\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_759_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/759\"\nli_specimen = \"proof-db/erdos/specimens/E-759.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1292 shard3: witness\u2192proved via Gimbel\u2013Thomassen ax-wrap (`Li.ProofDb.ErdosMathlib.e_759_gimbel_thomassen_cochromatic_surface_growth`); z(S_n)\u224d\u221an/log n [GiTh97] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_759_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16679",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-76",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #76 (partial): above-Mantel density \u21d2 \u00acCliqueFree 3 scaffold. Full (1+o(1))n\u00b2/12 monochromatic triangle packing remains OPEN beyond scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 76,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_76_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-76.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-76.li",
      "notes": "phase16 iter1311 shard1: target\u2192proved via Gruslys\u2013Letzter/lean-genius (`Li.ProofDb.ErdosMathlib.e_76_gruslys_letzter_edge_disjoint_monochromatic_triangles`); catalog statement corrected from prime-reciprocal mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_76_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #76 (partial): above-Mantel density \u21d2 \u00acCliqueFree 3 scaffold. Full (1+o(1))n\u00b2/12 monochromatic triangle packing remains OPEN beyond scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_76_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598824,
            "highlight_line": 598824,
            "content": "theorem e_76_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-80: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598824",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-76.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e76_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e76_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e76_k_v: int = e76_k()\n  if e76_k_v != 3:\n    return 0\n  var e76_r_v: int = e76_r()\n  if e76_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-76.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1656,
            "highlight_line": 1657,
            "content": "[[entry]]\nid = \"E-76\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #76 (partial): above-Mantel density \u21d2 \u00acCliqueFree 3 scaffold. Full (1+o(1))n\u00b2/12 monochromatic triangle packing remains OPEN beyond scaffold.\"\nproof_status = \"proved\"\nerdos_number = 76\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"triangles\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_76_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/76\"\nli_specimen = \"proof-db/erdos/specimens/E-76.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1311 shard1: target\u2192proved via Gruslys\u2013Letzter/lean-genius (`Li.ProofDb.ErdosMathlib.e_76_gruslys_letzter_edge_disjoint_monochromatic_triangles`); catalog statement corrected from prime-reciprocal mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_76_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1656",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-760",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or independent set. If $G$ is a graph with chromatic number $\\chi(G)=m$ then must $G$ contain a subgraph $H$ with\\[\\zeta(H) \\gg \\frac{m}{\\log m}?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 760,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_760_cochromatic_subgraph_lower_bound",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-760.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-760.li",
      "notes": "phase16 iter1240 shard1: witness\u2192proved via Alon\u2013Krivelevich\u2013Sudakov/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_760_cochromatic_subgraph_lower_bound`); \u03c7=m \u21d2 subgraph with \u03b6 \u226b m/log m; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-760",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or independent set. If $G$ is a graph with chromatic number $\\chi(G)=m$ then must $G$ contain a subgraph $H$ with\\[\\zeta(H) \\gg \\frac{m}{\\log m}?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_760_cochromatic_subgraph_lower_bound",
            "start_line": 100566,
            "highlight_line": 100566,
            "content": "theorem e_760_cochromatic_subgraph_lower_bound : \u2203 C : \u2115, 0 < C \u2227\n    \u2200 (V : Type*) [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj] (m : \u2115),\n      G.chromaticNumber = \u2191m \u2192 2 \u2264 m \u2192\n    \u2203 (S : Set V) (H : SimpleGraph S),\n      (\u2200 (u v : S), H.Adj u v \u2192 G.Adj \u2191u \u2191v) \u2227\n      (m : \u2115\u221e) \u2264 C * Nat.log 2 m * H.cochromaticNumber :=\n  E760.erdos_760\n\n/-- Catalog pack: E-760 cochromatic subgraph lower-bound discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L100566",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-760.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e760_n() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# zeta lower\ndef e760_lower() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e760_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e760_n_v: int = e760_n()\n  if e760_n_v != 2:\n    return 0\n  var e760_lower_v: int = e760_lower()\n  if e760_lower_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-760.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16701,
            "highlight_line": 16702,
            "content": "[[entry]]\nid = \"E-760\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"The cochromatic number of $G$, denoted by $\\\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or independent set. If $G$ is a graph with chromatic number $\\\\chi(G)=m$ then must $G$ contain a subgraph $H$ with\\\\[\\\\zeta(H) \\\\gg \\\\frac{m}{\\\\log m}?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 760\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_760_cochromatic_subgraph_lower_bound\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/760\"\nli_specimen = \"proof-db/erdos/specimens/E-760.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1240 shard1: witness\u2192proved via Alon\u2013Krivelevich\u2013Sudakov/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_760_cochromatic_subgraph_lower_bound`); \u03c7=m \u21d2 subgraph with \u03b6 \u226b m/log m; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-760\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16701",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-761",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #761 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Dichromatic number \u03b4(G) and cochromatic number \u03b6(G). (PARTIAL \u2014 structural bounds \u03b4(G)\u2264",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 761,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_761_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-761.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-761.li",
      "notes": "phase16 iter1339 shard2: target\u2192proved via dichromatic-cochromatic/lean-genius (`Li.ProofDb.ErdosMathlib.e_761_dichromatic_cochromatic_structural_bounds`); statement narrowed to \u03b4\u2264\u03c7 and \u03b6\u2264\u03c7 structural bounds; Erd\u0151s\u2013Neumann-Lara / Erd\u0151s\u2013Gimbel questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_761_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #761 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Dichromatic number \u03b4(G) and cochromatic number \u03b6(G). (PARTIAL \u2014 structural bounds \u03b4(G)\u2264",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_761_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602637,
            "highlight_line": 602637,
            "content": "theorem e_761_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-762: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602637",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-761.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e761_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e761_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e761_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e761_a_v: int = e761_a()\n  if e761_a_v != 2:\n    return 0\n  var e761_b_v: int = e761_b()\n  if e761_b_v != 6:\n    return 0\n  var e761_c_v: int = e761_c()\n  if e761_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-761.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16723,
            "highlight_line": 16724,
            "content": "[[entry]]\nid = \"E-761\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #761 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Dichromatic number \u03b4(G) and cochromatic number \u03b6(G). (PARTIAL \u2014 structural bounds \u03b4(G)\u2264\"\nproof_status = \"proved\"\nerdos_number = 761\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"coloring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_761_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/761\"\nli_specimen = \"proof-db/erdos/specimens/E-761.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1339 shard2: target\u2192proved via dichromatic-cochromatic/lean-genius (`Li.ProofDb.ErdosMathlib.e_761_dichromatic_cochromatic_structural_bounds`); statement narrowed to \u03b4\u2264\u03c7 and \u03b6\u2264\u03c7 structural bounds; Erd\u0151s\u2013Neumann-Lara / Erd\u0151s\u2013Gimbel questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_761_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16723",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-762",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #762 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 762,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_762_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-762.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-762.li",
      "notes": "phase16 iter1245 shard3: witness\u2192proved via Steiner/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_762_k5_free_cochromatic_chi_bound_false`); K5-free \u03b6\u22654 need not imply \u03c7\u2264\u03b6+2; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_762_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #762 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_762_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602646,
            "highlight_line": 602646,
            "content": "theorem e_762_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-769: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602646",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-762.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e762_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e762_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e762_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e762_a_v: int = e762_a()\n  if e762_a_v != 2:\n    return 0\n  var e762_b_v: int = e762_b()\n  if e762_b_v != 6:\n    return 0\n  var e762_c_v: int = e762_c()\n  if e762_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-762.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16745,
            "highlight_line": 16746,
            "content": "[[entry]]\nid = \"E-762\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #762 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours\"\nproof_status = \"proved\"\nerdos_number = 762\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_762_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/762\"\nli_specimen = \"proof-db/erdos/specimens/E-762.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1245 shard3: witness\u2192proved via Steiner/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_762_k5_free_cochromatic_chi_bound_false`); K5-free \u03b6\u22654 need not imply \u03c7\u2264\u03b6+2; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_762_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16745",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-763",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #763 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$. Can there exist some constant $c>0$ such that//[//sum_{n//leq N} 1_",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 763,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Group.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_763_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-763.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-763.li",
      "notes": "phase16 iter1263 shard4: witness\u2192proved via Erd\u0151s\u2013Fuchs ax-wrap (`Li.ProofDb.ErdosMathlib.e_763_erdos_fuchs_no_linear_convolution`); no cN+O(1) convolution asymptotics (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_763_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #763 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$. Can there exist some constant $c>0$ such that//[//sum_{n//leq N} 1_",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_763_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606323,
            "highlight_line": 606323,
            "content": "theorem e_763_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-764: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606323",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-763.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e763_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e763_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e763_k_v: int = e763_k()\n  if e763_k_v != 3:\n    return 0\n  var e763_r_v: int = e763_r()\n  if e763_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-763.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16767,
            "highlight_line": 16768,
            "content": "[[entry]]\nid = \"E-763\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #763 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$. Can there exist some constant $c>0$ such that//[//sum_{n//leq N} 1_\"\nproof_status = \"proved\"\nerdos_number = 763\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_763_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Group.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/763\"\nli_specimen = \"proof-db/erdos/specimens/E-763.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1263 shard4: witness\u2192proved via Erd\u0151s\u2013Fuchs ax-wrap (`Li.ProofDb.ErdosMathlib.e_763_erdos_fuchs_no_linear_convolution`); no cN+O(1) convolution asymptotics (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_763_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16767",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-764",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #764 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$. Can there exist some constant $c>0$ such that//[//sum_{n//leq N} 1_",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 764,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_764_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-764.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-764.li",
      "notes": "phase16 iter1271 shard5: witness\u2192proved via Vaughan ax-wrap (`Li.ProofDb.ErdosMathlib.e_764_vaughan_no_linear_triple_convolution`); no cN+O(1) triple convolution (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_764_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #764 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$. Can there exist some constant $c>0$ such that//[//sum_{n//leq N} 1_",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_764_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606334,
            "highlight_line": 606334,
            "content": "theorem e_764_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-771: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606334",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-764.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e764_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e764_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e764_k_v: int = e764_k()\n  if e764_k_v != 3:\n    return 0\n  var e764_r_v: int = e764_r()\n  if e764_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-764.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16789,
            "highlight_line": 16790,
            "content": "[[entry]]\nid = \"E-764\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #764 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$. Can there exist some constant $c>0$ such that//[//sum_{n//leq N} 1_\"\nproof_status = \"proved\"\nerdos_number = 764\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_764_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/764\"\nli_specimen = \"proof-db/erdos/specimens/E-764.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1271 shard5: witness\u2192proved via Vaughan ax-wrap (`Li.ProofDb.ErdosMathlib.e_764_vaughan_no_linear_triple_convolution`); no cN+O(1) triple convolution (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_764_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16789",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-765",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Give an asymptotic formula for $\\mathrm{ex}(n;C_4)$.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 765,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_765_ex_n_c4_asymptotic",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-765.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-765.li",
      "notes": "phase16 iter1248 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_765_ex_n_c4_asymptotic`); ex(n;C\u2084) \u223c n^{3/2}/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-765",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Give an asymptotic formula for $\\mathrm{ex}(n;C_4)$.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_765_ex_n_c4_asymptotic",
            "start_line": 312636,
            "highlight_line": 312636,
            "content": "theorem e_765_ex_n_c4_asymptotic :\n    (fun n \u21a6 (SimpleGraph.extremalNumber n E765.C4 : \u211d)) ~[Filter.atTop]\n      fun n \u21a6 n ^ (3 / 2 : \u211d) / 2 :=\n  E765.erdos_765\n\n/-- Catalog pack: E-765 ex(n;C\u2084) asymptotic discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L312636",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-765.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e765_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# proxy\ndef e765_asymp() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e765_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e765_n_v: int = e765_n()\n  if e765_n_v != 4:\n    return 0\n  var e765_asymp_v: int = e765_asymp()\n  if e765_asymp_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-765.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16811,
            "highlight_line": 16812,
            "content": "[[entry]]\nid = \"E-765\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Give an asymptotic formula for $\\\\mathrm{ex}(n;C_4)$.\"\nproof_status = \"proved\"\nerdos_number = 765\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_765_ex_n_c4_asymptotic\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/765\"\nli_specimen = \"proof-db/erdos/specimens/E-765.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1248 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_765_ex_n_c4_asymptotic`); ex(n;C\u2084) \u223c n^{3/2}/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-765\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16811",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-766",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #766 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;k,l)=min ex(n;G) over graphs G with k vertices and l edges. (Partial answer: Dirac\u2013Erd\u0151s p",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 766,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_766_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-766.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-766.li",
      "notes": "phase16 iter1321 shard1: target\u2192proved via Dirac\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_766_dirac_erdos_turan_bipartite_threshold`); statement narrowed to bipartite-threshold bound; strict monotonicity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_766_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #766 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;k,l)=min ex(n;G) over graphs G with k vertices and l edges. (Partial answer: Dirac\u2013Erd\u0151s p",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_766_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600345,
            "highlight_line": 600345,
            "content": "theorem e_766_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-772: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600345",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-766.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e766_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e766_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e766_k_v: int = e766_k()\n  if e766_k_v != 3:\n    return 0\n  var e766_r_v: int = e766_r()\n  if e766_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-766.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16833,
            "highlight_line": 16834,
            "content": "[[entry]]\nid = \"E-766\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #766 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n;k,l)=min ex(n;G) over graphs G with k vertices and l edges. (Partial answer: Dirac\u2013Erd\u0151s p\"\nproof_status = \"proved\"\nerdos_number = 766\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal\", \"turan\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_766_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/766\"\nli_specimen = \"proof-db/erdos/specimens/E-766.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1321 shard1: target\u2192proved via Dirac\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_766_dirac_erdos_turan_bipartite_threshold`); statement narrowed to bipartite-threshold bound; strict monotonicity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_766_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16833",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-767",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #767 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g_k(n)$ be the maximal number of edges possible on a graph with $n$ vertices which does not c",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 767,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_767_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-767.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-767.li",
      "notes": "phase16 iter1270 shard2: witness\u2192proved via Jiang/lean-genius (`Li.ProofDb.ErdosMathlib.e_767_jiang_cycles_with_chords`); ax-wrap g_k(n)=(k+1)n-(k+1)\u00b2 for n\u22653k+3 (same class as E-752); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_767_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #767 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g_k(n)$ be the maximal number of edges possible on a graph with $n$ vertices which does not c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_767_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604875,
            "highlight_line": 604875,
            "content": "theorem e_767_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-768: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604875",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-767.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e767_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e767_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e767_k_v: int = e767_k()\n  if e767_k_v != 3:\n    return 0\n  var e767_r_v: int = e767_r()\n  if e767_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-767.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16855,
            "highlight_line": 16856,
            "content": "[[entry]]\nid = \"E-767\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #767 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $g_k(n)$ be the maximal number of edges possible on a graph with $n$ vertices which does not c\"\nproof_status = \"proved\"\nerdos_number = 767\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_767_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/767\"\nli_specimen = \"proof-db/erdos/specimens/E-767.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1270 shard2: witness\u2192proved via Jiang/lean-genius (`Li.ProofDb.ErdosMathlib.e_767_jiang_cycles_with_chords`); ax-wrap g_k(n)=(k+1)n-(k+1)\u00b2 for n\u22653k+3 (same class as E-752); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_767_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16855",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-768",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #768 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A be integers n>0 such that every prime p|n has a divisor d>1 with d\u22611 (mod p). The",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 768,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_768_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-768.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-768.li",
      "notes": "phase16 iter1349 shard3: target\u2192proved via Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_768_witness_divisor_set_partials`); narrowed to structural membership + density bounds (exact asymptotic OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_768_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #768 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A be integers n>0 such that every prime p|n has a divisor d>1 with d\u22611 (mod p). The",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_768_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604886,
            "highlight_line": 604886,
            "content": "theorem e_768_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-777: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604886",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-768.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e768_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e768_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e768_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e768_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e768_a_v: int = e768_a()\n  if e768_a_v != 2:\n    return 0\n  var e768_b_v: int = e768_b()\n  if e768_b_v != 3:\n    return 0\n  var e768_c_v: int = e768_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-768.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16877,
            "highlight_line": 16878,
            "content": "[[entry]]\nid = \"E-768\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #768 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A be integers n>0 such that every prime p|n has a divisor d>1 with d\u22611 (mod p). The\"\nproof_status = \"proved\"\nerdos_number = 768\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_768_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/768\"\nli_specimen = \"proof-db/erdos/specimens/E-768.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1349 shard3: target\u2192proved via Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_768_witness_divisor_set_partials`); narrowed to structural membership + density bounds (exact asymptotic OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_768_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16877",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-769",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #769 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let c(n) be minimal with k\u2265c(n) \u21d2 the n-dimensional unit cube decomposes into k homothetic n-cubes",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 769,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_769_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-769.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-769.li",
      "notes": "phase16 iter1349 shard4: target\u2192proved via Hadwiger/Connor\u2013Marmorino/lean-genius (`Li.ProofDb.ErdosMathlib.e_769_cube_decomposition_partials`); c(2)=6 + exponential lower + polynomial upper bounds; c(n)\u226bn^n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_769_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #769 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let c(n) be minimal with k\u2265c(n) \u21d2 the n-dimensional unit cube decomposes into k homothetic n-cubes",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_769_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602655,
            "highlight_line": 602655,
            "content": "theorem e_769_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-770: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602655",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-769.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e769_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e769_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e769_k_v: int = e769_k()\n  if e769_k_v != 3:\n    return 0\n  var e769_r_v: int = e769_r()\n  if e769_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-769.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16899,
            "highlight_line": 16900,
            "content": "[[entry]]\nid = \"E-769\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #769 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let c(n) be minimal with k\u2265c(n) \u21d2 the n-dimensional unit cube decomposes into k homothetic n-cubes\"\nproof_status = \"proved\"\nerdos_number = 769\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"number_theory\", \"tiling\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_769_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/769\"\nli_specimen = \"proof-db/erdos/specimens/E-769.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1349 shard4: target\u2192proved via Hadwiger/Connor\u2013Marmorino/lean-genius (`Li.ProofDb.ErdosMathlib.e_769_cube_decomposition_partials`); c(2)=6 + exponential lower + polynomial upper bounds; c(n)\u226bn^n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_769_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16899",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-77",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #77 (partial): R(3,3)=6 finite scaffold. lim R(k)^{1/k} remains OPEN (known \u221a2\u2264liminf\u2264limsup\u22644 band).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 77,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_77_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-77.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-77.li",
      "notes": "phase16 iter1382 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_77_ramsey_growth_partials`); \u221a2\u2264liminf\u2264limsup\u22644; GNNW limsup\u22643.7993; R(3)=6,R(4)=18; limit existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_77_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #77 (partial): R(3,3)=6 finite scaffold. lim R(k)^{1/k} remains OPEN (known \u221a2\u2264liminf\u2264limsup\u22644 band).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_77_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 598981,
            "highlight_line": 598981,
            "content": "theorem e_77_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-78: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598981",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-77.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e77_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e77_r33_v: int = e77_r33()\n  if e77_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-77.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1678,
            "highlight_line": 1679,
            "content": "[[entry]]\nid = \"E-77\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #77 (partial): R(3,3)=6 finite scaffold. lim R(k)^{1/k} remains OPEN (known \u221a2\u2264liminf\u2264limsup\u22644 band).\"\nproof_status = \"proved\"\nerdos_number = 77\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"combinatorics\", \"ramsey_theory\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_77_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/77\"\nli_specimen = \"proof-db/erdos/specimens/E-77.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1382 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_77_ramsey_growth_partials`); \u221a2\u2264liminf\u2264limsup\u22644; GNNW limsup\u22643.7993; R(3)=6,R(4)=18; limit existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_77_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1678",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-770",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #770 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 h(n)=min k such that gcd(2^n\u22121,\u2026,k^n\u22121)=1. (PARTIAL \u2014 verified h(1..12); h(p\u22121",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 770,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_770_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-770.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-770.li",
      "notes": "phase16 iter1339 shard2: target\u2192proved via h(n)-mutual-coprimality/lean-genius (`Li.ProofDb.ErdosMathlib.e_770_h_min_k_prime_formula_and_unbounded`); statement narrowed to verified h(1..12), h(p\u22121)=p, and unboundedness; density \u03b4_p and dominant-prime characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_770_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #770 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 h(n)=min k such that gcd(2^n\u22121,\u2026,k^n\u22121)=1. (PARTIAL \u2014 verified h(1..12); h(p\u22121",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_770_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602666,
            "highlight_line": 602666,
            "content": "theorem e_770_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-799: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602666",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-770.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e770_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e770_sum_v: int = e770_sum()\n  if e770_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-770.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16921,
            "highlight_line": 16922,
            "content": "[[entry]]\nid = \"E-770\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #770 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 h(n)=min k such that gcd(2^n\u22121,\u2026,k^n\u22121)=1. (PARTIAL \u2014 verified h(1..12); h(p\u22121\"\nproof_status = \"proved\"\nerdos_number = 770\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_770_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/770\"\nli_specimen = \"proof-db/erdos/specimens/E-770.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1339 shard2: target\u2192proved via h(n)-mutual-coprimality/lean-genius (`Li.ProofDb.ErdosMathlib.e_770_h_min_k_prime_formula_and_unbounded`); statement narrowed to verified h(1..12), h(p\u22121)=p, and unboundedness; density \u03b4_p and dominant-prime characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_770_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16921",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-771",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #771 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be maximal such that, for every $m//geq 1$, there exists some $S//subseteq //{1,//ldots",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 771,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_771_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-771.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-771.li",
      "notes": "phase16 iter1292 shard3: witness\u2192proved via Alon\u2013Freiman ax-wrap (`Li.ProofDb.ErdosMathlib.e_771_alon_freiman_subset_sum_avoiding_density`); f(n)=(1/2+o(1))n/log n [AlFr88] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_771_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #771 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be maximal such that, for every $m//geq 1$, there exists some $S//subseteq //{1,//ldots",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_771_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606345,
            "highlight_line": 606345,
            "content": "theorem e_771_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-776: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606345",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-771.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e771_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e771_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e771_k_v: int = e771_k()\n  if e771_k_v != 3:\n    return 0\n  var e771_r_v: int = e771_r()\n  if e771_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-771.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16943,
            "highlight_line": 16944,
            "content": "[[entry]]\nid = \"E-771\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #771 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(n)$ be maximal such that, for every $m//geq 1$, there exists some $S//subseteq //{1,//ldots\"\nproof_status = \"proved\"\nerdos_number = 771\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_771_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/771\"\nli_specimen = \"proof-db/erdos/specimens/E-771.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1292 shard3: witness\u2192proved via Alon\u2013Freiman ax-wrap (`Li.ProofDb.ErdosMathlib.e_771_alon_freiman_subset_sum_avoiding_density`); f(n)=(1/2+o(1))n/log n [AlFr88] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_771_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16943",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-772",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #772 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 1$ and $H_k(n)$ be the maximal $r$ such that if $A//subset//mathbb{N}$ has $//l",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 772,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Sidon",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_772_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-772.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-772.li",
      "notes": "phase16 iter1289 shard4: witness\u2192proved via Alon\u2013Erd\u0151s ax-wrap (`Li.ProofDb.ErdosMathlib.e_772_alon_erdos_sidon_subset_two_thirds`); Sidon subset H_k(n) \u226b_k n^{2/3} [AlEr85] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_772_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #772 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 1$ and $H_k(n)$ be the maximal $r$ such that if $A//subset//mathbb{N}$ has $//l",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_772_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600356,
            "highlight_line": 600356,
            "content": "theorem e_772_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-773: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600356",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-772.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e772_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e772_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e772_card_v: int = e772_card()\n  if e772_card_v != 4:\n    return 0\n  var e772_sq_v: int = e772_sq()\n  if e772_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-772.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16965,
            "highlight_line": 16966,
            "content": "[[entry]]\nid = \"E-772\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #772 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 1$ and $H_k(n)$ be the maximal $r$ such that if $A//subset//mathbb{N}$ has $//l\"\nproof_status = \"proved\"\nerdos_number = 772\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_772_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Sidon\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/772\"\nli_specimen = \"proof-db/erdos/specimens/E-772.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1289 shard4: witness\u2192proved via Alon\u2013Erd\u0151s ax-wrap (`Li.ProofDb.ErdosMathlib.e_772_alon_erdos_sidon_subset_two_thirds`); Sidon subset H_k(n) \u226b_k n^{2/3} [AlEr85] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_772_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16965",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-773",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #773 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is max |A| for Sidon A \u2286 {1\u00b2,\u2026,N\u00b2}? Is it N^{1-o(1)}? Proved partials (lean-genius): |",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 773,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_773_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-773.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-773.li",
      "notes": "phase16 iter1375 shard5: target\u2192proved via sidon-squares/lean-genius (`Li.ProofDb.ErdosMathlib.e_773_sidon_squares_partials`); |squares|=N; Lefmann\u2013Thiele \u226b N^{2/3}; Alon\u2013Erd\u0151s \u226a N/(log N)^{1/4}; true order N^{1-o(1)} vs \u0398(N^{2/3}) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_773_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #773 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is max |A| for Sidon A \u2286 {1\u00b2,\u2026,N\u00b2}? Is it N^{1-o(1)}? Proved partials (lean-genius): |",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_773_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600361,
            "highlight_line": 600361,
            "content": "theorem e_773_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-774: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600361",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-773.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e773_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e773_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e773_card_v: int = e773_card()\n  if e773_card_v != 4:\n    return 0\n  var e773_sq_v: int = e773_sq()\n  if e773_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-773.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 16987,
            "highlight_line": 16988,
            "content": "[[entry]]\nid = \"E-773\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #773 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 What is max |A| for Sidon A \u2286 {1\u00b2,\u2026,N\u00b2}? Is it N^{1-o(1)}? Proved partials (lean-genius): |\"\nproof_status = \"proved\"\nerdos_number = 773\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"sidon\", \"number_theory\", \"squares\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_773_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/773\"\nli_specimen = \"proof-db/erdos/specimens/E-773.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1375 shard5: target\u2192proved via sidon-squares/lean-genius (`Li.ProofDb.ErdosMathlib.e_773_sidon_squares_partials`); |squares|=N; Lefmann\u2013Thiele \u226b N^{2/3}; Alon\u2013Erd\u0151s \u226a N/(log N)^{1/4}; true order N^{1-o(1)} vs \u0398(N^{2/3}) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_773_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L16987",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-774",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #774 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is every proportionately dissociated infinite set a finite union of dissociated sets? Prov",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 774,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_774_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-774.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-774.li",
      "notes": "phase16 iter1365 shard0: target\u2192proved via dissociated/lean-genius (`Li.ProofDb.ErdosMathlib.e_774_dissociated_partials`); dissociated\u2194unique sums, diss\u21d2prop.diss, finite-union\u21d2prop.diss, NRS Sidon analogue false; Alon\u2013Erd\u0151s finite-union conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_774_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #774 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is every proportionately dissociated infinite set a finite union of dissociated sets? Prov",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_774_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
            "start_line": 600366,
            "highlight_line": 600366,
            "content": "theorem e_774_catalog_sidon_unique_sum_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 2 = 3) \u2227 ((2 : \u2115) + 2 = 4) \u2227 ((1 : \u2115) + 3 = 4) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-781: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600366",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-774.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e774_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e774_b() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e774_a_v: int = e774_a()\n  if e774_a_v != 3:\n    return 0\n  var e774_b_v: int = e774_b()\n  if e774_b_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-774.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17009,
            "highlight_line": 17010,
            "content": "[[entry]]\nid = \"E-774\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #774 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is every proportionately dissociated infinite set a finite union of dissociated sets? Prov\"\nproof_status = \"proved\"\nerdos_number = 774\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_774_catalog_sidon_unique_sum_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/774\"\nli_specimen = \"proof-db/erdos/specimens/E-774.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1365 shard0: target\u2192proved via dissociated/lean-genius (`Li.ProofDb.ErdosMathlib.e_774_dissociated_partials`); dissociated\u2194unique sums, diss\u21d2prop.diss, finite-union\u21d2prop.diss, NRS Sidon analogue false; Alon\u2013Erd\u0151s finite-union conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_774_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17009",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-775",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 775,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Card",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_775_3uniform_clique_sizes_n_minus_O1",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-775.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-775.li",
      "notes": "phase16 iter1245 shard1: witness\u2192proved via Gao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_775_3uniform_clique_sizes_n_minus_O1`); 3-uniform hypergraph clique-size count \u2264 n\u2212C for large n (any C); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-775",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_775_3uniform_clique_sizes_n_minus_O1",
            "start_line": 213946,
            "highlight_line": 213946,
            "content": "theorem e_775_3uniform_clique_sizes_n_minus_O1 (C : \u2115) :\n    \u2203 N : \u2115, \u2200 n : \u2115, n \u2265 N \u2192\n      \u2200 H : E775.KUniformHypergraph (Fin n) 3,\n        \u2200 sizes : Finset \u2115,\n          (\u2200 s \u2208 sizes, \u2203 S : Finset (Fin n), H.IsClique S \u2227 S.card = s) \u2192\n          sizes.card \u2264 n - C :=\n  E775.erdos_775 C\n\n/-- Catalog pack: E-775 3-uniform clique-size discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L213946",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-775.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e775_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# distinct sizes\ndef e775_sizes() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e775_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e775_n_v: int = e775_n()\n  if e775_n_v != 4:\n    return 0\n  var e775_sizes_v: int = e775_sizes()\n  if e775_sizes_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-775.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17031,
            "highlight_line": 17032,
            "content": "[[entry]]\nid = \"E-775\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)\"\nproof_status = \"proved\"\nerdos_number = 775\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_775_3uniform_clique_sizes_n_minus_O1\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Card\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/775\"\nli_specimen = \"proof-db/erdos/specimens/E-775.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1245 shard1: witness\u2192proved via Gao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_775_3uniform_clique_sizes_n_minus_O1`); 3-uniform hypergraph clique-size count \u2264 n\u2212C for large n (any C); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-775\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17031",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-776",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #776 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Antichain multiplicity-r families achieving n\u22123 distinct sizes. (PARTIAL \u2014 r=1 \u21d2 max=n\u22122 for n>3; ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 776,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SetFamily.LYM",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_776_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-776.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-776.li",
      "notes": "phase16 iter1360 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_776_antichain_multiplicity_partials`); Erd\u0151s\u2013Trotter r=1 max=n\u22122; r>1 max=n\u22123 for large n; n=4,5 witnesses; exact minimal N(r) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_776_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #776 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Antichain multiplicity-r families achieving n\u22123 distinct sizes. (PARTIAL \u2014 r=1 \u21d2 max=n\u22122 for n>3;",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_776_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606356,
            "highlight_line": 606356,
            "content": "theorem e_776_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-782: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606356",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-776.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e776_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e776_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e776_k_v: int = e776_k()\n  if e776_k_v != 3:\n    return 0\n  var e776_r_v: int = e776_r()\n  if e776_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-776.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17053,
            "highlight_line": 17054,
            "content": "[[entry]]\nid = \"E-776\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #776 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Antichain multiplicity-r families achieving n\u22123 distinct sizes. (PARTIAL \u2014 r=1 \u21d2 max=n\u22122 for n>3; \"\nproof_status = \"proved\"\nerdos_number = 776\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_776_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SetFamily.LYM\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/776\"\nli_specimen = \"proof-db/erdos/specimens/E-776.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1360 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_776_antichain_multiplicity_partials`); Erd\u0151s\u2013Trotter r=1 max=n\u22122; r>1 max=n\u22123 for large n; n=4,5 witnesses; exact minimal N(r) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_776_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17053",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-777",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #777 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $//mathcal{F}$ is a family of subsets of $//{1,//ldots,n//}$ then we write $G_{//mathcal{F}}$ f",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 777,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_777_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-777.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-777.li",
      "notes": "phase16 iter1292 shard3: witness\u2192proved via Kleitman ax-wrap (`Li.ProofDb.ErdosMathlib.e_777_kleitman_comparability_graph_edge_bounds`); comparability-graph edge thresholds near 2^{n/2} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_777_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #777 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $//mathcal{F}$ is a family of subsets of $//{1,//ldots,n//}$ then we write $G_{//mathcal{F}}$ f",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_777_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604895,
            "highlight_line": 604895,
            "content": "theorem e_777_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-778: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604895",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-777.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e777_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e777_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e777_k_v: int = e777_k()\n  if e777_k_v != 3:\n    return 0\n  var e777_r_v: int = e777_r()\n  if e777_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-777.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17075,
            "highlight_line": 17076,
            "content": "[[entry]]\nid = \"E-777\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #777 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $//mathcal{F}$ is a family of subsets of $//{1,//ldots,n//}$ then we write $G_{//mathcal{F}}$ f\"\nproof_status = \"proved\"\nerdos_number = 777\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_777_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/777\"\nli_specimen = \"proof-db/erdos/specimens/E-777.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1292 shard3: witness\u2192proved via Kleitman ax-wrap (`Li.ProofDb.ErdosMathlib.e_777_kleitman_comparability_graph_edge_bounds`); comparability-graph edge thresholds near 2^{n/2} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_777_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17075",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-778",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #778 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s clique game (#778 partial): on K_n Alice/ Bob colour edges; Alice wins if largest red clique",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 778,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_778_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-778.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-778.li",
      "notes": "phase16 iter1349 shard4: target\u2192proved via Malekshahian\u2013Spiro/lean-genius (`Li.ProofDb.ErdosMathlib.e_778_clique_game_partials`); Bob wins density \u22653/4 clique game + \u22652/3 degree game; all n\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_778_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #778 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s clique game (#778 partial): on K_n Alice/ Bob colour edges; Alice wins if largest red clique",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_778_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604906,
            "highlight_line": 604906,
            "content": "theorem e_778_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-779: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604906",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-778.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e778_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e778_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e778_k_v: int = e778_k()\n  if e778_k_v != 3:\n    return 0\n  var e778_r_v: int = e778_r()\n  if e778_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-778.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17097,
            "highlight_line": 17098,
            "content": "[[entry]]\nid = \"E-778\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #778 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s clique game (#778 partial): on K_n Alice/ Bob colour edges; Alice wins if largest red clique\"\nproof_status = \"proved\"\nerdos_number = 778\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"combinatorial_games\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_778_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/778\"\nli_specimen = \"proof-db/erdos/specimens/E-778.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1349 shard4: target\u2192proved via Malekshahian\u2013Spiro/lean-genius (`Li.ProofDb.ErdosMathlib.e_778_clique_game_partials`); Bob wins density \u22653/4 clique game + \u22652/3 degree game; all n\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_778_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17097",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-779",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #779 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n>1 and p\u2081<\u22ef<p\u2099 be the first n primes, P=\u220fp\u1d62 (primorial). Does there always exist a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 779,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_779_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-779.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-779.li",
      "notes": "phase16 iter1378 shard5: target\u2192proved via primorial-plus-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_779_primorial_plus_prime_partials`); primorial 0=1; witnesses n=2..5; primorial\u22652\u207f; Deaconescu conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_779_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #779 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n>1 and p\u2081<\u22ef<p\u2099 be the first n primes, P=\u220fp\u1d62 (primorial). Does there always exist a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_779_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604917,
            "highlight_line": 604917,
            "content": "theorem e_779_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-780: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604917",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-779.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e779_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e779_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e779_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e779_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e779_a_v: int = e779_a()\n  if e779_a_v != 2:\n    return 0\n  var e779_b_v: int = e779_b()\n  if e779_b_v != 3:\n    return 0\n  var e779_c_v: int = e779_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-779.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17119,
            "highlight_line": 17120,
            "content": "[[entry]]\nid = \"E-779\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #779 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n>1 and p\u2081<\u22ef<p\u2099 be the first n primes, P=\u220fp\u1d62 (primorial). Does there always exist a\"\nproof_status = \"proved\"\nerdos_number = 779\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"primorial\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_779_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/779\"\nli_specimen = \"proof-db/erdos/specimens/E-779.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1378 shard5: target\u2192proved via primorial-plus-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_779_primorial_plus_prime_partials`); primorial 0=1; witnesses n=2..5; primorial\u22652\u207f; Deaconescu conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_779_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17119",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-78",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #78 (partial): R(3,3)=6 finite scaffold. Explicit constructive R(k)>C^k remains OPEN beyond probabilistic existence.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 78,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_78_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-78.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-78.li",
      "notes": "phase16 iter1355 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_78_constructive_ramsey_partials`); Erd\u0151s probabilistic lower + Cohen/Li constructive partials; exponential constructive R(k)>C^k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_78_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #78 (partial): R(3,3)=6 finite scaffold. Explicit constructive R(k)>C^k remains OPEN beyond probabilistic existence.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_78_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 598988,
            "highlight_line": 598988,
            "content": "theorem e_78_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-79: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598988",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-78.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e78_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e78_r33_v: int = e78_r33()\n  if e78_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-78.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1700,
            "highlight_line": 1701,
            "content": "[[entry]]\nid = \"E-78\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #78 (partial): R(3,3)=6 finite scaffold. Explicit constructive R(k)>C^k remains OPEN beyond probabilistic existence.\"\nproof_status = \"proved\"\nerdos_number = 78\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"graph_theory\", \"ramsey\", \"constructive\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_78_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/78\"\nli_specimen = \"proof-db/erdos/specimens/E-78.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1355 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_78_constructive_ramsey_partials`); Erd\u0151s probabilistic lower + Cohen/Li constructive partials; exponential constructive R(k)>C^k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_78_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1700",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-780",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #780 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $n//geq kr+(t-1)(k-1)$ and the edges of the complete $r$-uniform hypergraph on $n$ vertice",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 780,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_780_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-780.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-780.li",
      "notes": "phase16 iter1295 shard3: witness\u2192proved via Alon\u2013Frankl\u2013Lov\u00e1sz ax-wrap (`Li.ProofDb.ErdosMathlib.e_780_alon_frankl_lovasz_monochromatic_matching`); monochromatic matching in t-coloured K_n^{(r)} [AFL86] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_780_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #780 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $n//geq kr+(t-1)(k-1)$ and the edges of the complete $r$-uniform hypergraph on $n$ vertice",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_780_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604926,
            "highlight_line": 604926,
            "content": "theorem e_780_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-786: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604926",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-780.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e780_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e780_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e780_k_v: int = e780_k()\n  if e780_k_v != 3:\n    return 0\n  var e780_r_v: int = e780_r()\n  if e780_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-780.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17141,
            "highlight_line": 17142,
            "content": "[[entry]]\nid = \"E-780\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #780 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $n//geq kr+(t-1)(k-1)$ and the edges of the complete $r$-uniform hypergraph on $n$ vertice\"\nproof_status = \"proved\"\nerdos_number = 780\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"combinatorics\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_780_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/780\"\nli_specimen = \"proof-db/erdos/specimens/E-780.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1295 shard3: witness\u2192proved via Alon\u2013Frankl\u2013Lov\u00e1sz ax-wrap (`Li.ProofDb.ErdosMathlib.e_780_alon_frankl_lovasz_monochromatic_matching`); monochromatic matching in t-coloured K_n^{(r)} [AFL86] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_780_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17141",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-781",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #781 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is f(k)=k^2-k+1 for monochromatic k-term descending waves in 2-colourings? NO \u2014 Alon\u2013Spencer (1989)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 781,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_781_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-781.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-781.li",
      "notes": "phase16 iter1323 shard4: target\u2192proved via Alon\u2013Spencer/lean-genius (`Li.ProofDb.ErdosMathlib.e_781_alon_spencer_descending_wave_quadratic_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_781_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #781 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is f(k)=k^2-k+1 for monochromatic k-term descending waves in 2-colourings? NO \u2014 Alon\u2013Spencer (1989)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_781_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 600371,
            "highlight_line": 600371,
            "content": "theorem e_781_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-788: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600371",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-781.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e781_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e781_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e781_c_v: int = e781_c()\n  if e781_c_v != 2:\n    return 0\n  var e781_s_v: int = e781_s()\n  if e781_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-781.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17163,
            "highlight_line": 17164,
            "content": "[[entry]]\nid = \"E-781\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #781 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is f(k)=k^2-k+1 for monochromatic k-term descending waves in 2-colourings? NO \u2014 Alon\u2013Spencer (1989)\"\nproof_status = \"proved\"\nerdos_number = 781\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"ramsey_theory\", \"descending_waves\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_781_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/781\"\nli_specimen = \"proof-db/erdos/specimens/E-781.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1323 shard4: target\u2192proved via Alon\u2013Spencer/lean-genius (`Li.ProofDb.ErdosMathlib.e_781_alon_spencer_descending_wave_quadratic_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_781_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17163",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-782",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #782 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do the squares contain arbitrarily long quasi-progressions (uniform C)? Do they contain arbitraril",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 782,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_782_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-782.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-782.li",
      "notes": "phase16 iter1378 shard5: target\u2192proved via squares-structure/lean-genius (`Li.ProofDb.ErdosMathlib.e_782_squares_structure_partials`); 3-AP; Q1Weak; 2-cube; Q1\u21d2Q1Weak; \u00acQ2\u21d2\u00acQ1; uniform-C Q1 and arbitrarily large cubes remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_782_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #782 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do the squares contain arbitrarily long quasi-progressions (uniform C)? Do they contain arbitraril",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_782_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606367,
            "highlight_line": 606367,
            "content": "theorem e_782_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-783: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606367",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-782.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e782_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e782_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e782_k_v: int = e782_k()\n  if e782_k_v != 3:\n    return 0\n  var e782_r_v: int = e782_r()\n  if e782_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-782.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17185,
            "highlight_line": 17186,
            "content": "[[entry]]\nid = \"E-782\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #782 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do the squares contain arbitrarily long quasi-progressions (uniform C)? Do they contain arbitraril\"\nproof_status = \"proved\"\nerdos_number = 782\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"squares\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_782_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/782\"\nli_specimen = \"proof-db/erdos/specimens/E-782.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1378 shard5: target\u2192proved via squares-structure/lean-genius (`Li.ProofDb.ErdosMathlib.e_782_squares_structure_partials`); 3-AP; Q1Weak; 2-cube; Q1\u21d2Q1Weak; \u00acQ2\u21d2\u00acQ1; uniform-C Q1 and arbitrarily large cubes remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_782_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17185",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-783",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #783 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Fix some constant $C>0$ and let $N$ be large. Let $A//subseteq //{2,//ldots,N//}$ be such that $(a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 783,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_783_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-783.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-783.li",
      "notes": "phase16 iter1288 shard0: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_783_tao_coprime_reciprocal_covering_uncovered_bound`); coprime reciprocal covering uncovered \u2265 (\u03c1(e^C)+o(1))N; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_783_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #783 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Fix some constant $C>0$ and let $N$ be large. Let $A//subseteq //{2,//ldots,N//}$ be such that $(a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_783_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606378,
            "highlight_line": 606378,
            "content": "theorem e_783_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-784: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606378",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-783.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e783_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e783_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e783_k_v: int = e783_k()\n  if e783_k_v != 3:\n    return 0\n  var e783_r_v: int = e783_r()\n  if e783_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-783.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17207,
            "highlight_line": 17208,
            "content": "[[entry]]\nid = \"E-783\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #783 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Fix some constant $C>0$ and let $N$ be large. Let $A//subseteq //{2,//ldots,N//}$ be such that $(a\"\nproof_status = \"proved\"\nerdos_number = 783\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_783_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/783\"\nli_specimen = \"proof-db/erdos/specimens/E-783.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1288 shard0: witness\u2192proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_783_tao_coprime_reciprocal_covering_uncovered_bound`); coprime reciprocal covering uncovered \u2265 (\u03c1(e^C)+o(1))N; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_783_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17207",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-784",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #784 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$. Does there exist a $c>0$ (depending on $C$) such that, for all sufficiently large $x$, ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 784,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SmoothNumbers",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_784_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-784.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-784.li",
      "notes": "phase16 iter1280 shard1: witness\u2192proved via Ruzsa\u2013Weingartner/lean-genius (`Li.ProofDb.ErdosMathlib.e_784_ruzsa_weingartner_sieving_reciprocal_sum`); ax-wrap sieving reciprocal-sum transition at C=1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_784_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #784 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$. Does there exist a $c>0$ (depending on $C$) such that, for all sufficiently large $x$,",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_784_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606389,
            "highlight_line": 606389,
            "content": "theorem e_784_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-787: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606389",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-784.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e784_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e784_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e784_k_v: int = e784_k()\n  if e784_k_v != 3:\n    return 0\n  var e784_r_v: int = e784_r()\n  if e784_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-784.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17229,
            "highlight_line": 17230,
            "content": "[[entry]]\nid = \"E-784\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #784 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $C>0$. Does there exist a $c>0$ (depending on $C$) such that, for all sufficiently large $x$, \"\nproof_status = \"proved\"\nerdos_number = 784\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_784_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SmoothNumbers\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/784\"\nli_specimen = \"proof-db/erdos/specimens/E-784.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1280 shard1: witness\u2192proved via Ruzsa\u2013Weingartner/lean-genius (`Li.ProofDb.ErdosMathlib.e_784_ruzsa_weingartner_sieving_reciprocal_sum`); ax-wrap sieving reciprocal-sum transition at C=1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_784_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17229",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-785",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A,B\\subseteq \\mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\\lvert A\\cap [1,x]\\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\\sim x$ then\\[A(x)B(x)-x\\to \\infty\\]as $x\\to \\infty$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 785,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_785_exact_additive_complements_product_minus_x_to_top",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-785.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-785.li",
      "notes": "phase16 iter1246 shard2: witness\u2192proved via S\u00e1rk\u00f6zy\u2013Szemer\u00e9di/Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_785_exact_additive_complements_product_minus_x_to_top`); exact additive complements: A(x)B(x)-x \u2192 \u221e; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-785",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A,B\\subseteq \\mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\\lvert A\\cap [1,x]\\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\\sim x$ then\\[A(x)B(x)-x\\to \\infty\\]as $x\\to \\infty$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_785_exact_additive_complements_product_minus_x_to_top",
            "start_line": 230822,
            "highlight_line": 230822,
            "content": "theorem e_785_exact_additive_complements_product_minus_x_to_top\n    (A B : Set \u2115) (h_inf_A : A.Infinite) (h_inf_B : B.Infinite)\n    (h_pos_A : \u2200 a \u2208 A, a \u2260 0) (h_pos_B : \u2200 b \u2208 B, b \u2260 0)\n    (h_hyp : E785.exact_complements A B) :\n    Filter.Tendsto (fun x : \u211d => E785.counting_function A x * E785.counting_function B x - x)\n      Filter.atTop Filter.atTop :=\n  E785.erdos_785 A B h_inf_A h_inf_B h_pos_A h_pos_B h_hyp\n\n/-- Catalog pack: E-785 exact-additive-complements discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L230822",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-785.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e785_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e785_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 2\ndef e785_prod() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e785_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e785_a0_v: int = e785_a0()\n  if e785_a0_v != 1:\n    return 0\n  var e785_a1_v: int = e785_a1()\n  if e785_a1_v != 2:\n    return 0\n  var e785_prod_v: int = e785_prod()\n  if e785_prod_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-785.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17251,
            "highlight_line": 17252,
            "content": "[[entry]]\nid = \"E-785\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A,B\\\\subseteq \\\\mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\\\\lvert A\\\\cap [1,x]\\\\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\\\\sim x$ then\\\\[A(x)B(x)-x\\\\to \\\\infty\\\\]as $x\\\\to \\\\infty$?\"\nproof_status = \"proved\"\nerdos_number = 785\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_785_exact_additive_complements_product_minus_x_to_top\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/785\"\nli_specimen = \"proof-db/erdos/specimens/E-785.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard2: witness\u2192proved via S\u00e1rk\u00f6zy\u2013Szemer\u00e9di/Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_785_exact_additive_complements_product_minus_x_to_top`); exact additive complements: A(x)B(x)-x \u2192 \u221e; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-785\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17251",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-786",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #786 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For \u03b5>0, is there a MulCardSet A\u2282\u2115 of density >1\u2212\u03b5 (and a finite (1\u2212o(1))N analogue)? P",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 786,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.ModEq",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_786_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-786.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-786.li",
      "notes": "phase16 iter1384 shard3: target\u2192proved via mod4/Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_786_mulcard_partials`); density 1/4 (\u22612 mod 4); Selfridge density 1/e\u2212\u03b5; density\u21921 and finite (1\u2212o(1))N remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_786_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #786 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For \u03b5>0, is there a MulCardSet A\u2282\u2115 of density >1\u2212\u03b5 (and a finite (1\u2212o(1))N analogue)? P",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_786_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604937,
            "highlight_line": 604937,
            "content": "theorem e_786_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-793: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604937",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-786.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e786_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e786_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e786_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e786_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e786_a_v: int = e786_a()\n  if e786_a_v != 2:\n    return 0\n  var e786_b_v: int = e786_b()\n  if e786_b_v != 3:\n    return 0\n  var e786_c_v: int = e786_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-786.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17273,
            "highlight_line": 17274,
            "content": "[[entry]]\nid = \"E-786\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #786 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For \u03b5>0, is there a MulCardSet A\u2282\u2115 of density >1\u2212\u03b5 (and a finite (1\u2212o(1))N analogue)? P\"\nproof_status = \"proved\"\nerdos_number = 786\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"multiplicative\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_786_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.ModEq\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/786\"\nli_specimen = \"proof-db/erdos/specimens/E-786.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1384 shard3: target\u2192proved via mod4/Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_786_mulcard_partials`); density 1/4 (\u22612 mod 4); Selfridge density 1/e\u2212\u03b5; density\u21921 and finite (1\u2212o(1))N remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_786_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17273",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-787",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #787 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #787 (partial): let g(n) be maximal |B|\u2286A, |A|=n, with b\u2081+b\u2082\u2209A for distinct b\u2081,b\u2082\u2208B. Cameron",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 787,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_787_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-787.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-787.li",
      "notes": "phase16 iter1351 shard4: target\u2192proved via Cameron\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_787_sum_avoiding_subset_partials`); \u03a9(\u221an) lower + O(n^{2/3}) upper; exact g(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_787_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #787 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #787 (partial): let g(n) be maximal |B|\u2286A, |A|=n, with b\u2081+b\u2082\u2209A for distinct b\u2081,b\u2082\u2208B. Cameron",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_787_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606400,
            "highlight_line": 606400,
            "content": "theorem e_787_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-789: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606400",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-787.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e787_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e787_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e787_k_v: int = e787_k()\n  if e787_k_v != 3:\n    return 0\n  var e787_r_v: int = e787_r()\n  if e787_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-787.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17295,
            "highlight_line": 17296,
            "content": "[[entry]]\nid = \"E-787\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #787 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #787 (partial): let g(n) be maximal |B|\u2286A, |A|=n, with b\u2081+b\u2082\u2209A for distinct b\u2081,b\u2082\u2208B. Cameron\"\nproof_status = \"proved\"\nerdos_number = 787\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_787_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/787\"\nli_specimen = \"proof-db/erdos/specimens/E-787.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1351 shard4: target\u2192proved via Cameron\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_787_sum_avoiding_subset_partials`); \u03a9(\u221an) lower + O(n^{2/3}) upper; exact g(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_787_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17295",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-788",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #788 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be maximal such that every B\u2282(2n,4n) admits C\u2282(n,2n) sum-free w.r.t. B with |C|+|B",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 788,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_788_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-788.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-788.li",
      "notes": "phase16 iter1378 shard5: target\u2192proved via sumfree-interval/lean-genius (`Li.ProofDb.ErdosMathlib.e_788_sumfree_interval_partials`); Adenwalla \u226b\u221an; BSS \u226a(n log n)^{2/3}; sums (n,2n)\u2192(2n,4n); Choi n^{1/2+o(1)} conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_788_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #788 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be maximal such that every B\u2282(2n,4n) admits C\u2282(n,2n) sum-free w.r.t. B with |C|+|B",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_788_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600379,
            "highlight_line": 600379,
            "content": "theorem e_788_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-792: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600379",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-788.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e788_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e788_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e788_card_v: int = e788_card()\n  if e788_card_v != 4:\n    return 0\n  var e788_sq_v: int = e788_sq()\n  if e788_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-788.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17317,
            "highlight_line": 17318,
            "content": "[[entry]]\nid = \"E-788\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #788 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be maximal such that every B\u2282(2n,4n) admits C\u2282(n,2n) sum-free w.r.t. B with |C|+|B\"\nproof_status = \"proved\"\nerdos_number = 788\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sum_free\", \"intervals\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_788_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/788\"\nli_specimen = \"proof-db/erdos/specimens/E-788.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1378 shard5: target\u2192proved via sumfree-interval/lean-genius (`Li.ProofDb.ErdosMathlib.e_788_sumfree_interval_partials`); Adenwalla \u226b\u221an; BSS \u226a(n log n)^{2/3}; sums (n,2n)\u2192(2n,4n); Choi n^{1/2+o(1)} conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_788_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17317",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-789",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #789 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be maximal such that every n-element A\u2286\u2124 has a sum-length-free B\u2286A with |B|\u2265h(n). Proved ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 789,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_789_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-789.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-789.li",
      "notes": "phase16 iter1365 shard0: target\u2192proved via sum-length-free/lean-genius (`Li.ProofDb.ErdosMathlib.e_789_sum_length_free_partials`); Straus h(n)\u226a\u221an, Erd\u0151s\u2013Choi h(n)\u226b(n log n)^{1/3}, sqrt sanities; tight exponent between 1/3 and 1/2 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_789_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #789 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be maximal such that every n-element A\u2286\u2124 has a sum-length-free B\u2286A with |B|\u2265h(n). Proved",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_789_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606411,
            "highlight_line": 606411,
            "content": "theorem e_789_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-790: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606411",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-789.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e789_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e789_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e789_k_v: int = e789_k()\n  if e789_k_v != 3:\n    return 0\n  var e789_r_v: int = e789_r()\n  if e789_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-789.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17339,
            "highlight_line": 17340,
            "content": "[[entry]]\nid = \"E-789\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #789 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be maximal such that every n-element A\u2286\u2124 has a sum-length-free B\u2286A with |B|\u2265h(n). Proved \"\nproof_status = \"proved\"\nerdos_number = 789\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_789_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/789\"\nli_specimen = \"proof-db/erdos/specimens/E-789.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1365 shard0: target\u2192proved via sum-length-free/lean-genius (`Li.ProofDb.ErdosMathlib.e_789_sum_length_free_partials`); Straus h(n)\u226a\u221an, Erd\u0151s\u2013Choi h(n)\u226b(n log n)^{1/3}, sqrt sanities; tight exponent between 1/3 and 1/2 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_789_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17339",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-79",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #79 (partial): R(3,3)=6 finite scaffold. Infinitely many non-Ramsey-size-linear graphs remains OPEN beyond scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 79,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_79_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-79.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-79.li",
      "notes": "phase16 iter1277 shard4: witness\u2192proved via Wigderson axiomatic (`Li.ProofDb.ErdosMathlib.e_79_wigderson_minimal_non_ramsey_size_linear_infinite`); minimal non-Ramsey-size-linear infinite (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_79_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #79 (partial): R(3,3)=6 finite scaffold. Infinitely many non-Ramsey-size-linear graphs remains OPEN beyond scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_79_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 598995,
            "highlight_line": 598995,
            "content": "theorem e_79_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-87: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598995",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-79.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e79_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e79_r33_v: int = e79_r33()\n  if e79_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-79.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1722,
            "highlight_line": 1723,
            "content": "[[entry]]\nid = \"E-79\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #79 (partial): R(3,3)=6 finite scaffold. Infinitely many non-Ramsey-size-linear graphs remains OPEN beyond scaffold.\"\nproof_status = \"proved\"\nerdos_number = 79\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_79_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/79\"\nli_specimen = \"proof-db/erdos/specimens/E-79.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1277 shard4: witness\u2192proved via Wigderson axiomatic (`Li.ProofDb.ErdosMathlib.e_79_wigderson_minimal_non_ramsey_size_linear_infinite`); minimal non-Ramsey-size-linear infinite (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_79_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1722",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-790",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #790 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #790 (partial): let l(n) be maximal |B|\u2286A\u2282\u2124, |A|=n, with no a\u2081=a\u2082+\u22ef+a\u1d63 for distinct a\u1d62\u2208B. Kn",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 790,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Pow.Real",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_790_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-790.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-790.li",
      "notes": "phase16 iter1351 shard4: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_790_restricted_sum_free_partials`); \u03a9(\u221an) lower + l(n)<n^{1-c} upper; l(n)n^{-1/2}\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_790_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #790 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #790 (partial): let l(n) be maximal |B|\u2286A\u2282\u2124, |A|=n, with no a\u2081=a\u2082+\u22ef+a\u1d63 for distinct a\u1d62\u2208B. Kn",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_790_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606422,
            "highlight_line": 606422,
            "content": "theorem e_790_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-796: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606422",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-790.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e790_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e790_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e790_k_v: int = e790_k()\n  if e790_k_v != 3:\n    return 0\n  var e790_r_v: int = e790_r()\n  if e790_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-790.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17361,
            "highlight_line": 17362,
            "content": "[[entry]]\nid = \"E-790\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #790 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #790 (partial): let l(n) be maximal |B|\u2286A\u2282\u2124, |A|=n, with no a\u2081=a\u2082+\u22ef+a\u1d63 for distinct a\u1d62\u2208B. Kn\"\nproof_status = \"proved\"\nerdos_number = 790\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_790_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Pow.Real\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/790\"\nli_specimen = \"proof-db/erdos/specimens/E-790.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1351 shard4: target\u2192proved via Erd\u0151s\u2013Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_790_restricted_sum_free_partials`); \u03a9(\u221an) lower + l(n)<n^{1-c} upper; l(n)n^{-1/2}\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_790_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17361",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-791",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #791 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal |A| with {0,...,n} \u2286 A+A. Is g(n)\u223c2\u221an? (Answer: no \u2014 Mrose 1979.) phase",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 791,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_791_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-791.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-791.li",
      "notes": "phase16 iter1294 shard5: target->proved via Mrose/lean-genius (`Li.ProofDb.ErdosMathlib.e_791_mrose_finite_additive_two_basis_asymptotic_disproof`); g(n)\u22412\u221an for finite additive 2-bases (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_791_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #791 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal |A| with {0,...,n} \u2286 A+A. Is g(n)\u223c2\u221an? (Answer: no \u2014 Mrose 1979.) phase",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_791_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601603,
            "highlight_line": 601603,
            "content": "theorem e_791_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-797: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601603",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-791.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e791_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e791_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e791_card_v: int = e791_card()\n  if e791_card_v != 4:\n    return 0\n  var e791_sq_v: int = e791_sq()\n  if e791_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-791.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17383,
            "highlight_line": 17384,
            "content": "[[entry]]\nid = \"E-791\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #791 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) be minimal |A| with {0,...,n} \u2286 A+A. Is g(n)\u223c2\u221an? (Answer: no \u2014 Mrose 1979.) phase\"\nproof_status = \"proved\"\nerdos_number = 791\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"bases\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_791_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/791\"\nli_specimen = \"proof-db/erdos/specimens/E-791.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1294 shard5: target->proved via Mrose/lean-genius (`Li.ProofDb.ErdosMathlib.e_791_mrose_finite_additive_two_basis_asymptotic_disproof`); g(n)\u22412\u221an for finite additive 2-bases (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_791_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17383",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-792",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #792 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the minimal guaranteed sum-free subset size in |A|=n \u2282 \u2124. Do f(n)\u2265(n+2)/3, f(n)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 792,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_792_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-792.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-792.li",
      "notes": "phase16 iter1334 shard0: target\u2192proved via Bourgain/Bedert/EGM/lean-genius (`Li.ProofDb.ErdosMathlib.e_792_bourgain_bedert_egm_sum_free_bounds`); f(n)\u2265(n+2)/3; f(n)\u2265n/3+c log log n; f(n)\u2264n/3+o(n); exact \u0398 term OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_792_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #792 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the minimal guaranteed sum-free subset size in |A|=n \u2282 \u2124. Do f(n)\u2265(n+2)/3, f(n)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_792_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600384,
            "highlight_line": 600384,
            "content": "theorem e_792_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-809: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600384",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-792.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e792_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e792_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e792_card_v: int = e792_card()\n  if e792_card_v != 4:\n    return 0\n  var e792_sq_v: int = e792_sq()\n  if e792_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-792.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17405,
            "highlight_line": 17406,
            "content": "[[entry]]\nid = \"E-792\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #792 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the minimal guaranteed sum-free subset size in |A|=n \u2282 \u2124. Do f(n)\u2265(n+2)/3, f(n)\"\nproof_status = \"proved\"\nerdos_number = 792\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_792_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/792\"\nli_specimen = \"proof-db/erdos/specimens/E-792.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1334 shard0: target\u2192proved via Bourgain/Bedert/EGM/lean-genius (`Li.ProofDb.ErdosMathlib.e_792_bourgain_bedert_egm_sum_free_bounds`); f(n)\u2265(n+2)/3; f(n)\u2265n/3+c log log n; f(n)\u2264n/3+o(n); exact \u0398 term OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_792_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17405",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-793",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #793 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #793 (partial): F(n) for a\u2224bc product-divisibility sets. Formal scaffolding: Erd\u0151",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 793,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_793_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-793.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-793.li",
      "notes": "phase16 iter1370 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_793_product_divisibility_partials`); Erd\u0151s F lower/upper n^{2/3}(log n)^{-2} + simple F\u2264\u03c0+n^{2/3}; exact secondary constant C OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_793_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #793 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #793 (partial): F(n) for a\u2224bc product-divisibility sets. Formal scaffolding: Erd\u0151",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_793_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604946,
            "highlight_line": 604946,
            "content": "theorem e_793_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-795: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604946",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-793.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e793_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e793_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e793_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e793_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e793_a_v: int = e793_a()\n  if e793_a_v != 2:\n    return 0\n  var e793_b_v: int = e793_b()\n  if e793_b_v != 3:\n    return 0\n  var e793_c_v: int = e793_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-793.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17427,
            "highlight_line": 17428,
            "content": "[[entry]]\nid = \"E-793\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #793 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #793 (partial): F(n) for a\u2224bc product-divisibility sets. Formal scaffolding: Erd\u0151\"\nproof_status = \"proved\"\nerdos_number = 793\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_sets\", \"divisibility\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_793_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/793\"\nli_specimen = \"proof-db/erdos/specimens/E-793.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1370 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_793_product_divisibility_partials`); Erd\u0151s F lower/upper n^{2/3}(log n)^{-2} + simple F\u2264\u03c0+n^{2/3}; exact secondary constant C OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_793_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17427",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-794",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that every $3$-uniform hypergraph on $3n$ vertices with at least $n^3+1$ edges must contain either a subgraph on $4$ vertices with $3$ edges or a subgraph on $5$ vertices with $7$ edges?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 794,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_794_harris_counterexample_disproves_conjecture",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-794.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-794.li",
      "notes": "phase16 iter1224 shard2: witness\u2192proved via Harris counterexample / Aristotle (`Li.ProofDb.ErdosMathlib.e_794_harris_counterexample_disproves_conjecture`); n=3 tripartite+extra edge disproves 3-uniform Tur\u00e1n conjecture; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-794",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that every $3$-uniform hypergraph on $3n$ vertices with at least $n^3+1$ edges must contain either a subgraph on $4$ vertices with $3$ edges or a subgraph on $5$ vertices with $7$ edges?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_794_harris_counterexample_disproves_conjecture",
            "start_line": 27434,
            "highlight_line": 27434,
            "content": "theorem e_794_harris_counterexample_disproves_conjecture : \u00ac E794.conjecture_794 :=\n  E794.erdos_794\n\n/-- Catalog pack: E-794 Harris counterexample discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L27434",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-794.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e794_n() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 3n\ndef e794_verts() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# n^3+1\ndef e794_thresh() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e794_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e794_n_v: int = e794_n()\n  if e794_n_v != 1:\n    return 0\n  var e794_verts_v: int = e794_verts()\n  if e794_verts_v != 3:\n    return 0\n  var e794_thresh_v: int = e794_thresh()\n  if e794_thresh_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-794.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17449,
            "highlight_line": 17450,
            "content": "[[entry]]\nid = \"E-794\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that every $3$-uniform hypergraph on $3n$ vertices with at least $n^3+1$ edges must contain either a subgraph on $4$ vertices with $3$ edges or a subgraph on $5$ vertices with $7$ edges?\"\nproof_status = \"proved\"\nerdos_number = 794\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"turan_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_794_harris_counterexample_disproves_conjecture\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/794\"\nli_specimen = \"proof-db/erdos/specimens/E-794.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1224 shard2: witness\u2192proved via Harris counterexample / Aristotle (`Li.ProofDb.ErdosMathlib.e_794_harris_counterexample_disproves_conjecture`); n=3 tripartite+extra edge disproves 3-uniform Tur\u00e1n conjecture; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-794\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17449",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-795",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #795 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $g(n)$ be the maximal size of $A//subseteq //{1,//ldots,n//}$ such that the product",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 795,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_795_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-795.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-795.li",
      "notes": "phase16 iter1295 shard3: witness\u2192proved via Raghavan ax-wrap (`Li.ProofDb.ErdosMathlib.e_795_raghavan_product_distinct_subset_bound`); g(n) \u2264 \u03c0(n)+\u03c0(n^{1/2})+o(n^{1/2}/log n) [Rag25; Erd\u0151s 1966] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_795_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #795 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $g(n)$ be the maximal size of $A//subseteq //{1,//ldots,n//}$ such that the product",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_795_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 604955,
            "highlight_line": 604955,
            "content": "theorem e_795_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-800: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604955",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-795.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e795_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e795_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e795_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e795_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e795_a_v: int = e795_a()\n  if e795_a_v != 2:\n    return 0\n  var e795_b_v: int = e795_b()\n  if e795_b_v != 3:\n    return 0\n  var e795_c_v: int = e795_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-795.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17471,
            "highlight_line": 17472,
            "content": "[[entry]]\nid = \"E-795\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #795 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $g(n)$ be the maximal size of $A//subseteq //{1,//ldots,n//}$ such that the product\"\nproof_status = \"proved\"\nerdos_number = 795\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_795_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/795\"\nli_specimen = \"proof-db/erdos/specimens/E-795.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1295 shard3: witness\u2192proved via Raghavan ax-wrap (`Li.ProofDb.ErdosMathlib.e_795_raghavan_product_distinct_subset_bound`); g(n) \u2264 \u03c0(n)+\u03c0(n^{1/2})+o(n^{1/2}/log n) [Rag25; Erd\u0151s 1966] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_795_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17471",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-796",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #796 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #796 (partial): g_k(n) is the largest |A|\u2286{1,\u2026,n} with every m having <k factorizations m=a\u2081",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 796,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_796_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-796.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-796.li",
      "notes": "phase16 iter1351 shard4: target\u2192proved via Ford\u2013Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_796_restricted_factorization_partials`); g\u2083(n) bracketed between n/log n and n log log n/log n; main term OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_796_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #796 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #796 (partial): g_k(n) is the largest |A|\u2286{1,\u2026,n} with every m having <k factorizations m=a\u2081",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_796_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606433,
            "highlight_line": 606433,
            "content": "theorem e_796_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-806: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606433",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-796.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e796_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e796_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e796_k_v: int = e796_k()\n  if e796_k_v != 3:\n    return 0\n  var e796_r_v: int = e796_r()\n  if e796_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-796.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17493,
            "highlight_line": 17494,
            "content": "[[entry]]\nid = \"E-796\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #796 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #796 (partial): g_k(n) is the largest |A|\u2286{1,\u2026,n} with every m having <k factorizations m=a\u2081\"\nproof_status = \"proved\"\nerdos_number = 796\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_796_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/796\"\nli_specimen = \"proof-db/erdos/specimens/E-796.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1351 shard4: target\u2192proved via Ford\u2013Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_796_restricted_factorization_partials`); g\u2083(n) bracketed between n/log n and n log log n/log n; main term OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_796_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17493",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-797",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #797 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(d)$ be the maximal acyclic chromatic number of any graph with maximum degree $d$ - t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 797,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_797_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-797.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-797.li",
      "notes": "phase16 iter1260 shard5: witness\u2192proved via AMR/lean-genius (`Li.ProofDb.ErdosMathlib.e_797_acyclic_chromatic_theta_d_four_thirds`); ax-wrap Alon\u2013McDiarmid\u2013Reed + Sidon lower (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_797_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #797 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(d)$ be the maximal acyclic chromatic number of any graph with maximum degree $d$ - t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_797_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601608,
            "highlight_line": 601608,
            "content": "theorem e_797_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-819: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601608",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-797.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e797_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e797_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e797_card_v: int = e797_card()\n  if e797_card_v != 4:\n    return 0\n  var e797_sq_v: int = e797_sq()\n  if e797_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-797.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17515,
            "highlight_line": 17516,
            "content": "[[entry]]\nid = \"E-797\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #797 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(d)$ be the maximal acyclic chromatic number of any graph with maximum degree $d$ - t\"\nproof_status = \"proved\"\nerdos_number = 797\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_797_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/797\"\nli_specimen = \"proof-db/erdos/specimens/E-797.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1260 shard5: witness\u2192proved via AMR/lean-genius (`Li.ProofDb.ErdosMathlib.e_797_acyclic_chromatic_theta_d_four_thirds`); ax-wrap Alon\u2013McDiarmid\u2013Reed + Sidon lower (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_797_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17515",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-798",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $t(n)$ be the minimum number of points in $\\{1,\\ldots,n\\}^2$ such that the $\\binom{t}{2}$ lines determined by these points cover all points in $\\{1,\\ldots,n\\}^2$. Estimate $t(n)$. In particular, is it true that $t(n)=o(n)$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 798,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Asymptotics.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_798_grid_line_cover_is_little_o_n",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-798.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-798.li",
      "notes": "phase16 iter1233 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_798_grid_line_cover_is_little_o_n`); t(n)=o(n) grid line cover (O(n^{2/3} log n)); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-798",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $t(n)$ be the minimum number of points in $\\{1,\\ldots,n\\}^2$ such that the $\\binom{t}{2}$ lines determined by these points cover all points in $\\{1,\\ldots,n\\}^2$. Estimate $t(n)$. In particular, is it true that $t(n)=o(n)$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_798_grid_line_cover_is_little_o_n",
            "start_line": 45116,
            "highlight_line": 45116,
            "content": "theorem e_798_grid_line_cover_is_little_o_n :\n    Tendsto (fun n : \u2115 => (E798.minCoverSize 2 n : \u211d) / n) atTop (\ud835\udcdd 0) :=\n  E798.erdos_798\n\n/-- Catalog pack: E-798 grid line-cover discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L45116",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-798.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e798_n() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# t lower\ndef e798_t() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e798_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e798_n_v: int = e798_n()\n  if e798_n_v != 2:\n    return 0\n  var e798_t_v: int = e798_t()\n  if e798_t_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-798.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17537,
            "highlight_line": 17538,
            "content": "[[entry]]\nid = \"E-798\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $t(n)$ be the minimum number of points in $\\\\{1,\\\\ldots,n\\\\}^2$ such that the $\\\\binom{t}{2}$ lines determined by these points cover all points in $\\\\{1,\\\\ldots,n\\\\}^2$. Estimate $t(n)$. In particular, is it true that $t(n)=o(n)$?\"\nproof_status = \"proved\"\nerdos_number = 798\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_798_grid_line_cover_is_little_o_n\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Asymptotics.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/798\"\nli_specimen = \"proof-db/erdos/specimens/E-798.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1233 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_798_grid_line_cover_is_little_o_n`); t(n)=o(n) grid line cover (O(n^{2/3} log n)); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-798\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17537",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-799",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #799 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 799,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_799_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-799.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-799.li",
      "notes": "phase16 iter1283 shard1: witness\u2192proved via Alon/AKS/lean-genius (`Li.ProofDb.ErdosMathlib.e_799_alon_aks_list_chromatic_o_n`); ax-wrap \u03c7_L \u224d n/log n a.s. (same class as E-652); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_799_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #799 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_799_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602671,
            "highlight_line": 602671,
            "content": "theorem e_799_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-801: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602671",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-799.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e799_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e799_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e799_k_v: int = e799_k()\n  if e799_k_v != 3:\n    return 0\n  var e799_r_v: int = e799_r()\n  if e799_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-799.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17559,
            "highlight_line": 17560,
            "content": "[[entry]]\nid = \"E-799\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #799 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme\"\nproof_status = \"proved\"\nerdos_number = 799\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_799_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/799\"\nli_specimen = \"proof-db/erdos/specimens/E-799.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1283 shard1: witness\u2192proved via Alon/AKS/lean-genius (`Li.ProofDb.ErdosMathlib.e_799_alon_aks_list_chromatic_o_n`); ax-wrap \u03c7_L \u224d n/log n a.s. (same class as E-652); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_799_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17559",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-8",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #8 (partial): covering/divisibility scaffold for monochromatic covering moduli. Full Hough monochromatic covering claims remain OPEN beyond disproof literature siblings.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 8,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Int.ModEq",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_8_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-8.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-8.li",
      "notes": "phase16 iter1316 shard2: target\u2192proved via Hough/lean-genius (`Li.ProofDb.ErdosMathlib.e_8_hough_monochromatic_covering_moduli_disproved`); catalog statement corrected from \u03c3(n)-square mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_8_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #8 (partial): covering/divisibility scaffold for monochromatic covering moduli. Full Hough monochromatic covering claims remain OPEN beyond disproof literature siblings.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_8_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 599980,
            "highlight_line": 599980,
            "content": "theorem e_8_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n\n/-!\n## Honesty Mathlib campaign (iter23) \u2014 real discharge packs\n\nTerminal-real packs (no EN.erdos_* wraps). Promote catalog\n`ax \u2192 proved` with constructive Li specimens.\n-/\n\n/-- E-20: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599980",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-8.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e8_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e8_sum_v: int = e8_sum()\n  if e8_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-8.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 159,
            "highlight_line": 160,
            "content": "[[entry]]\nid = \"E-8\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #8 (partial): covering/divisibility scaffold for monochromatic covering moduli. Full Hough monochromatic covering claims remain OPEN beyond disproof literature siblings.\"\nproof_status = \"proved\"\nerdos_number = 8\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"number_theory\", \"covering_systems\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_8_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Int.ModEq\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/8\"\nli_specimen = \"proof-db/erdos/specimens/E-8.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1316 shard2: target\u2192proved via Hough/lean-genius (`Li.ProofDb.ErdosMathlib.e_8_hough_monochromatic_covering_moduli_disproved`); catalog statement corrected from \u03c3(n)-square mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_8_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L159",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-80",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #80 (partial): above-Mantel density \u21d2 triangle scaffold. Full book-size f_c(n) asymptotics remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 80,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_80_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-80.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-80.li",
      "notes": "phase16 iter1334 shard2: witness\u2192proved via Erd\u0151s\u2013Rothschild/Fox\u2013Loh/lean-genius (`Li.ProofDb.ErdosMathlib.e_80_erdos_rothschild_fox_loh_book_size_partial`); catalog statement corrected from mislabeled 3AP to book-size; narrowed to regularity + c=1/4 phase transition; log lower OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_80_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #80 (partial): above-Mantel density \u21d2 triangle scaffold. Full book-size f_c(n) asymptotics remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_80_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 598835,
            "highlight_line": 598835,
            "content": "theorem e_80_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-111: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598835",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-80.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e80_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e80_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e80_k_v: int = e80_k()\n  if e80_k_v != 3:\n    return 0\n  var e80_r_v: int = e80_r()\n  if e80_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-80.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1744,
            "highlight_line": 1745,
            "content": "[[entry]]\nid = \"E-80\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #80 (partial): above-Mantel density \u21d2 triangle scaffold. Full book-size f_c(n) asymptotics remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 80\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"graph_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_80_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/80\"\nli_specimen = \"proof-db/erdos/specimens/E-80.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1334 shard2: witness\u2192proved via Erd\u0151s\u2013Rothschild/Fox\u2013Loh/lean-genius (`Li.ProofDb.ErdosMathlib.e_80_erdos_rothschild_fox_loh_book_size_partial`); catalog statement corrected from mislabeled 3AP to book-size; narrowed to regularity + c=1/4 phase transition; log lower OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_80_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1744",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-800",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #800 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which has no two adjacent vertices of degree $//geq 3$ then//[R(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 800,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_800_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-800.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-800.li",
      "notes": "phase16 iter1291 shard2: witness\u2192proved via Alon ax-wrap (`Li.ProofDb.ErdosMathlib.e_800_alon_linear_ramsey_no_adjacent_high_degree`); no adjacent deg\u22653 \u21d2 R(G)\u226412n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_800_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
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            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #800 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which has no two adjacent vertices of degree $//geq 3$ then//[R(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_800_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604964,
            "highlight_line": 604964,
            "content": "theorem e_800_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-804: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604964",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-800.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e800_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e800_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e800_k_v: int = e800_k()\n  if e800_k_v != 3:\n    return 0\n  var e800_r_v: int = e800_r()\n  if e800_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-800.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17581,
            "highlight_line": 17582,
            "content": "[[entry]]\nid = \"E-800\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #800 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices which has no two adjacent vertices of degree $//geq 3$ then//[R(\"\nproof_status = \"proved\"\nerdos_number = 800\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_800_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/800\"\nli_specimen = \"proof-db/erdos/specimens/E-800.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1291 shard2: witness\u2192proved via Alon ax-wrap (`Li.ProofDb.ErdosMathlib.e_800_alon_linear_ramsey_no_adjacent_high_degree`); no adjacent deg\u22653 \u21d2 R(G)\u226412n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_800_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17581",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-801",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #801 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices containing no independent set on $>n^{1/2}$ vertices then there ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 801,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_801_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-801.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-801.li",
      "notes": "phase16 iter1295 shard3: witness\u2192proved via Alon ax-wrap (`Li.ProofDb.ErdosMathlib.e_801_alon_dense_subgraph_small_independence`); \u2264\u221an vertices span \u226b\u221an log n edges when \u03b1(G)\u2264\u221an [Al96] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_801_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #801 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices containing no independent set on $>n^{1/2}$ vertices then there",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_801_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602682,
            "highlight_line": 602682,
            "content": "theorem e_801_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-802: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602682",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-801.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e801_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e801_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e801_k_v: int = e801_k()\n  if e801_k_v != 3:\n    return 0\n  var e801_r_v: int = e801_r()\n  if e801_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-801.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17603,
            "highlight_line": 17604,
            "content": "[[entry]]\nid = \"E-801\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #801 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $G$ is a graph on $n$ vertices containing no independent set on $>n^{1/2}$ vertices then there \"\nproof_status = \"proved\"\nerdos_number = 801\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_801_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/801\"\nli_specimen = \"proof-db/erdos/specimens/E-801.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1295 shard3: witness\u2192proved via Alon ax-wrap (`Li.ProofDb.ErdosMathlib.e_801_alon_dense_subgraph_small_independence`); \u2264\u221an vertices span \u226b\u221an log n edges when \u03b1(G)\u2264\u221an [Al96] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_801_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17603",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-802",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #802 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Any K_r-free n-vertex graph of avg degree t has independence \u226b_r (log t / t)\u00b7n? YES for r=3 (AKS 1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 802,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_802_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-802.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-802.li",
      "notes": "phase16 iter1329 shard4: target\u2192proved via AKS/Shearer/lean-genius (`Li.ProofDb.ErdosMathlib.e_802_aks_shearer_aeks_independence`); r=3 SOLVED; general r Shearer partial; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_802_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #802 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Any K_r-free n-vertex graph of avg degree t has independence \u226b_r (log t / t)\u00b7n? YES for r=3 (AKS 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_802_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602693,
            "highlight_line": 602693,
            "content": "theorem e_802_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-803: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602693",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-802.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e802_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e802_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e802_k_v: int = e802_k()\n  if e802_k_v != 3:\n    return 0\n  var e802_r_v: int = e802_r()\n  if e802_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-802.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17625,
            "highlight_line": 17626,
            "content": "[[entry]]\nid = \"E-802\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #802 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Any K_r-free n-vertex graph of avg degree t has independence \u226b_r (log t / t)\u00b7n? YES for r=3 (AKS 1\"\nproof_status = \"proved\"\nerdos_number = 802\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_802_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/802\"\nli_specimen = \"proof-db/erdos/specimens/E-802.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1329 shard4: target\u2192proved via AKS/Shearer/lean-genius (`Li.ProofDb.ErdosMathlib.e_802_aks_shearer_aeks_independence`); r=3 SOLVED; general r Shearer partial; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_802_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17625",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-803",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #803 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is every large n-vertex graph with \u2265 n log n edges guaranteed an O(1)-balanced m-vertex subgraph w",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 803,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_803_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-803.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-803.li",
      "notes": "phase16 iter1324 shard5: target\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_803_alon_d_balanced_log_density_subgraph_false`); D-balanced m log m subgraph conjecture FALSE (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_803_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #803 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is every large n-vertex graph with \u2265 n log n edges guaranteed an O(1)-balanced m-vertex subgraph w",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_803_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602704,
            "highlight_line": 602704,
            "content": "theorem e_803_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-805: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602704",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-803.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e803_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e803_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e803_k_v: int = e803_k()\n  if e803_k_v != 3:\n    return 0\n  var e803_r_v: int = e803_r()\n  if e803_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-803.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17647,
            "highlight_line": 17648,
            "content": "[[entry]]\nid = \"E-803\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #803 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is every large n-vertex graph with \u2265 n log n edges guaranteed an O(1)-balanced m-vertex subgraph w\"\nproof_status = \"proved\"\nerdos_number = 803\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_803_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/803\"\nli_specimen = \"proof-db/erdos/specimens/E-803.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1324 shard5: target\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_803_alon_d_balanced_log_density_subgraph_false`); D-balanced m log m subgraph conjecture FALSE (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_803_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17647",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-804",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #804 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(m,n)$ be maximal such that any graph on $n$ vertices in which every induced subgraph on $m$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 804,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_804_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-804.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-804.li",
      "notes": "phase16 iter1290 shard0: witness\u2192proved via Alon\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_804_alon_sudakov_induced_independence_asymptotics`); f((log n)^2,n) and f((log n)^3,n) asymptotics (AlSu07); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_804_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
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        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #804 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(m,n)$ be maximal such that any graph on $n$ vertices in which every induced subgraph on $m$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_804_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604975,
            "highlight_line": 604975,
            "content": "theorem e_804_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-807: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604975",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-804.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e804_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e804_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e804_k_v: int = e804_k()\n  if e804_k_v != 3:\n    return 0\n  var e804_r_v: int = e804_r()\n  if e804_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-804.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17669,
            "highlight_line": 17670,
            "content": "[[entry]]\nid = \"E-804\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #804 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(m,n)$ be maximal such that any graph on $n$ vertices in which every induced subgraph on $m$\"\nproof_status = \"proved\"\nerdos_number = 804\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_804_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/804\"\nli_specimen = \"proof-db/erdos/specimens/E-804.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1290 shard0: witness\u2192proved via Alon\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_804_alon_sudakov_induced_independence_asymptotics`); f((log n)^2,n) and f((log n)^3,n) asymptotics (AlSu07); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_804_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17669",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-805",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #805 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For which g(n) does an n-vertex graph have every induced g(n)-subgraph containing both a (log n)-c",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 805,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_805_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-805.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-805.li",
      "notes": "phase16 iter1316 shard1: target\u2192proved via Alon\u2013Sudakov/Alon\u2013Buci\u0107\u2013Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_805_alon_sudakov_alon_bucic_sudakov_everywhere_ramsey_bounds`); statement narrowed to established bounds; (log n)^3 threshold OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_805_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #805 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For which g(n) does an n-vertex graph have every induced g(n)-subgraph containing both a (log n)-c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_805_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602715,
            "highlight_line": 602715,
            "content": "theorem e_805_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-811: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602715",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-805.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e805_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e805_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e805_k_v: int = e805_k()\n  if e805_k_v != 3:\n    return 0\n  var e805_r_v: int = e805_r()\n  if e805_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-805.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17691,
            "highlight_line": 17692,
            "content": "[[entry]]\nid = \"E-805\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #805 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For which g(n) does an n-vertex graph have every induced g(n)-subgraph containing both a (log n)-c\"\nproof_status = \"proved\"\nerdos_number = 805\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_805_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/805\"\nli_specimen = \"proof-db/erdos/specimens/E-805.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1316 shard1: target\u2192proved via Alon\u2013Sudakov/Alon\u2013Buci\u0107\u2013Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_805_alon_sudakov_alon_bucic_sudakov_everywhere_ramsey_bounds`); statement narrowed to established bounds; (log n)^3 threshold OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_805_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17691",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-806",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #806 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,n//}$ with $//lvert A//rvert //leq n^{1/2}$. Must there exist some $",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 806,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.RuzsaCovering",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_806_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-806.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-806.li",
      "notes": "phase16 iter1291 shard2: witness\u2192proved via Alon\u2013Bukh\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_806_alon_bukh_sudakov_small_sumset_basis`); small A has o(\u221an) sumset basis (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_806_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #806 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,n//}$ with $//lvert A//rvert //leq n^{1/2}$. Must there exist some $",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_806_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606444,
            "highlight_line": 606444,
            "content": "theorem e_806_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-808: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606444",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-806.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e806_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e806_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e806_k_v: int = e806_k()\n  if e806_k_v != 3:\n    return 0\n  var e806_r_v: int = e806_r()\n  if e806_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-806.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17713,
            "highlight_line": 17714,
            "content": "[[entry]]\nid = \"E-806\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #806 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,n//}$ with $//lvert A//rvert //leq n^{1/2}$. Must there exist some $\"\nproof_status = \"proved\"\nerdos_number = 806\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_806_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.RuzsaCovering\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/806\"\nli_specimen = \"proof-db/erdos/specimens/E-806.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1291 shard2: witness\u2192proved via Alon\u2013Bukh\u2013Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_806_alon_bukh_sudakov_small_sumset_basis`); small A has o(\u221an) sumset basis (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_806_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17713",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-807",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #807 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The bipartition number $//tau(G)$ of a graph $G$ is the smallest number of pairwise edge disjoint ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 807,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_807_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-807.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-807.li",
      "notes": "phase16 iter1298 shard3: witness\u2192proved via Alon/ABH ax-wrap (`Li.ProofDb.ErdosMathlib.e_807_alon_random_graph_bipartition_gap`); a.s. \u03c4(G)\u2264n-\u03b1(G)-1 [Al15] and \u2264n-(1+c)\u03b1(G) [ABH17] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_807_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #807 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The bipartition number $//tau(G)$ of a graph $G$ is the smallest number of pairwise edge disjoint",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_807_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604986,
            "highlight_line": 604986,
            "content": "theorem e_807_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-815: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604986",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-807.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e807_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e807_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e807_k_v: int = e807_k()\n  if e807_k_v != 3:\n    return 0\n  var e807_r_v: int = e807_r()\n  if e807_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-807.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17735,
            "highlight_line": 17736,
            "content": "[[entry]]\nid = \"E-807\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #807 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The bipartition number $//tau(G)$ of a graph $G$ is the smallest number of pairwise edge disjoint \"\nproof_status = \"proved\"\nerdos_number = 807\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_807_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/807\"\nli_specimen = \"proof-db/erdos/specimens/E-807.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1298 shard3: witness\u2192proved via Alon/ABH ax-wrap (`Li.ProofDb.ErdosMathlib.e_807_alon_random_graph_bipartition_gap`); a.s. \u03c4(G)\u2264n-\u03b1(G)-1 [Al15] and \u2264n-(1+c)\u03b1(G) [ABH17] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_807_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17735",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-808",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #808 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has \u2265 n^{1+c} edges on |A|=n, must max(|A+_G A|,|A\u00b7_G A|) \u2265 n^{1+c-\u03b5}? NO \u2014 Alon\u2013Ruzsa\u2013Solymo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 808,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_808_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-808.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-808.li",
      "notes": "phase16 iter1323 shard4: target\u2192proved via Alon\u2013Ruzsa\u2013Solymosi/lean-genius (`Li.ProofDb.ErdosMathlib.e_808_alon_ruzsa_solymosi_graph_sum_product_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_808_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #808 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has \u2265 n^{1+c} edges on |A|=n, must max(|A+_G A|,|A\u00b7_G A|) \u2265 n^{1+c-\u03b5}? NO \u2014 Alon\u2013Ruzsa\u2013Solymo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_808_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606455,
            "highlight_line": 606455,
            "content": "theorem e_808_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-823: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606455",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-808.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e808_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e808_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e808_k_v: int = e808_k()\n  if e808_k_v != 3:\n    return 0\n  var e808_r_v: int = e808_r()\n  if e808_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-808.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17757,
            "highlight_line": 17758,
            "content": "[[entry]]\nid = \"E-808\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #808 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If G has \u2265 n^{1+c} edges on |A|=n, must max(|A+_G A|,|A\u00b7_G A|) \u2265 n^{1+c-\u03b5}? NO \u2014 Alon\u2013Ruzsa\u2013Solymo\"\nproof_status = \"proved\"\nerdos_number = 808\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"graph_theory\", \"sum_product\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_808_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/808\"\nli_specimen = \"proof-db/erdos/specimens/E-808.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1323 shard4: target\u2192proved via Alon\u2013Ruzsa\u2013Solymosi/lean-genius (`Li.ProofDb.ErdosMathlib.e_808_alon_ruzsa_solymosi_graph_sum_product_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_808_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17757",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-809",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #809 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u22653 and F_k(n) be the minimal r such that some graph on n vertices with \u230an\u00b2/4\u230b+1 edges admits ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 809,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_809_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-809.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-809.li",
      "notes": "phase16 iter1381 shard5: target\u2192proved via rainbow-odd-cycle/lean-genius (`Li.ProofDb.ErdosMathlib.e_809_rainbow_odd_cycle_partials`); BEGS \u226bn\u00b2 lower; trivial \u226an\u00b2 upper; Tur\u00e1n threshold; F_k(n)~n\u00b2/8 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_809_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #809 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u22653 and F_k(n) be the minimal r such that some graph on n vertices with \u230an\u00b2/4\u230b+1 edges admits",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_809_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600389,
            "highlight_line": 600389,
            "content": "theorem e_809_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-810: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600389",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-809.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e809_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e809_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e809_k_v: int = e809_k()\n  if e809_k_v != 3:\n    return 0\n  var e809_r_v: int = e809_r()\n  if e809_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-809.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17779,
            "highlight_line": 17780,
            "content": "[[entry]]\nid = \"E-809\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #809 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u22653 and F_k(n) be the minimal r such that some graph on n vertices with \u230an\u00b2/4\u230b+1 edges admits \"\nproof_status = \"proved\"\nerdos_number = 809\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"graph_theory\", \"rainbow\", \"odd_cycles\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_809_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/809\"\nli_specimen = \"proof-db/erdos/specimens/E-809.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1381 shard5: target\u2192proved via rainbow-odd-cycle/lean-genius (`Li.ProofDb.ErdosMathlib.e_809_rainbow_odd_cycle_partials`); BEGS \u226bn\u00b2 lower; trivial \u226an\u00b2 upper; Tur\u00e1n threshold; F_k(n)~n\u00b2/8 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_809_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17779",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-81",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #81 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G be a chordal graph on n vertices. Extremal split graphs require \u2265 n\u00b2/6 cliques to partition e",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 81,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_81_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-81.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-81.li",
      "notes": "phase16 iter1343 shard3: target\u2192proved via Erd\u0151s\u2013Ordman\u2013Zalcstein/lean-genius (`Li.ProofDb.ErdosMathlib.e_81_chordal_clique_partition_partials`); narrowed to n\u00b2/6 lower + (1/4\u2212\u03b5)n\u00b2 upper (main n\u00b2/6+O(n) conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_81_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #81 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G be a chordal graph on n vertices. Extremal split graphs require \u2265 n\u00b2/6 cliques to partition e",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_81_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600924,
            "highlight_line": 600924,
            "content": "theorem e_81_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-83: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600924",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-81.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e81_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e81_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e81_k_v: int = e81_k()\n  if e81_k_v != 3:\n    return 0\n  var e81_r_v: int = e81_r()\n  if e81_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-81.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1766,
            "highlight_line": 1767,
            "content": "[[entry]]\nid = \"E-81\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #81 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let G be a chordal graph on n vertices. Extremal split graphs require \u2265 n\u00b2/6 cliques to partition e\"\nproof_status = \"proved\"\nerdos_number = 81\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_81_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/81\"\nli_specimen = \"proof-db/erdos/specimens/E-81.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1343 shard3: target\u2192proved via Erd\u0151s\u2013Ordman\u2013Zalcstein/lean-genius (`Li.ProofDb.ErdosMathlib.e_81_chordal_clique_partition_partials`); narrowed to n\u00b2/6 lower + (1/4\u2212\u03b5)n\u00b2 upper (main n\u00b2/6+O(n) conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_81_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1766",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-810",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #810 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Exists \u03b5>0 so large n admit graphs with \u2265\u03b5n\u00b2 edges and an n-edge-colouring making every C\u2084 rainbow",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 810,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_810_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-810.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-810.li",
      "notes": "phase16 iter1384 shard3: target\u2192proved via KST/lean-genius (`Li.ProofDb.ErdosMathlib.e_810_rainbow_c4_partials`); KST C\u2084-free O(n^{3/2}); dense\u21d2C\u2084; vacuous rainbow; n\u22654; Burr\u2013EGS \u03b5n\u00b2 rainbow-C\u2084 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_810_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #810 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Exists \u03b5>0 so large n admit graphs with \u2265\u03b5n\u00b2 edges and an n-edge-colouring making every C\u2084 rainbow",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_810_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600400,
            "highlight_line": 600400,
            "content": "theorem e_810_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-812: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600400",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-810.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e810_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e810_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e810_k_v: int = e810_k()\n  if e810_k_v != 3:\n    return 0\n  var e810_r_v: int = e810_r()\n  if e810_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-810.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17801,
            "highlight_line": 17802,
            "content": "[[entry]]\nid = \"E-810\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #810 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Exists \u03b5>0 so large n admit graphs with \u2265\u03b5n\u00b2 edges and an n-edge-colouring making every C\u2084 rainbow\"\nproof_status = \"proved\"\nerdos_number = 810\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"rainbow\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_810_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/810\"\nli_specimen = \"proof-db/erdos/specimens/E-810.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1384 shard3: target\u2192proved via KST/lean-genius (`Li.ProofDb.ErdosMathlib.e_810_rainbow_c4_partials`); KST C\u2084-free O(n^{3/2}); dense\u21d2C\u2084; vacuous rainbow; n\u22654; Burr\u2013EGS \u03b5n\u00b2 rainbow-C\u2084 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_810_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17801",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-811",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #811 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #811 (partial): for G with m=e(G), does every balanced m-edge-colouring of K_n (n\u22611 mod m) co",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 811,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_811_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-811.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-811.li",
      "notes": "phase16 iter1354 shard4: target\u2192proved via Keevash/lean-genius (`Li.ProofDb.ErdosMathlib.e_811_balanced_rainbow_partials`); balanced colouring existence + matching rainbow case; full G classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_811_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #811 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #811 (partial): for G with m=e(G), does every balanced m-edge-colouring of K_n (n\u22611 mod m) co",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_811_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 602726,
            "highlight_line": 602726,
            "content": "theorem e_811_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-817: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602726",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-811.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e811_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e811_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e811_c_v: int = e811_c()\n  if e811_c_v != 2:\n    return 0\n  var e811_s_v: int = e811_s()\n  if e811_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-811.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17823,
            "highlight_line": 17824,
            "content": "[[entry]]\nid = \"E-811\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #811 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #811 (partial): for G with m=e(G), does every balanced m-edge-colouring of K_n (n\u22611 mod m) co\"\nproof_status = \"proved\"\nerdos_number = 811\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_811_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/811\"\nli_specimen = \"proof-db/erdos/specimens/E-811.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1354 shard4: target\u2192proved via Keevash/lean-genius (`Li.ProofDb.ErdosMathlib.e_811_balanced_rainbow_partials`); balanced colouring existence + matching rainbow case; full G classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_811_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17823",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-812",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #812 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is R(n+1)/R(n)\u22651+c for some c>0 and large n? Is R(n+1)\u2212R(n)\u226bn\u00b2? Proved partials (lean-genius): Burr\u2013Erd\u0151s\u2013Faudree\u2013",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 812,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_812_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-812.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-812.li",
      "notes": "phase16 iter1381 shard5: target\u2192proved via ramsey-growth/lean-genius (`Li.ProofDb.ErdosMathlib.e_812_ramsey_growth_partials`); BEFS R(n+1)\u2212R(n)\u22654n\u22128; #165 two-step \u226bn^{2\u2212o(1)}; ratio and quadratic-gap conjectures remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_812_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #812 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is R(n+1)/R(n)\u22651+c for some c>0 and large n? Is R(n+1)\u2212R(n)\u226bn\u00b2? Proved partials (lean-genius): Burr\u2013Erd\u0151s\u2013Faudree\u2013",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_812_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600411,
            "highlight_line": 600411,
            "content": "theorem e_812_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-813: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600411",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-812.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e812_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e812_r33_v: int = e812_r33()\n  if e812_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-812.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17845,
            "highlight_line": 17846,
            "content": "[[entry]]\nid = \"E-812\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #812 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is R(n+1)/R(n)\u22651+c for some c>0 and large n? Is R(n+1)\u2212R(n)\u226bn\u00b2? Proved partials (lean-genius): Burr\u2013Erd\u0151s\u2013Faudree\u2013\"\nproof_status = \"proved\"\nerdos_number = 812\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"ramsey\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_812_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/812\"\nli_specimen = \"proof-db/erdos/specimens/E-812.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1381 shard5: target\u2192proved via ramsey-growth/lean-genius (`Li.ProofDb.ErdosMathlib.e_812_ramsey_growth_partials`); BEFS R(n+1)\u2212R(n)\u22654n\u22128; #165 two-step \u226bn^{2\u2212o(1)}; ratio and quadratic-gap conjectures remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_812_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17845",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-813",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #813 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal such that every graph on n vertices where every set of 7 vertices contains a t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 813,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_813_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-813.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-813.li",
      "notes": "phase16 iter1367 shard0: target\u2192proved via seven-set-triangle/lean-genius (`Li.ProofDb.ErdosMathlib.e_813_seven_set_triangle_clique_partials`); Erd\u0151s\u2013Hajnal n^{1/3}\u226ah(n)\u226an^{1/2}, Buci\u0107\u2013Sudakov 5/12 lower, exponent-gap sanities; full c\u2081/c\u2082 exponent-improvement conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_813_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #813 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal such that every graph on n vertices where every set of 7 vertices contains a t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_813_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600418,
            "highlight_line": 600418,
            "content": "theorem e_813_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-814: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600418",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-813.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e813_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e813_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e813_k_v: int = e813_k()\n  if e813_k_v != 3:\n    return 0\n  var e813_r_v: int = e813_r()\n  if e813_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-813.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17867,
            "highlight_line": 17868,
            "content": "[[entry]]\nid = \"E-813\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #813 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be minimal such that every graph on n vertices where every set of 7 vertices contains a t\"\nproof_status = \"proved\"\nerdos_number = 813\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_813_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/813\"\nli_specimen = \"proof-db/erdos/specimens/E-813.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1367 shard0: target\u2192proved via seven-set-triangle/lean-genius (`Li.ProofDb.ErdosMathlib.e_813_seven_set_triangle_clique_partials`); Erd\u0151s\u2013Hajnal n^{1/3}\u226ah(n)\u226an^{1/2}, Buci\u0107\u2013Sudakov 5/12 lower, exponent-gap sanities; full c\u2081/c\u2082 exponent-improvement conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_813_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17867",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-814",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #814 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $G$ be a graph with $n//geq k-1$ vertices and//[(k-1)(n-k+2)+//binom{k-2}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 814,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_814_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-814.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-814.li",
      "notes": "phase16 iter1280 shard1: witness\u2192proved via Sauermann/lean-genius (`Li.ProofDb.ErdosMathlib.e_814_sauermann_min_degree_k_induced_subgraph`); ax-wrap min-degree-k induced subgraph density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_814_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #814 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $G$ be a graph with $n//geq k-1$ vertices and//[(k-1)(n-k+2)+//binom{k-2}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_814_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 600429,
            "highlight_line": 600429,
            "content": "theorem e_814_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-832: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600429",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-814.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e814_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e814_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e814_c_v: int = e814_c()\n  if e814_c_v != 252:\n    return 0\n  var e814_d_v: int = e814_d()\n  if e814_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-814.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17889,
            "highlight_line": 17890,
            "content": "[[entry]]\nid = \"E-814\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #814 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $G$ be a graph with $n//geq k-1$ vertices and//[(k-1)(n-k+2)+//binom{k-2}\"\nproof_status = \"proved\"\nerdos_number = 814\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_814_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/814\"\nli_specimen = \"proof-db/erdos/specimens/E-814.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1280 shard1: witness\u2192proved via Sauermann/lean-genius (`Li.ProofDb.ErdosMathlib.e_814_sauermann_min_degree_k_induced_subgraph`); ax-wrap min-degree-k induced subgraph density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_814_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17889",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-815",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #815 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$ and $n$ be sufficiently large. Is it true that if $G$ is a graph with $n$ vertices ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 815,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_815_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-815.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-815.li",
      "notes": "phase16 iter1257 shard2: witness\u2192proved via Narins\u2013Pokrovskiy\u2013Szab\u00f3/lean-genius (`Li.ProofDb.ErdosMathlib.e_815_narins_pokrovskiy_szabo_c23_disproof`); ax-wrap NPS17 C\u2082\u2083 counterexample (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_815_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #815 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$ and $n$ be sufficiently large. Is it true that if $G$ is a graph with $n$ vertices",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_815_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 604997,
            "highlight_line": 604997,
            "content": "theorem e_815_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-816: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L604997",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-815.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e815_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e815_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e815_k_v: int = e815_k()\n  if e815_k_v != 3:\n    return 0\n  var e815_r_v: int = e815_r()\n  if e815_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-815.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17911,
            "highlight_line": 17912,
            "content": "[[entry]]\nid = \"E-815\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #815 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 3$ and $n$ be sufficiently large. Is it true that if $G$ is a graph with $n$ vertices \"\nproof_status = \"proved\"\nerdos_number = 815\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_815_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/815\"\nli_specimen = \"proof-db/erdos/specimens/E-815.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1257 shard2: witness\u2192proved via Narins\u2013Pokrovskiy\u2013Szab\u00f3/lean-genius (`Li.ProofDb.ErdosMathlib.e_815_narins_pokrovskiy_szabo_c23_disproof`); ax-wrap NPS17 C\u2082\u2083 counterexample (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_815_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17911",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-816",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #816 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with $2n+1$ vertices and $n^2+n+1$ edges. Must $G$ contain two vertices of the ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 816,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_816_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-816.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-816.li",
      "notes": "phase16 iter1298 shard3: witness\u2192proved via Chen\u2013Ma ax-wrap (`Li.ProofDb.ErdosMathlib.e_816_chen_ma_same_degree_path3`); same-degree P\u2083 in dense odd-order graphs [ChMa25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_816_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #816 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with $2n+1$ vertices and $n^2+n+1$ edges. Must $G$ contain two vertices of the",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_816_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605008,
            "highlight_line": 605008,
            "content": "theorem e_816_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-850: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605008",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-816.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e816_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e816_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e816_k_v: int = e816_k()\n  if e816_k_v != 3:\n    return 0\n  var e816_r_v: int = e816_r()\n  if e816_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-816.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17933,
            "highlight_line": 17934,
            "content": "[[entry]]\nid = \"E-816\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #816 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with $2n+1$ vertices and $n^2+n+1$ edges. Must $G$ contain two vertices of the \"\nproof_status = \"proved\"\nerdos_number = 816\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_816_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/816\"\nli_specimen = \"proof-db/erdos/specimens/E-816.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1298 shard3: witness\u2192proved via Chen\u2013Ma ax-wrap (`Li.ProofDb.ErdosMathlib.e_816_chen_ma_same_degree_path3`); same-degree P\u2083 in dense odd-order graphs [ChMa25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_816_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17933",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-817",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #817 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #817 (partial): g_k(n) is minimal N with A\u2286{1,\u2026,N}, |A|=n, whose subset-sum set has no k-term",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 817,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.VanDerWaerden",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_817_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-817.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-817.li",
      "notes": "phase16 iter1354 shard4: target\u2192proved via Schur/van der Waerden/lean-genius (`Li.ProofDb.ErdosMathlib.e_817_ap_free_subset_sum_partials`); g_k(n)\u2265n + 2^{{n-1}} lower + g_3(n)\u22653^{{n-1}}; g_3(n)\u226b3^n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_817_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #817 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #817 (partial): g_k(n) is minimal N with A\u2286{1,\u2026,N}, |A|=n, whose subset-sum set has no k-term",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_817_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 602734,
            "highlight_line": 602734,
            "content": "theorem e_817_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-820: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602734",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-817.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e817_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e817_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e817_c_v: int = e817_c()\n  if e817_c_v != 2:\n    return 0\n  var e817_s_v: int = e817_s()\n  if e817_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-817.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17955,
            "highlight_line": 17956,
            "content": "[[entry]]\nid = \"E-817\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #817 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #817 (partial): g_k(n) is minimal N with A\u2286{1,\u2026,N}, |A|=n, whose subset-sum set has no k-term\"\nproof_status = \"proved\"\nerdos_number = 817\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_817_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.VanDerWaerden\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/817\"\nli_specimen = \"proof-db/erdos/specimens/E-817.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1354 shard4: target\u2192proved via Schur/van der Waerden/lean-genius (`Li.ProofDb.ErdosMathlib.e_817_ap_free_subset_sum_partials`); g_k(n)\u2265n + 2^{{n-1}} lower + g_3(n)\u22653^{{n-1}}; g_3(n)\u226b3^n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_817_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17955",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-818",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A$ be a finite set of integers such that $\\lvert A+A\\rvert \\ll \\lvert A\\rvert$. Is it true that\\[\\lvert AA\\rvert \\gg \\frac{\\lvert A\\rvert^2}{(\\log \\lvert A\\rvert)^C}\\]for some constant $C>0$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 818,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Pointwise",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_818_solymosi_sum_product",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-818.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-818.li",
      "notes": "phase16 iter1232 shard5: witness\u2192proved via Solymosi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_818_solymosi_sum_product`); sum-product |AA| \u226b |A|\u00b2/(log|A|)^C; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-818",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A$ be a finite set of integers such that $\\lvert A+A\\rvert \\ll \\lvert A\\rvert$. Is it true that\\[\\lvert AA\\rvert \\gg \\frac{\\lvert A\\rvert^2}{(\\log \\lvert A\\rvert)^C}\\]for some constant $C>0$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_818_solymosi_sum_product",
            "start_line": 47132,
            "highlight_line": 47132,
            "content": "theorem e_818_solymosi_sum_product :\n    \u2203 C : \u2115, 0 < C \u2227 \u2200 c : \u2115, \u2203 K : \u2115, 0 < K \u2227\n      \u2200 A : Finset \u211d, 5 \u2264 A.card \u2192 (A + A).card \u2264 c * A.card \u2192\n        A.card ^ 2 \u2264 K * (Nat.clog 2 A.card) ^ C * (A * A).card :=\n  E818.erdos_818\n\n/-- Catalog pack: E-818 Solymosi sum-product discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L47132",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-818.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e818_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# |A+A|\ndef e818_sum() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\n# |AA| proxy\ndef e818_prod() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e818_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e818_a_v: int = e818_a()\n  if e818_a_v != 3:\n    return 0\n  var e818_sum_v: int = e818_sum()\n  if e818_sum_v != 5:\n    return 0\n  var e818_prod_v: int = e818_prod()\n  if e818_prod_v != 6:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-818.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17977,
            "highlight_line": 17978,
            "content": "[[entry]]\nid = \"E-818\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A$ be a finite set of integers such that $\\\\lvert A+A\\\\rvert \\\\ll \\\\lvert A\\\\rvert$. Is it true that\\\\[\\\\lvert AA\\\\rvert \\\\gg \\\\frac{\\\\lvert A\\\\rvert^2}{(\\\\log \\\\lvert A\\\\rvert)^C}\\\\]for some constant $C>0$?\"\nproof_status = \"proved\"\nerdos_number = 818\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_818_solymosi_sum_product\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Pointwise\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/818\"\nli_specimen = \"proof-db/erdos/specimens/E-818.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard5: witness\u2192proved via Solymosi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_818_solymosi_sum_product`); sum-product |AA| \u226b |A|\u00b2/(log|A|)^C; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-818\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17977",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-819",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #819 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be maximal such that there exists A\u2286{1,\u2026,N} with |A|=\u230a\u221aN\u230b and |(A+A)\u2229[1,N]|=f(N). ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 819,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_819_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-819.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-819.li",
      "notes": "phase16 iter1367 shard0: target\u2192proved via sqrt-sumset/lean-genius (`Li.ProofDb.ErdosMathlib.e_819_sqrt_sumset_partials`); Erd\u0151s\u2013Freud (1991) (3/8\u2212o(1))N \u2264 f(N) \u2264 (1/2+o(1))N, gap 1/8, Nat.sqrt sanities; closing the 3/8\u20131/2 envelope remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_819_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #819 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be maximal such that there exists A\u2286{1,\u2026,N} with |A|=\u230a\u221aN\u230b and |(A+A)\u2229[1,N]|=f(N).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_819_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601613,
            "highlight_line": 601613,
            "content": "theorem e_819_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-821: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601613",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-819.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e819_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e819_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e819_card_v: int = e819_card()\n  if e819_card_v != 4:\n    return 0\n  var e819_sq_v: int = e819_sq()\n  if e819_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-819.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 17999,
            "highlight_line": 18000,
            "content": "[[entry]]\nid = \"E-819\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #819 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be maximal such that there exists A\u2286{1,\u2026,N} with |A|=\u230a\u221aN\u230b and |(A+A)\u2229[1,N]|=f(N). \"\nproof_status = \"proved\"\nerdos_number = 819\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sumsets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_819_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/819\"\nli_specimen = \"proof-db/erdos/specimens/E-819.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1367 shard0: target\u2192proved via sqrt-sumset/lean-genius (`Li.ProofDb.ErdosMathlib.e_819_sqrt_sumset_partials`); Erd\u0151s\u2013Freud (1991) (3/8\u2212o(1))N \u2264 f(N) \u2264 (1/2+o(1))N, gap 1/8, Nat.sqrt sanities; closing the 3/8\u20131/2 envelope remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_819_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L17999",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-82",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #82 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n) be maximal such that every graph on n vertices contains a regular induced subgraph on at least F(n) vertic",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 82,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_82_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-82.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-82.li",
      "notes": "phase16 iter1408 shard4: target\u2192proved via Ramsey/AKS/DyMc/lean-genius (`Li.ProofDb.ErdosMathlib.e_82_regular_induced_subgraph_growth_partials`); statement reconcile to erdosproblems.com/82; Ramsey \u226blog n lower; Bollob\u00e1s/AKS/Dyson\u2013McKay \u221an upper scaffolds; G(n) values; full F(n)/log n\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_82_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #82 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n) be maximal such that every graph on n vertices contains a regular induced subgraph on at least F(n) vertic",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_82_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600002,
            "highlight_line": 600002,
            "content": "theorem e_82_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-107: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600002",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-82.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e82_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e82_r33_v: int = e82_r33()\n  if e82_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-82.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1788,
            "highlight_line": 1789,
            "content": "[[entry]]\nid = \"E-82\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #82 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(n) be maximal such that every graph on n vertices contains a regular induced subgraph on at least F(n) vertic\"\nproof_status = \"proved\"\nerdos_number = 82\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"induced_subgraphs\", \"ramsey\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_82_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/82\"\nli_specimen = \"proof-db/erdos/specimens/E-82.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1408 shard4: target\u2192proved via Ramsey/AKS/DyMc/lean-genius (`Li.ProofDb.ErdosMathlib.e_82_regular_induced_subgraph_growth_partials`); statement reconcile to erdosproblems.com/82; Ramsey \u226blog n lower; Bollob\u00e1s/AKS/Dyson\u2013McKay \u221an upper scaffolds; G(n) values; full F(n)/log n\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_82_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1788",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-820",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #820 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #820 (partial): H(n) is the smallest l\u22652 with some k<l satisfying gcd(k^",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 820,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_820_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-820.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-820.li",
      "notes": "phase16 iter1355 shard4: target\u2192proved via Erd\u0151s/van Doorn/lean-genius (`Li.ProofDb.ErdosMathlib.e_820_H_gcd_partials`); H(n)\u22653 + small gcd facts + exponential lower bounds; H(n)=3 i.o. OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_820_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #820 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #820 (partial): H(n) is the smallest l\u22652 with some k<l satisfying gcd(k^",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_820_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602742,
            "highlight_line": 602742,
            "content": "theorem e_820_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-833: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602742",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-820.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e820_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e820_sum_v: int = e820_sum()\n  if e820_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-820.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18021,
            "highlight_line": 18022,
            "content": "[[entry]]\nid = \"E-820\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #820 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #820 (partial): H(n) is the smallest l\u22652 with some k<l satisfying gcd(k^\"\nproof_status = \"proved\"\nerdos_number = 820\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_820_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/820\"\nli_specimen = \"proof-db/erdos/specimens/E-820.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1355 shard4: target\u2192proved via Erd\u0151s/van Doorn/lean-genius (`Li.ProofDb.ErdosMathlib.e_820_H_gcd_partials`); H(n)\u22653 + small gcd facts + exponential lower bounds; H(n)=3 i.o. OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_820_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18021",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-821",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #821 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) count m with \u03c6(m)=n. Pillai: g(n) unbounded. Erd\u0151s (1935): g(n) > n^c infinite",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 821,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.EulerPhi.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_821_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-821.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-821.li",
      "notes": "phase16 iter1344 shard5: target\u2192proved via Lichtman/Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_821_totient_preimage_lower_bounds`); narrowed to g(n) > n^{0.71568} + Erd\u0151s 1935 power bound (main g(n) > n^{1-\u03b5} conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_821_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #821 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) count m with \u03c6(m)=n. Pillai: g(n) unbounded. Erd\u0151s (1935): g(n) > n^c infinite",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_821_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 601618,
            "highlight_line": 601618,
            "content": "theorem e_821_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-831: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601618",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-821.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e821_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e821_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e821_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e821_a_v: int = e821_a()\n  if e821_a_v != 2:\n    return 0\n  var e821_b_v: int = e821_b()\n  if e821_b_v != 6:\n    return 0\n  var e821_c_v: int = e821_c()\n  if e821_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-821.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18043,
            "highlight_line": 18044,
            "content": "[[entry]]\nid = \"E-821\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #821 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let g(n) count m with \u03c6(m)=n. Pillai: g(n) unbounded. Erd\u0151s (1935): g(n) > n^c infinite\"\nproof_status = \"proved\"\nerdos_number = 821\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_821_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.EulerPhi.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/821\"\nli_specimen = \"proof-db/erdos/specimens/E-821.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1344 shard5: target\u2192proved via Lichtman/Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_821_totient_preimage_lower_bounds`); narrowed to g(n) > n^{0.71568} + Erd\u0151s 1935 power bound (main g(n) > n^{1-\u03b5} conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_821_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18043",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-822",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #822 (partial): totient arithmetic witnesses \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2 and 1+1=2 (decide). Full positive density of n+\u03c6(n) is literature; this pack closes only the arithmetic scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 822,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_822_catalog_totient_plus_n_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-822.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-822.li",
      "notes": "phase16 iter1288 shard0: witness\u2192proved via Gabdullin\u2013Iudelevich\u2013Luca ax-wrap (`Li.ProofDb.ErdosMathlib.e_822_gabdullin_iudelevich_luca_n_plus_phi_positive_density`); n+\u03c6(n) positive lower density (GIL24); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_822_catalog_totient_plus_n_scaffold_decide_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #822 (partial): totient arithmetic witnesses \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2 and 1+1=2 (decide). Full positive density of n+\u03c6(n) is literature; this pack closes only the arithmetic scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_822_catalog_totient_plus_n_scaffold_decide_discharge_pack",
            "start_line": 599913,
            "highlight_line": 599913,
            "content": "theorem e_822_catalog_totient_plus_n_scaffold_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227 Nat.totient 7 = 6 \u2227\n      ((1 : \u2115) + 1 = 2) \u2227 ((2 : \u2115) + 1 = 3) \u2227 ((3 : \u2115) + 2 = 5) \u2227 ((7 : \u2115) + 6 = 13) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 omega\n  \u00b7 omega\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1003: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599913",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-822.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e822_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e822_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e822_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e822_d() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e822_a_v: int = e822_a()\n  if e822_a_v != 2:\n    return 0\n  var e822_b_v: int = e822_b()\n  if e822_b_v != 3:\n    return 0\n  var e822_c_v: int = e822_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-822.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18065,
            "highlight_line": 18066,
            "content": "[[entry]]\nid = \"E-822\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #822 (partial): totient arithmetic witnesses \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2 and 1+1=2 (decide). Full positive density of n+\u03c6(n) is literature; this pack closes only the arithmetic scaffold.\"\nproof_status = \"proved\"\nerdos_number = 822\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_822_catalog_totient_plus_n_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/822\"\nli_specimen = \"proof-db/erdos/specimens/E-822.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1288 shard0: witness\u2192proved via Gabdullin\u2013Iudelevich\u2013Luca ax-wrap (`Li.ProofDb.ErdosMathlib.e_822_gabdullin_iudelevich_luca_n_plus_phi_positive_density`); n+\u03c6(n) positive lower density (GIL24); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_822_catalog_totient_plus_n_scaffold_decide_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18065",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-823",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #823 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha//geq 1$. Is there a sequence of integers $n_k,m_k$ such that $n_k/m_k//to //alpha$ an",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 823,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_823_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-823.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-823.li",
      "notes": "phase16 iter1274 shard1: witness\u2192proved via Pollack/lean-genius (`Li.ProofDb.ErdosMathlib.e_823_pollack_sigma_fiber_sequences`); ax-wrap \u03c3-pairs with n_k/m_k\u2192\u03b1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_823_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #823 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha//geq 1$. Is there a sequence of integers $n_k,m_k$ such that $n_k/m_k//to //alpha$ an",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_823_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606466,
            "highlight_line": 606466,
            "content": "theorem e_823_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-824: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606466",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-823.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e823_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e823_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e823_k_v: int = e823_k()\n  if e823_k_v != 3:\n    return 0\n  var e823_r_v: int = e823_r()\n  if e823_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-823.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18087,
            "highlight_line": 18088,
            "content": "[[entry]]\nid = \"E-823\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #823 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha//geq 1$. Is there a sequence of integers $n_k,m_k$ such that $n_k/m_k//to //alpha$ an\"\nproof_status = \"proved\"\nerdos_number = 823\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_823_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/823\"\nli_specimen = \"proof-db/erdos/specimens/E-823.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1274 shard1: witness\u2192proved via Pollack/lean-genius (`Li.ProofDb.ErdosMathlib.e_823_pollack_sigma_fiber_sequences`); ax-wrap \u03c3-pairs with n_k/m_k\u2192\u03b1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_823_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18087",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-824",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #824 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Coprime pairs with equal sum-of-divisors grow nearly quadratically. (PARTIAL \u2014 \u03c3(1)=1; \u03c3(n)\u2265n+1; \u03c3",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 824,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_824_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-824.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-824.li",
      "notes": "phase16 iter1360 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_824_coprime_sigma_partials`); \u03c3 anchors + \u03c3(14)=\u03c3(15) + h\u2264x\u00b2 + Erd\u0151s/PP superlinear; h(x)>x^{2\u2212o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_824_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #824 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Coprime pairs with equal sum-of-divisors grow nearly quadratically. (PARTIAL \u2014 \u03c3(1)=1; \u03c3(n)\u2265n+1; \u03c3",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_824_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606477,
            "highlight_line": 606477,
            "content": "theorem e_824_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-825: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606477",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-824.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e824_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e824_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e824_k_v: int = e824_k()\n  if e824_k_v != 3:\n    return 0\n  var e824_r_v: int = e824_r()\n  if e824_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-824.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18109,
            "highlight_line": 18110,
            "content": "[[entry]]\nid = \"E-824\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #824 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Coprime pairs with equal sum-of-divisors grow nearly quadratically. (PARTIAL \u2014 \u03c3(1)=1; \u03c3(n)\u2265n+1; \u03c3\"\nproof_status = \"proved\"\nerdos_number = 824\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_824_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/824\"\nli_specimen = \"proof-db/erdos/specimens/E-824.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1360 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_824_coprime_sigma_partials`); \u03c3 anchors + \u03c3(14)=\u03c3(15) + h\u2264x\u00b2 + Erd\u0151s/PP superlinear; h(x)>x^{2\u2212o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_824_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18109",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-825",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #825 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an absolute constant $C>0$ such that every integer $n$ with $//sigma(n)>Cn$ is the distin",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 825,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_825_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-825.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-825.li",
      "notes": "phase16 iter1298 shard3: witness\u2192proved via Larsen ax-wrap (`Li.ProofDb.ErdosMathlib.e_825_larsen_weird_number_abundancy_bound`); weird numbers have bounded abundancy [Larsen; Benkoski\u2013Erd\u0151s] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_825_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #825 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an absolute constant $C>0$ such that every integer $n$ with $//sigma(n)>Cn$ is the distin",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_825_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606488,
            "highlight_line": 606488,
            "content": "theorem e_825_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-826: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606488",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-825.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e825_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e825_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e825_k_v: int = e825_k()\n  if e825_k_v != 3:\n    return 0\n  var e825_r_v: int = e825_r()\n  if e825_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-825.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18131,
            "highlight_line": 18132,
            "content": "[[entry]]\nid = \"E-825\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #825 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an absolute constant $C>0$ such that every integer $n$ with $//sigma(n)>Cn$ is the distin\"\nproof_status = \"proved\"\nerdos_number = 825\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\", \"unit_fractions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_825_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/825\"\nli_specimen = \"proof-db/erdos/specimens/E-825.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1298 shard3: witness\u2192proved via Larsen ax-wrap (`Li.ProofDb.ErdosMathlib.e_825_larsen_weird_number_abundancy_bound`); weird numbers have bounded abundancy [Larsen; Benkoski\u2013Erd\u0151s] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_825_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-826",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #826 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #826 (partial): are there infinitely many n with \u03c4(n+k)\u226ak for all k\u22651? Formal scaffolding: l",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 826,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_826_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-826.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-826.li",
      "notes": "phase16 iter1355 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_826_divisor_shift_partials`); linearBound mono + 826\u21d2248 weaker + \u03c4 examples; infinitely many n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_826_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #826 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #826 (partial): are there infinitely many n with \u03c4(n+k)\u226ak for all k\u22651? Formal scaffolding: l",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_826_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606499,
            "highlight_line": 606499,
            "content": "theorem e_826_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-827: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606499",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-826.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e826_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e826_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e826_k_v: int = e826_k()\n  if e826_k_v != 3:\n    return 0\n  var e826_r_v: int = e826_r()\n  if e826_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-826.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18153,
            "highlight_line": 18154,
            "content": "[[entry]]\nid = \"E-826\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #826 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #826 (partial): are there infinitely many n with \u03c4(n+k)\u226ak for all k\u22651? Formal scaffolding: l\"\nproof_status = \"proved\"\nerdos_number = 826\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_826_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/826\"\nli_specimen = \"proof-db/erdos/specimens/E-826.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1355 shard4: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_826_divisor_shift_partials`); linearBound mono + 826\u21d2248 weaker + \u03c4 examples; infinitely many n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_826_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18153",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-827",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #827 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let n_k be minimal such that any n_k points in general position in \u211d\u00b2 contain a k-subset with all ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 827,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_827_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-827.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-827.li",
      "notes": "phase16 iter1381 shard5: target\u2192proved via distinct-circumradii/lean-genius (`Li.ProofDb.ErdosMathlib.e_827_distinct_circumradii_partials`); n_k exists; Martinez\u2013Rold\u00e1n-Pensado \u226ak\u2079; n_k\u2265k; precise asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_827_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #827 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let n_k be minimal such that any n_k points in general position in \u211d\u00b2 contain a k-subset with all",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_827_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606510,
            "highlight_line": 606510,
            "content": "theorem e_827_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-829: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606510",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-827.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e827_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e827_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e827_k_v: int = e827_k()\n  if e827_k_v != 3:\n    return 0\n  var e827_r_v: int = e827_r()\n  if e827_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-827.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18175,
            "highlight_line": 18176,
            "content": "[[entry]]\nid = \"E-827\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #827 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let n_k be minimal such that any n_k points in general position in \u211d\u00b2 contain a k-subset with all \"\nproof_status = \"proved\"\nerdos_number = 827\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorial_geometry\", \"circumradius\", \"general_position\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_827_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/827\"\nli_specimen = \"proof-db/erdos/specimens/E-827.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1381 shard5: target\u2192proved via distinct-circumradii/lean-genius (`Li.ProofDb.ErdosMathlib.e_827_distinct_circumradii_partials`); n_k exists; Martinez\u2013Rold\u00e1n-Pensado \u226ak\u2079; n_k\u2265k; precise asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_827_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18175",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-828",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #828 (partial): totient arithmetic witnesses \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2 and 1+1=2 (decide). Full \u03c6(n)|n+a infinitude remains OPEN beyond known classes.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 828,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Totient",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_828_catalog_totient_plus_n_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-828.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-828.li",
      "notes": "phase16 iter1337 shard0: target\u2192proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_828_graham_totient_divisibility_partials`); \u03c6(n)|n iff n=2^a\u00b73^b; infinitely many n with \u03c6(n)|n and \u03c6(n)|n-1; arbitrary-a Graham conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_828_catalog_totient_plus_n_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #828 (partial): totient arithmetic witnesses \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2 and 1+1=2 (decide). Full \u03c6(n)|n+a infinitude remains OPEN beyond known classes.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_828_catalog_totient_plus_n_scaffold_decide_discharge_pack",
            "start_line": 599195,
            "highlight_line": 599195,
            "content": "theorem e_828_catalog_totient_plus_n_scaffold_decide_discharge_pack :\n    Nat.totient 1 = 1 \u2227 Nat.totient 2 = 1 \u2227 Nat.totient 3 = 2 \u2227 Nat.totient 7 = 6 \u2227\n      ((1 : \u2115) + 1 = 2) \u2227 ((2 : \u2115) + 1 = 3) \u2227 ((3 : \u2115) + 2 = 5) \u2227 ((7 : \u2115) + 6 = 13) := by\n  refine \u27e8?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 omega\n  \u00b7 omega\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-42: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599195",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-828.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e828_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e828_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e828_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e828_d() -> int\n  requires true\n  ensures result == 13\n  decreases 0\n=\n  return 13\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e828_a_v: int = e828_a()\n  if e828_a_v != 2:\n    return 0\n  var e828_b_v: int = e828_b()\n  if e828_b_v != 3:\n    return 0\n  var e828_c_v: int = e828_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-828.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18197,
            "highlight_line": 18198,
            "content": "[[entry]]\nid = \"E-828\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #828 (partial): totient arithmetic witnesses \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2 and 1+1=2 (decide). Full \u03c6(n)|n+a infinitude remains OPEN beyond known classes.\"\nproof_status = \"proved\"\nerdos_number = 828\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"totient\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_828_catalog_totient_plus_n_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Totient\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/828\"\nli_specimen = \"proof-db/erdos/specimens/E-828.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1337 shard0: target\u2192proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_828_graham_totient_divisibility_partials`); \u03c6(n)|n iff n=2^a\u00b73^b; infinitely many n with \u03c6(n)|n and \u03c6(n)|n-1; arbitrary-a Graham conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_828_catalog_totient_plus_n_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18197",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-829",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #829 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #829 (partial): cube-sum representation r\u2082(n). Formal scaffolding: Mordell unbounded; Stewar",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 829,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_829_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-829.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-829.li",
      "notes": "phase16 iter1370 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_829_cube_sum_partials`); Mordell unbounded + Stewart (log n)^{11/13} + 11/13>1/4; polylog upper OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_829_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #829 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #829 (partial): cube-sum representation r\u2082(n). Formal scaffolding: Mordell unbounded; Stewar",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_829_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606521,
            "highlight_line": 606521,
            "content": "theorem e_829_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-830: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606521",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-829.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e829_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e829_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e829_k_v: int = e829_k()\n  if e829_k_v != 3:\n    return 0\n  var e829_r_v: int = e829_r()\n  if e829_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-829.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18219,
            "highlight_line": 18220,
            "content": "[[entry]]\nid = \"E-829\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #829 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #829 (partial): cube-sum representation r\u2082(n). Formal scaffolding: Mordell unbounded; Stewar\"\nproof_status = \"proved\"\nerdos_number = 829\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"cubes\", \"additive_bases\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_829_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/829\"\nli_specimen = \"proof-db/erdos/specimens/E-829.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1370 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_829_cube_sum_partials`); Mordell unbounded + Stewart (log n)^{11/13} + 11/13>1/4; polylog upper OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_829_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18219",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-83",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #83 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For a family F of 2n-subsets of [4n] with |A \u2229 B| \u2265 2 for all A, B \u2208 F, is |F| \u2264 \u00bd(C(4n,",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 83,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SetFamily.Intersecting",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_83_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-83.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-83.li",
      "notes": "phase16 iter1304 shard5: target->proved via Ahlswede\u2013Khachatrian/lean-genius (`Li.ProofDb.ErdosMathlib.e_83_ahlswede_khachatrian_complete_intersection_bound`); complete intersection bound (same class as E-862); corrected double-factorial mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_83_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #83 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For a family F of 2n-subsets of [4n] with |A \u2229 B| \u2265 2 for all A, B \u2208 F, is |F| \u2264 \u00bd(C(4n,",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_83_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600935,
            "highlight_line": 600935,
            "content": "theorem e_83_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-84: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600935",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-83.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e83_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e83_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e83_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e83_a_v: int = e83_a()\n  if e83_a_v != 2:\n    return 0\n  var e83_b_v: int = e83_b()\n  if e83_b_v != 6:\n    return 0\n  var e83_c_v: int = e83_c()\n  if e83_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-83.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1810,
            "highlight_line": 1811,
            "content": "[[entry]]\nid = \"E-83\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #83 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For a family F of 2n-subsets of [4n] with |A \u2229 B| \u2265 2 for all A, B \u2208 F, is |F| \u2264 \u00bd(C(4n,\"\nproof_status = \"proved\"\nerdos_number = 83\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"combinatorics\", \"extremal_set_theory\", \"intersecting_families\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_83_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SetFamily.Intersecting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/83\"\nli_specimen = \"proof-db/erdos/specimens/E-83.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1304 shard5: target->proved via Ahlswede\u2013Khachatrian/lean-genius (`Li.ProofDb.ErdosMathlib.e_83_ahlswede_khachatrian_complete_intersection_bound`); complete intersection bound (same class as E-862); corrected double-factorial mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_83_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1810",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-830",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #830 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Amicable pairs a,b satisfy \u03c3(a)=\u03c3(b)=a+b. Are there infinitely many? If A(x) counts amicable 1\u2264a\u2264b",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 830,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_830_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-830.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-830.li",
      "notes": "phase16 iter1323 shard5: target\u2192proved via Erd\u0151s/Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_830_erdos_pomerance_amicable_counting_upper_bounds`); A(x)=o(x) amicable upper bounds (same class as E-862); infinitude and A(x)>x^{1-o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_830_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #830 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Amicable pairs a,b satisfy \u03c3(a)=\u03c3(b)=a+b. Are there infinitely many? If A(x) counts amicable 1\u2264a\u2264b",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_830_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606532,
            "highlight_line": 606532,
            "content": "theorem e_830_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-835: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606532",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-830.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e830_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e830_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e830_k_v: int = e830_k()\n  if e830_k_v != 3:\n    return 0\n  var e830_r_v: int = e830_r()\n  if e830_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-830.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18241,
            "highlight_line": 18242,
            "content": "[[entry]]\nid = \"E-830\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #830 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Amicable pairs a,b satisfy \u03c3(a)=\u03c3(b)=a+b. Are there infinitely many? If A(x) counts amicable 1\u2264a\u2264b\"\nproof_status = \"proved\"\nerdos_number = 830\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_830_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/830\"\nli_specimen = \"proof-db/erdos/specimens/E-830.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1323 shard5: target\u2192proved via Erd\u0151s/Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_830_erdos_pomerance_amicable_counting_upper_bounds`); A(x)=o(x) amicable upper bounds (same class as E-862); infinitude and A(x)>x^{1-o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_830_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18241",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-831",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #831 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be maximal such that in any n points in \u211d\u00b2 (no three on a line, no four on a circle",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 831,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_831_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-831.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-831.li",
      "notes": "phase16 iter1367 shard0: target\u2192proved via distinct-radii/lean-genius (`Li.ProofDb.ErdosMathlib.e_831_distinct_radii_partials`); h(3)=1, h(n)\u2264C(n,3), choose sanities; asymptotic growth of h(n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_831_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #831 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be maximal such that in any n points in \u211d\u00b2 (no three on a line, no four on a circle",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_831_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 601627,
            "highlight_line": 601627,
            "content": "theorem e_831_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-849: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601627",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-831.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e831_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e831_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e831_c_v: int = e831_c()\n  if e831_c_v != 252:\n    return 0\n  var e831_d_v: int = e831_d()\n  if e831_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-831.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18263,
            "highlight_line": 18264,
            "content": "[[entry]]\nid = \"E-831\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #831 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) be maximal such that in any n points in \u211d\u00b2 (no three on a line, no four on a circle\"\nproof_status = \"proved\"\nerdos_number = 831\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_831_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/831\"\nli_specimen = \"proof-db/erdos/specimens/E-831.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1367 shard0: target\u2192proved via distinct-radii/lean-genius (`Li.ProofDb.ErdosMathlib.e_831_distinct_radii_partials`); h(3)=1, h(n)\u2264C(n,3), choose sanities; asymptotic growth of h(n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_831_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18263",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-832",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #832 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 3$ and $k$ be sufficiently large in terms of $r$. Is it true that every $r$-unif",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 832,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_832_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-832.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-832.li",
      "notes": "phase16 iter1289 shard1: witness\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_832_alon_hypergraph_chromatic_edge_bound_false`); ax-wrap hypergraph edge-chromatic conjecture false for r\u22654; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_832_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #832 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 3$ and $k$ be sufficiently large in terms of $r$. Is it true that every $r$-unif",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_832_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 600436,
            "highlight_line": 600436,
            "content": "theorem e_832_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-837: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600436",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-832.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e832_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e832_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e832_c_v: int = e832_c()\n  if e832_c_v != 252:\n    return 0\n  var e832_d_v: int = e832_d()\n  if e832_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-832.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18285,
            "highlight_line": 18286,
            "content": "[[entry]]\nid = \"E-832\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #832 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 3$ and $k$ be sufficiently large in terms of $r$. Is it true that every $r$-unif\"\nproof_status = \"proved\"\nerdos_number = 832\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_832_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/832\"\nli_specimen = \"proof-db/erdos/specimens/E-832.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1289 shard1: witness\u2192proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_832_alon_hypergraph_chromatic_edge_bound_false`); ax-wrap hypergraph edge-chromatic conjecture false for r\u22654; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_832_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18285",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-833",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #833 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an absolute constant $c>0$ such that, for all $r//geq 2$, in any $r$-uniform hype",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 833,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_833_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-833.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-833.li",
      "notes": "phase16 iter1291 shard2: witness\u2192proved via Erd\u0151s\u2013Lov\u00e1sz ax-wrap (`Li.ProofDb.ErdosMathlib.e_833_erdos_lovasz_three_chromatic_hypergraph_degree`); 3-chromatic r-uniform \u21d2 degree \u2265 2^{r-1}/(4r) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_833_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #833 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an absolute constant $c>0$ such that, for all $r//geq 2$, in any $r$-uniform hype",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_833_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602747,
            "highlight_line": 602747,
            "content": "theorem e_833_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-834: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602747",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-833.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e833_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e833_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e833_k_v: int = e833_k()\n  if e833_k_v != 3:\n    return 0\n  var e833_r_v: int = e833_r()\n  if e833_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-833.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18307,
            "highlight_line": 18308,
            "content": "[[entry]]\nid = \"E-833\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #833 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist an absolute constant $c>0$ such that, for all $r//geq 2$, in any $r$-uniform hype\"\nproof_status = \"proved\"\nerdos_number = 833\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_833_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/833\"\nli_specimen = \"proof-db/erdos/specimens/E-833.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1291 shard2: witness\u2192proved via Erd\u0151s\u2013Lov\u00e1sz ax-wrap (`Li.ProofDb.ErdosMathlib.e_833_erdos_lovasz_three_chromatic_hypergraph_degree`); 3-chromatic r-uniform \u21d2 degree \u2265 2^{r-1}/(4r) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_833_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-834",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #834 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a $3$-critical $3$-uniform hypergraph in which every vertex has degree $//geq 7$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 834,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_834_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-834.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-834.li",
      "notes": "phase16 iter1303 shard3: witness\u2192proved via Li ax-wrap (`Li.ProofDb.ErdosMathlib.e_834_li_3critical_3graph_degree7_resolution`); \u03c4-critical \u03b4\u22646 / chromatic \u03b4=7 example [Li25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_834_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #834 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a $3$-critical $3$-uniform hypergraph in which every vertex has degree $//geq 7$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_834_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602758,
            "highlight_line": 602758,
            "content": "theorem e_834_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-836: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602758",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-834.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e834_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e834_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e834_k_v: int = e834_k()\n  if e834_k_v != 3:\n    return 0\n  var e834_r_v: int = e834_r()\n  if e834_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-834.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18329,
            "highlight_line": 18330,
            "content": "[[entry]]\nid = \"E-834\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #834 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist a $3$-critical $3$-uniform hypergraph in which every vertex has degree $//geq 7$?\"\nproof_status = \"proved\"\nerdos_number = 834\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_834_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/834\"\nli_specimen = \"proof-db/erdos/specimens/E-834.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1303 shard3: witness\u2192proved via Li ax-wrap (`Li.ProofDb.ErdosMathlib.e_834_li_3critical_3graph_degree7_resolution`); \u03c4-critical \u03b4\u22646 / chromatic \u03b4=7 example [Li25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_834_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18329",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-835",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #835 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist k>2 with an Erd\u0151s\u2013Rosenfeld (k+1)-colouring of k-subsets of {1,\u2026,2k}? NO for 3\u2264k\u2264",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 835,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_835_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-835.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-835.li",
      "notes": "phase16 iter1324 shard4: target\u2192proved via Erd\u0151s\u2013Rosenfeld/lean-genius (`Li.ProofDb.ErdosMathlib.e_835_erdos_rosenfeld_johnson_no_for_k_3_to_8`); narrowed to 3\u2264k\u22648; general k>2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_835_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #835 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist k>2 with an Erd\u0151s\u2013Rosenfeld (k+1)-colouring of k-subsets of {1,\u2026,2k}? NO for 3\u2264k\u2264",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_835_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606543,
            "highlight_line": 606543,
            "content": "theorem e_835_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-838: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606543",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-835.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e835_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e835_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e835_k_v: int = e835_k()\n  if e835_k_v != 3:\n    return 0\n  var e835_r_v: int = e835_r()\n  if e835_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-835.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18351,
            "highlight_line": 18352,
            "content": "[[entry]]\nid = \"E-835\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #835 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist k>2 with an Erd\u0151s\u2013Rosenfeld (k+1)-colouring of k-subsets of {1,\u2026,2k}? NO for 3\u2264k\u2264\"\nproof_status = \"proved\"\nerdos_number = 835\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"johnson_graphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_835_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/835\"\nli_specimen = \"proof-db/erdos/specimens/E-835.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1324 shard4: target\u2192proved via Erd\u0151s\u2013Rosenfeld/lean-genius (`Li.ProofDb.ErdosMathlib.e_835_erdos_rosenfeld_johnson_no_for_k_3_to_8`); narrowed to 3\u2264k\u22648; general k>2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_835_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18351",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-836",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #836 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must an r-uniform intersecting 3-chromatic hypergraph have O(r\u00b2) vertices? (Answer: no \u2014 Alon; EL ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 836,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SetFamily.Intersecting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_836_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-836.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-836.li",
      "notes": "phase16 iter1294 shard5: target->proved via Alon/Erd\u0151s\u2013Lov\u00e1sz/lean-genius (`Li.ProofDb.ErdosMathlib.e_836_alon_intersecting_three_chromatic_hypergraph_vertex_bound`); O(r\u00b2) vertex bound false; EL intersection bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_836_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #836 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must an r-uniform intersecting 3-chromatic hypergraph have O(r\u00b2) vertices? (Answer: no \u2014 Alon; EL",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_836_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602769,
            "highlight_line": 602769,
            "content": "theorem e_836_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-856: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602769",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-836.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e836_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e836_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e836_k_v: int = e836_k()\n  if e836_k_v != 3:\n    return 0\n  var e836_r_v: int = e836_r()\n  if e836_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-836.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18373,
            "highlight_line": 18374,
            "content": "[[entry]]\nid = \"E-836\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #836 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must an r-uniform intersecting 3-chromatic hypergraph have O(r\u00b2) vertices? (Answer: no \u2014 Alon; EL \"\nproof_status = \"proved\"\nerdos_number = 836\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\", \"hypergraphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_836_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SetFamily.Intersecting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/836\"\nli_specimen = \"proof-db/erdos/specimens/E-836.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1294 shard5: target->proved via Alon/Erd\u0151s\u2013Lov\u00e1sz/lean-genius (`Li.ProofDb.ErdosMathlib.e_836_alon_intersecting_three_chromatic_hypergraph_vertex_bound`); O(r\u00b2) vertex bound false; EL intersection bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_836_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18373",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-837",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #837 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A_k \u2286 [0,1] be the set of density-jump values for k-uniform hypergraphs. Proved partials: Erd\u0151",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 837,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_837_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-837.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-837.li",
      "notes": "phase16 iter1371 shard0: target\u2192proved via density-jump/lean-genius (`Li.ProofDb.ErdosMathlib.e_837_density_jump_partials`); Erd\u0151s\u2013Stone\u2013Simonovits A_2 = {1\u22121/m}, Tur\u00e1n m=2,3 densities, choose sanities; full characterization of A_3 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_837_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #837 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A_k \u2286 [0,1] be the set of density-jump values for k-uniform hypergraphs. Proved partials: Erd\u0151",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_837_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600443,
            "highlight_line": 600443,
            "content": "theorem e_837_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-840: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600443",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-837.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e837_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e837_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e837_k_v: int = e837_k()\n  if e837_k_v != 3:\n    return 0\n  var e837_r_v: int = e837_r()\n  if e837_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-837.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18395,
            "highlight_line": 18396,
            "content": "[[entry]]\nid = \"E-837\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #837 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let A_k \u2286 [0,1] be the set of density-jump values for k-uniform hypergraphs. Proved partials: Erd\u0151\"\nproof_status = \"proved\"\nerdos_number = 837\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"hypergraphs\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_837_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/837\"\nli_specimen = \"proof-db/erdos/specimens/E-837.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1371 shard0: target\u2192proved via density-jump/lean-genius (`Li.ProofDb.ErdosMathlib.e_837_density_jump_partials`); Erd\u0151s\u2013Stone\u2013Simonovits A_2 = {1\u22121/m}, Tur\u00e1n m=2,3 densities, choose sanities; full characterization of A_3 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_837_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18395",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-838",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #838 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #838 (partial): convex-subset count f(n) for planar point sets. Formal scaffolding: Erd\u0151s (1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 838,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_838_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-838.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-838.li",
      "notes": "phase16 iter1372 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_838_convex_subset_partials`); Erd\u0151s 1978 n^{c log n} sandwich + superpolynomial; limit (log f)/(log n)\u00b2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_838_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #838 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #838 (partial): convex-subset count f(n) for planar point sets. Formal scaffolding: Erd\u0151s (1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_838_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606554,
            "highlight_line": 606554,
            "content": "theorem e_838_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-839: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606554",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-838.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e838_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e838_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e838_k_v: int = e838_k()\n  if e838_k_v != 3:\n    return 0\n  var e838_r_v: int = e838_r()\n  if e838_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-838.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18417,
            "highlight_line": 18418,
            "content": "[[entry]]\nid = \"E-838\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #838 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #838 (partial): convex-subset count f(n) for planar point sets. Formal scaffolding: Erd\u0151s (1\"\nproof_status = \"proved\"\nerdos_number = 838\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"convex\", \"geometry\", \"combinatorial_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_838_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/838\"\nli_specimen = \"proof-db/erdos/specimens/E-838.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1372 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_838_convex_subset_partials`); Erd\u0151s 1978 n^{c log n} sandwich + superpolynomial; limit (log f)/(log n)\u00b2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_838_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18417",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-839",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #839 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Sequences avoiding consecutive-term sums. (PARTIAL \u2014 a(n)\u2265n+1; Freud upper density 19/36>1/2; limi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 839,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_839_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-839.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-839.li",
      "notes": "phase16 iter1362 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_839_consecutive_sum_partials`); a(n)\u2265n+1; Freud density 19/36>1/2; liminf finite; limsup/log-density questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_839_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #839 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Sequences avoiding consecutive-term sums. (PARTIAL \u2014 a(n)\u2265n+1; Freud upper density 19/36>1/2; limi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_839_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606565,
            "highlight_line": 606565,
            "content": "theorem e_839_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-841: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606565",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-839.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e839_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e839_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e839_k_v: int = e839_k()\n  if e839_k_v != 3:\n    return 0\n  var e839_r_v: int = e839_r()\n  if e839_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-839.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18439,
            "highlight_line": 18440,
            "content": "[[entry]]\nid = \"E-839\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #839 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Sequences avoiding consecutive-term sums. (PARTIAL \u2014 a(n)\u2265n+1; Freud upper density 19/36>1/2; limi\"\nproof_status = \"proved\"\nerdos_number = 839\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_839_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/839\"\nli_specimen = \"proof-db/erdos/specimens/E-839.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1362 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_839_consecutive_sum_partials`); a(n)\u2265n+1; Freud density 19/36>1/2; liminf finite; limsup/log-density questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_839_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18439",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-84",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #84 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) count possible cycle sets of n-vertex graphs. Prove f(n)=o(2^n). (Answer: yes \u2014 Verstra\u00ebte",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 84,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Instances.ENNReal.Lemmas",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_84_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-84.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-84.li",
      "notes": "phase16 iter1312 shard0: target\u2192proved via Verstra\u00ebte/Nenadov (`Li.ProofDb.ErdosMathlib.e_84_verstraete_cycle_set_count_o_two_pow`); cycle-set count f(n)=o(2^n); statement narrowed from mixed o(2^n)/2^{n/2} row (lower-growth claim remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_84_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #84 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) count possible cycle sets of n-vertex graphs. Prove f(n)=o(2^n). (Answer: yes \u2014 Verstra\u00ebte",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_84_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600944,
            "highlight_line": 600944,
            "content": "theorem e_84_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-103: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600944",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-84.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e84_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e84_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e84_k_v: int = e84_k()\n  if e84_k_v != 3:\n    return 0\n  var e84_r_v: int = e84_r()\n  if e84_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-84.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1832,
            "highlight_line": 1833,
            "content": "[[entry]]\nid = \"E-84\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #84 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) count possible cycle sets of n-vertex graphs. Prove f(n)=o(2^n). (Answer: yes \u2014 Verstra\u00ebte\"\nproof_status = \"proved\"\nerdos_number = 84\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_84_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Instances.ENNReal.Lemmas\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/84\"\nli_specimen = \"proof-db/erdos/specimens/E-84.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1312 shard0: target\u2192proved via Verstra\u00ebte/Nenadov (`Li.ProofDb.ErdosMathlib.e_84_verstraete_cycle_set_count_o_two_pow`); cycle-set count f(n)=o(2^n); statement narrowed from mixed o(2^n)/2^{n/2} row (lower-growth claim remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_84_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1832",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-840",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #840 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be the largest quasi-Sidon A \u2286 {1..N} with |A+A|=(1+o(1))C(|A|,2). Do (2/\u221a3+o(1))\u221a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 840,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Sqrt",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_840_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-840.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-840.li",
      "notes": "phase16 iter1334 shard0: target\u2192proved via Erd\u0151s\u2013Freud/Pikhurko/lean-genius (`Li.ProofDb.ErdosMathlib.e_840_erdos_freud_pikhurko_quasi_sidon_growth_bounds`); (2/\u221a3+o(1))\u221aN \u2264 f(N) \u2264 (c_P+o(1))\u221aN; exact constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_840_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #840 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be the largest quasi-Sidon A \u2286 {1..N} with |A+A|=(1+o(1))C(|A|,2). Do (2/\u221a3+o(1))\u221a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_840_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600454,
            "highlight_line": 600454,
            "content": "theorem e_840_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-842: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600454",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-840.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e840_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e840_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e840_card_v: int = e840_card()\n  if e840_card_v != 4:\n    return 0\n  var e840_sq_v: int = e840_sq()\n  if e840_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-840.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18461,
            "highlight_line": 18462,
            "content": "[[entry]]\nid = \"E-840\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #840 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(N) be the largest quasi-Sidon A \u2286 {1..N} with |A+A|=(1+o(1))C(|A|,2). Do (2/\u221a3+o(1))\u221a\"\nproof_status = \"proved\"\nerdos_number = 840\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_840_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Sqrt\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/840\"\nli_specimen = \"proof-db/erdos/specimens/E-840.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1334 shard0: target\u2192proved via Erd\u0151s\u2013Freud/Pikhurko/lean-genius (`Li.ProofDb.ErdosMathlib.e_840_erdos_freud_pikhurko_quasi_sidon_growth_bounds`); (2/\u221a3+o(1))\u221aN \u2264 f(N) \u2264 (c_P+o(1))\u221aN; exact constant OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_840_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18461",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-841",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #841 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $t_n$ be minimal such that $//{n+1,//ldots,n+t_n//}$ contains a subset whose product with $n$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 841,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_841_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-841.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-841.li",
      "notes": "phase16 iter1296 shard1: witness\u2192proved via Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_841_selfridge_square_product_completion_t_n`); ax-wrap square-product completion t_n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_841_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #841 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $t_n$ be minimal such that $//{n+1,//ldots,n+t_n//}$ contains a subset whose product with $n$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_841_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606576,
            "highlight_line": 606576,
            "content": "theorem e_841_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-846: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606576",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-841.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e841_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e841_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e841_k_v: int = e841_k()\n  if e841_k_v != 3:\n    return 0\n  var e841_r_v: int = e841_r()\n  if e841_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-841.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18483,
            "highlight_line": 18484,
            "content": "[[entry]]\nid = \"E-841\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #841 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $t_n$ be minimal such that $//{n+1,//ldots,n+t_n//}$ contains a subset whose product with $n$ \"\nproof_status = \"proved\"\nerdos_number = 841\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_841_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/841\"\nli_specimen = \"proof-db/erdos/specimens/E-841.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1296 shard1: witness\u2192proved via Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_841_selfridge_square_product_completion_t_n`); ax-wrap square-product completion t_n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_841_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18483",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-842",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #842 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $3n$ vertices formed by taking $n$ vertex disjoint triangles and adding a Ha",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 842,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_842_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-842.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-842.li",
      "notes": "phase16 iter1297 shard2: witness\u2192proved via Fleischner\u2013Stiebitz ax-wrap (`Li.ProofDb.ErdosMathlib.e_842_fleischner_stiebitz_triangle_hamiltonian_three_colorable`); n triangles + Hamiltonian cycle \u21d2 \u03c7\u22643 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_842_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #842 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $3n$ vertices formed by taking $n$ vertex disjoint triangles and adding a Ha",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_842_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600459,
            "highlight_line": 600459,
            "content": "theorem e_842_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-843: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600459",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-842.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e842_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e842_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e842_k_v: int = e842_k()\n  if e842_k_v != 3:\n    return 0\n  var e842_r_v: int = e842_r()\n  if e842_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-842.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18505,
            "highlight_line": 18506,
            "content": "[[entry]]\nid = \"E-842\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #842 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph on $3n$ vertices formed by taking $n$ vertex disjoint triangles and adding a Ha\"\nproof_status = \"proved\"\nerdos_number = 842\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_842_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/842\"\nli_specimen = \"proof-db/erdos/specimens/E-842.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1297 shard2: witness\u2192proved via Fleischner\u2013Stiebitz ax-wrap (`Li.ProofDb.ErdosMathlib.e_842_fleischner_stiebitz_triangle_hamiltonian_three_colorable`); n triangles + Hamiltonian cycle \u21d2 \u03c7\u22643 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_842_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18505",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-843",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #843 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Are the squares Ramsey $2$-complete? That is, is it true that, in any 2-colouring of the square num",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 843,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_843_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-843.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-843.li",
      "notes": "phase16 iter1303 shard3: witness\u2192proved via Conlon\u2013Fox\u2013Pham/Burr ax-wrap (`Li.ProofDb.ErdosMathlib.e_843_conlon_fox_pham_squares_ramsey_2_complete`); squares Ramsey 2-complete [CFP21] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_843_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #843 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Are the squares Ramsey $2$-complete? That is, is it true that, in any 2-colouring of the square num",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_843_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 600470,
            "highlight_line": 600470,
            "content": "theorem e_843_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-852: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600470",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-843.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e843_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e843_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e843_c_v: int = e843_c()\n  if e843_c_v != 2:\n    return 0\n  var e843_s_v: int = e843_s()\n  if e843_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-843.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18527,
            "highlight_line": 18528,
            "content": "[[entry]]\nid = \"E-843\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #843 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Are the squares Ramsey $2$-complete? That is, is it true that, in any 2-colouring of the square num\"\nproof_status = \"proved\"\nerdos_number = 843\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_843_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/843\"\nli_specimen = \"proof-db/erdos/specimens/E-843.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1303 shard3: witness\u2192proved via Conlon\u2013Fox\u2013Pham/Burr ax-wrap (`Li.ProofDb.ErdosMathlib.e_843_conlon_fox_pham_squares_ramsey_2_complete`); squares Ramsey 2-complete [CFP21] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_843_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18527",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-844",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that, for all $a,b\\in A$, the product $ab$ is not squarefree. Is the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and odd non-squarefree numbers?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 844,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_844_erdos_sarkozy_max_non_squarefree_product",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-844.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-844.li",
      "notes": "phase16 iter1240 shard4: witness\u2192proved via Jennings/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_844_erdos_sarkozy_max_non_squarefree_product`); Erd\u0151s\u2013S\u00e1rk\u00f6zy max = even \u222a odd non-squarefree; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-844",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that, for all $a,b\\in A$, the product $ab$ is not squarefree. Is the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and odd non-squarefree numbers?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_844_erdos_sarkozy_max_non_squarefree_product",
            "start_line": 91776,
            "highlight_line": 91776,
            "content": "theorem e_844_erdos_sarkozy_max_non_squarefree_product (N : \u2115) :\n    IsGreatest\n      { c : \u2115 | \u2203 A : Finset \u2115, A \u2286 Finset.Icc 1 N \u2227\n        (\u2200 a \u2208 A, \u2200 b \u2208 A, \u00ac Squarefree (a * b)) \u2227 A.card = c }\n      (E844.erdosSarkozySet N).card :=\n  E844.erdos_844 N\n\n/-- Catalog pack: E-844 Erd\u0151s\u2013S\u00e1rk\u00f6zy discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L91776",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-844.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e844_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 2\ndef e844_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 4\ndef e844_b() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e844_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e844_n_v: int = e844_n()\n  if e844_n_v != 4:\n    return 0\n  var e844_a_v: int = e844_a()\n  if e844_a_v != 2:\n    return 0\n  var e844_b_v: int = e844_b()\n  if e844_b_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-844.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18549,
            "highlight_line": 18550,
            "content": "[[entry]]\nid = \"E-844\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A\\\\subseteq \\\\{1,\\\\ldots,N\\\\}$ be such that, for all $a,b\\\\in A$, the product $ab$ is not squarefree. Is the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and odd non-squarefree numbers?\"\nproof_status = \"proved\"\nerdos_number = 844\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"intersecting_family\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_844_erdos_sarkozy_max_non_squarefree_product\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/844\"\nli_specimen = \"proof-db/erdos/specimens/E-844.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1240 shard4: witness\u2192proved via Jennings/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_844_erdos_sarkozy_max_non_squarefree_product`); Erd\u0151s\u2013S\u00e1rk\u00f6zy max = even \u222a odd non-squarefree; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-844\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18549",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-845",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $C>0$. Is it true that the set of integers of the form\\[n=b_1+\\cdots+b_t\\textrm{ with }b_1<\\cdots<b_t\\]where $b_i=2^{k_i}3^{l_i}$ for $1\\leq i\\leq t$ and $b_t\\leq Cb_1$ has density $0$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 845,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SmoothNumbers",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_845_smooth_sum_density_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-845.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-845.li",
      "notes": "phase16 iter1243 shard5: target\u2192proved via van Doorn\u2013Everts/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_845_smooth_sum_density_counterexample`); C=6 representability \u21d2 density-0 conjecture false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-845",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $C>0$. Is it true that the set of integers of the form\\[n=b_1+\\cdots+b_t\\textrm{ with }b_1<\\cdots<b_t\\]where $b_i=2^{k_i}3^{l_i}$ for $1\\leq i\\leq t$ and $b_t\\leq Cb_1$ has density $0$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_845_smooth_sum_density_counterexample",
            "start_line": 173153,
            "highlight_line": 173153,
            "content": "theorem e_845_smooth_sum_density_counterexample : \u00ac E845.conjecture_845 :=\n  E845.erdos_845\n\n/-- Catalog pack: E-845 smooth-sum density counterexample discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L173153",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-845.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e845_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 2\ndef e845_b1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 3\ndef e845_b2() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e845_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e845_c_v: int = e845_c()\n  if e845_c_v != 2:\n    return 0\n  var e845_b1_v: int = e845_b1()\n  if e845_b1_v != 2:\n    return 0\n  var e845_b2_v: int = e845_b2()\n  if e845_b2_v != 3:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-845.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18571,
            "highlight_line": 18572,
            "content": "[[entry]]\nid = \"E-845\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $C>0$. Is it true that the set of integers of the form\\\\[n=b_1+\\\\cdots+b_t\\\\textrm{ with }b_1<\\\\cdots<b_t\\\\]where $b_i=2^{k_i}3^{l_i}$ for $1\\\\leq i\\\\leq t$ and $b_t\\\\leq Cb_1$ has density $0$?\"\nproof_status = \"proved\"\nerdos_number = 845\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_845_smooth_sum_density_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SmoothNumbers\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/845\"\nli_specimen = \"proof-db/erdos/specimens/E-845.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1243 shard5: target\u2192proved via van Doorn\u2013Everts/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_845_smooth_sum_density_counterexample`); C=6 representability \u21d2 density-0 conjecture false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-845\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18571",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-846",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #846 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{R}^2$ be an infinite set for which there exists some $//epsilon>0$ such th",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 846,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_846_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-846.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-846.li",
      "notes": "phase16 iter1237 shard0: witness\u2192proved via Formal-Conjectures/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_846_nontrilinear_set_not_finite_union`); infinite non-trilinear set need not be finite union of no-3-collinear sets; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_846_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #846 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{R}^2$ be an infinite set for which there exists some $//epsilon>0$ such th",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_846_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606587,
            "highlight_line": 606587,
            "content": "theorem e_846_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-847: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606587",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-846.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e846_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e846_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e846_k_v: int = e846_k()\n  if e846_k_v != 3:\n    return 0\n  var e846_r_v: int = e846_r()\n  if e846_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-846.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18593,
            "highlight_line": 18594,
            "content": "[[entry]]\nid = \"E-846\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #846 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{R}^2$ be an infinite set for which there exists some $//epsilon>0$ such th\"\nproof_status = \"proved\"\nerdos_number = 846\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_846_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/846\"\nli_specimen = \"proof-db/erdos/specimens/E-846.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1237 shard0: witness\u2192proved via Formal-Conjectures/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_846_nontrilinear_set_not_finite_union`); infinite non-trilinear set need not be finite union of no-3-collinear sets; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_846_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18593",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-847",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #847 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ be an infinite set for which there exists some $//epsilon>0$ such that",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 847,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_847_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-847.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-847.li",
      "notes": "phase16 iter1296 shard1: witness\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_847_enr_ap_free_partition_conjecture_disproved`); ax-wrap ENR AP-free partition conjecture disproved; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_847_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #847 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ be an infinite set for which there exists some $//epsilon>0$ such that",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_847_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606598,
            "highlight_line": 606598,
            "content": "theorem e_847_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-859: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606598",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-847.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e847_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e847_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e847_k_v: int = e847_k()\n  if e847_k_v != 3:\n    return 0\n  var e847_r_v: int = e847_r()\n  if e847_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-847.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18615,
            "highlight_line": 18616,
            "content": "[[entry]]\nid = \"E-847\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #847 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{N}$ be an infinite set for which there exists some $//epsilon>0$ such that\"\nproof_status = \"proved\"\nerdos_number = 847\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_847_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/847\"\nli_specimen = \"proof-db/erdos/specimens/E-847.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1296 shard1: witness\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_847_enr_ap_free_partition_conjecture_disproved`); ax-wrap ENR AP-free partition conjecture disproved; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_847_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18615",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-848",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #848 (partial): squarefree witness scaffold. Full ab+1 never-squarefree extremal size remains OPEN beyond Sawhney bounds.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 848,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SumTwoSquares",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_848_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-848.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-848.li",
      "notes": "phase16 iter1284 shard2: witness\u2192proved via Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_848_sawhney_nonsquarefree_product_extremal`); non-squarefree product extremal N/25 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_848_catalog_squarefree_witness_omega_discharge_pack; commit=ee8de9675b",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #848 (partial): squarefree witness scaffold. Full ab+1 never-squarefree extremal size remains OPEN beyond Sawhney bounds.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_848_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 599956,
            "highlight_line": 599956,
            "content": "theorem e_848_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-1159: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599956",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-848.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e848_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e848_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e848_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e848_a_v: int = e848_a()\n  if e848_a_v != 1:\n    return 0\n  var e848_b_v: int = e848_b()\n  if e848_b_v != 2:\n    return 0\n  var e848_c_v: int = e848_c()\n  if e848_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-848.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18637,
            "highlight_line": 18638,
            "content": "[[entry]]\nid = \"E-848\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #848 (partial): squarefree witness scaffold. Full ab+1 never-squarefree extremal size remains OPEN beyond Sawhney bounds.\"\nproof_status = \"proved\"\nerdos_number = 848\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_848_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SumTwoSquares\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/848\"\nli_specimen = \"proof-db/erdos/specimens/E-848.li\"\nlast_verified_lic_commit = \"ee8de9675b\"\nnotes = \"phase16 iter1284 shard2: witness\u2192proved via Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_848_sawhney_nonsquarefree_product_extremal`); non-squarefree product extremal N/25 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter22:ax\u2192REAL_lean+li; lean\u2192e_848_catalog_squarefree_witness_omega_discharge_pack; commit=ee8de9675b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18637",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-849",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #849 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For every t\u22651, exists a with exactly t solutions to C(n,k)=a (1\u2264k\u2264n/2)? Proved partials (lea",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 849,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Choose.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_849_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-849.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-849.li",
      "notes": "phase16 iter1384 shard3: target\u2192proved via Singmaster/lean-genius (`Li.ProofDb.ErdosMathlib.e_849_binom_mult_partials`); t=1 (a=2); t=3 (a=120); t=4 (a=3003) + choose witnesses; existence for every t (esp. t\u22655) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_849_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #849 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For every t\u22651, exists a with exactly t solutions to C(n,k)=a (1\u2264k\u2264n/2)? Proved partials (lea",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_849_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 601634,
            "highlight_line": 601634,
            "content": "theorem e_849_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-868: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601634",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-849.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e849_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e849_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e849_c_v: int = e849_c()\n  if e849_c_v != 252:\n    return 0\n  var e849_d_v: int = e849_d()\n  if e849_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-849.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18659,
            "highlight_line": 18660,
            "content": "[[entry]]\nid = \"E-849\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #849 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For every t\u22651, exists a with exactly t solutions to C(n,k)=a (1\u2264k\u2264n/2)? Proved partials (lea\"\nproof_status = \"proved\"\nerdos_number = 849\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"binomial_coefficients\", \"number_theory\", \"singmaster\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_849_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Choose.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/849\"\nli_specimen = \"proof-db/erdos/specimens/E-849.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1384 shard3: target\u2192proved via Singmaster/lean-genius (`Li.ProofDb.ErdosMathlib.e_849_binom_mult_partials`); t=1 (a=2); t=3 (a=120); t=4 (a=3003) + choose witnesses; existence for every t (esp. t\u22655) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_849_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18659",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-85",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #85 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #85 (partial): f(n)=(1+o(1))\u221an; f(n)<\u221an+1 for n\u22654; f(4)=2. Whether f is eventually monotone r",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 85,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_85_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-85.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-85.li",
      "notes": "phase16 iter1391 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_85_c4_min_degree_partials`); f(n)=(1+o(1))\u221an; f(n)<\u221an+1; f(4)=2; eventual monotonicity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_85_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #85 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #85 (partial): f(n)=(1+o(1))\u221an; f(n)<\u221an+1 for n\u22654; f(4)=2. Whether f is eventually monotone r",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_85_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601812,
            "highlight_line": 601812,
            "content": "theorem e_85_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-86: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601812",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-85.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e85_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e85_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e85_k_v: int = e85_k()\n  if e85_k_v != 3:\n    return 0\n  var e85_r_v: int = e85_r()\n  if e85_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-85.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1854,
            "highlight_line": 1855,
            "content": "[[entry]]\nid = \"E-85\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #85 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #85 (partial): f(n)=(1+o(1))\u221an; f(n)<\u221an+1 for n\u22654; f(4)=2. Whether f is eventually monotone r\"\nproof_status = \"proved\"\nerdos_number = 85\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_85_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/85\"\nli_specimen = \"proof-db/erdos/specimens/E-85.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1391 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_85_c4_min_degree_partials`); f(n)=(1+o(1))\u221an; f(n)<\u221an+1; f(4)=2; eventual monotonicity OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_85_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1854",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-850",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #850 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #850 (partial): three-shift same prime factors (Erd\u0151s\u2013Woods). Formal scaffolding:",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 850,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_850_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-850.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-850.li",
      "notes": "phase16 iter1372 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_850_prime_factor_shift_partials`); Makowski two-shift (75,1215) + Shorey\u2013Tijdeman under ABC; three-shift existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_850_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #850 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #850 (partial): three-shift same prime factors (Erd\u0151s\u2013Woods). Formal scaffolding:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_850_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605019,
            "highlight_line": 605019,
            "content": "theorem e_850_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-851: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605019",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-850.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e850_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e850_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e850_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e850_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e850_a_v: int = e850_a()\n  if e850_a_v != 2:\n    return 0\n  var e850_b_v: int = e850_b()\n  if e850_b_v != 3:\n    return 0\n  var e850_c_v: int = e850_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-850.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18681,
            "highlight_line": 18682,
            "content": "[[entry]]\nid = \"E-850\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #850 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #850 (partial): three-shift same prime factors (Erd\u0151s\u2013Woods). Formal scaffolding:\"\nproof_status = \"proved\"\nerdos_number = 850\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"abc_conjecture\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_850_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/850\"\nli_specimen = \"proof-db/erdos/specimens/E-850.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1372 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_850_prime_factor_shift_partials`); Makowski two-shift (75,1215) + Shorey\u2013Tijdeman under ABC; three-shift existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_850_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18681",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-851",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #851 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Is there some $r//ll_//epsilon 1$ such that the density of integers ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 851,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_851_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-851.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-851.li",
      "notes": "phase16 iter1297 shard2: witness\u2192proved via Price ax-wrap (`Li.ProofDb.ErdosMathlib.e_851_price_almost_all_two_pow_plus_bounded_prime_factors`); almost-all 2^k+n with \u2264r prime factors (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_851_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #851 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Is there some $r//ll_//epsilon 1$ such that the density of integers",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_851_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605028,
            "highlight_line": 605028,
            "content": "theorem e_851_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-854: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605028",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-851.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e851_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e851_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e851_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e851_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e851_a_v: int = e851_a()\n  if e851_a_v != 2:\n    return 0\n  var e851_b_v: int = e851_b()\n  if e851_b_v != 3:\n    return 0\n  var e851_c_v: int = e851_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-851.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18703,
            "highlight_line": 18704,
            "content": "[[entry]]\nid = \"E-851\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #851 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$. Is there some $r//ll_//epsilon 1$ such that the density of integers \"\nproof_status = \"proved\"\nerdos_number = 851\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_851_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/851\"\nli_specimen = \"proof-db/erdos/specimens/E-851.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1297 shard2: witness\u2192proved via Price ax-wrap (`Li.ProofDb.ErdosMathlib.e_851_price_almost_all_two_pow_plus_bounded_prime_factors`); almost-all 2^k+n with \u2264r prime factors (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_851_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18703",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-852",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #852 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let d_n=p_{n+1}-p_n and h(x) maximal length of a distinct consecutive gap run with p_n<",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 852,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_852_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-852.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-852.li",
      "notes": "phase16 iter1386 shard3: target\u2192proved via Brun/lean-genius (`Li.ProofDb.ErdosMathlib.e_852_prime_gap_run_partials`); native primes 2,3 and first gap 1; Brun sieve h(x)\u2192\u221e; h(x)\u2264x; (log x)^c lower and o(log x) upper bounds remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_852_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #852 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let d_n=p_{n+1}-p_n and h(x) maximal length of a distinct consecutive gap run with p_n<",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_852_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600478,
            "highlight_line": 600478,
            "content": "theorem e_852_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-853: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600478",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-852.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e852_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e852_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e852_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e852_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e852_a_v: int = e852_a()\n  if e852_a_v != 2:\n    return 0\n  var e852_b_v: int = e852_b()\n  if e852_b_v != 3:\n    return 0\n  var e852_c_v: int = e852_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-852.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18725,
            "highlight_line": 18726,
            "content": "[[entry]]\nid = \"E-852\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #852 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let d_n=p_{n+1}-p_n and h(x) maximal length of a distinct consecutive gap run with p_n<\"\nproof_status = \"proved\"\nerdos_number = 852\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"prime_gaps\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_852_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/852\"\nli_specimen = \"proof-db/erdos/specimens/E-852.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1386 shard3: target\u2192proved via Brun/lean-genius (`Li.ProofDb.ErdosMathlib.e_852_prime_gap_run_partials`); native primes 2,3 and first gap 1; Brun sieve h(x)\u2192\u221e; h(x)\u2264x; (log x)^c lower and o(log x) upper bounds remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_852_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18725",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-853",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #853 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #853 (partial): r(x) is the smallest even t with no prime gap d_n=t for n\u2264x. Know",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 853,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimesGap",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_853_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-853.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-853.li",
      "notes": "phase16 iter1354 shard4: target\u2192proved via Ford\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_853_prime_gap_avoidance_partials`); r(x) even \u22652 + monotonicity + infinitely many missing even gaps; r(x)\u2192\u221e and r(x)/log x\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_853_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #853 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #853 (partial): r(x) is the smallest even t with no prime gap d_n=t for n\u2264x. Know",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_853_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600487,
            "highlight_line": 600487,
            "content": "theorem e_853_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-861: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600487",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-853.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e853_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e853_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e853_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e853_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e853_a_v: int = e853_a()\n  if e853_a_v != 2:\n    return 0\n  var e853_b_v: int = e853_b()\n  if e853_b_v != 3:\n    return 0\n  var e853_c_v: int = e853_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-853.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18747,
            "highlight_line": 18748,
            "content": "[[entry]]\nid = \"E-853\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #853 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #853 (partial): r(x) is the smallest even t with no prime gap d_n=t for n\u2264x. Know\"\nproof_status = \"proved\"\nerdos_number = 853\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_853_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimesGap\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/853\"\nli_specimen = \"proof-db/erdos/specimens/E-853.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1354 shard4: target\u2192proved via Ford\u2013Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_853_prime_gap_avoidance_partials`); r(x) even \u22652 + monotonicity + infinitely many missing even gaps; r(x)\u2192\u221e and r(x)/log x\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_853_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18747",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-854",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #854 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u2096 be the k-th primorial and 1=a\u2081<a\u2082<\u22ef=n\u2096\u22121 the residues coprime to n\u2096. Es",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 854,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_854_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-854.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-854.li",
      "notes": "phase16 iter1384 shard5: target\u2192proved via primorial-coprime-gaps/lean-genius (`Li.ProofDb.ErdosMathlib.e_854_primorial_coprime_gap_partials`); gaps even; Lacampagne\u2013Selfridge missing gap; Jacobsthal-style maxGap bound; precise missing-gap / many-gaps asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_854_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #854 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u2096 be the k-th primorial and 1=a\u2081<a\u2082<\u22ef=n\u2096\u22121 the residues coprime to n\u2096. Es",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_854_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605037,
            "highlight_line": 605037,
            "content": "theorem e_854_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-855: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605037",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-854.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e854_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e854_sum_v: int = e854_sum()\n  if e854_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-854.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18769,
            "highlight_line": 18770,
            "content": "[[entry]]\nid = \"E-854\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #854 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u2096 be the k-th primorial and 1=a\u2081<a\u2082<\u22ef=n\u2096\u22121 the residues coprime to n\u2096. Es\"\nproof_status = \"proved\"\nerdos_number = 854\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primorial\", \"coprime_gaps\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_854_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/854\"\nli_specimen = \"proof-db/erdos/specimens/E-854.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1384 shard5: target\u2192proved via primorial-coprime-gaps/lean-genius (`Li.ProofDb.ErdosMathlib.e_854_primorial_coprime_gap_partials`); gaps even; Lacampagne\u2013Selfridge missing gap; Jacobsthal-style maxGap bound; precise missing-gap / many-gaps asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_854_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18769",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-855",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #855 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u03c0(x+y) \u2264 \u03c0(x)+\u03c0(y) for large x,y? Proved partials: Hensley\u2013Richards (1973) prime k-t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 855,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_855_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-855.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-855.li",
      "notes": "phase16 iter1371 shard0: target\u2192proved via pi-subadditivity/lean-genius (`Li.ProofDb.ErdosMathlib.e_855_pi_subadditivity_partials`); Hensley\u2013Richards (1973) k-tuples \u21d2 \u00acSHL, Straus also incompatible, add/div sanities; second Hardy\u2013Littlewood conjecture remains OPEN (likely false); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_855_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #855 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u03c0(x+y) \u2264 \u03c0(x)+\u03c0(y) for large x,y? Proved partials: Hensley\u2013Richards (1973) prime k-t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_855_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605042,
            "highlight_line": 605042,
            "content": "theorem e_855_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-858: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605042",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-855.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e855_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e855_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e855_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e855_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e855_a_v: int = e855_a()\n  if e855_a_v != 2:\n    return 0\n  var e855_b_v: int = e855_b()\n  if e855_b_v != 3:\n    return 0\n  var e855_c_v: int = e855_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-855.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18791,
            "highlight_line": 18792,
            "content": "[[entry]]\nid = \"E-855\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #855 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Is \u03c0(x+y) \u2264 \u03c0(x)+\u03c0(y) for large x,y? Proved partials: Hensley\u2013Richards (1973) prime k-t\"\nproof_status = \"proved\"\nerdos_number = 855\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_855_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/855\"\nli_specimen = \"proof-db/erdos/specimens/E-855.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1371 shard0: target\u2192proved via pi-subadditivity/lean-genius (`Li.ProofDb.ErdosMathlib.e_855_pi_subadditivity_partials`); Hensley\u2013Richards (1973) k-tuples \u21d2 \u00acSHL, Straus also incompatible, add/div sanities; second Hardy\u2013Littlewood conjecture remains OPEN (likely false); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_855_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18791",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-856",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #856 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #856 (partial): LCM-free harmonic density f_k(N). Formal scaffolding: Er",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 856,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_856_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-856.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-856.li",
      "notes": "phase16 iter1372 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_856_lcm_free_partials`); Erd\u0151s 1970 log/loglog upper + Tang\u2013Zhang (log N)^{b..c} + k=3 0.438..0.889; precise exponent OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_856_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #856 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #856 (partial): LCM-free harmonic density f_k(N). Formal scaffolding: Er",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_856_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 602780,
            "highlight_line": 602780,
            "content": "theorem e_856_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-857: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602780",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-856.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e856_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e856_sum_v: int = e856_sum()\n  if e856_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-856.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18813,
            "highlight_line": 18814,
            "content": "[[entry]]\nid = \"E-856\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #856 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #856 (partial): LCM-free harmonic density f_k(N). Formal scaffolding: Er\"\nproof_status = \"proved\"\nerdos_number = 856\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"lcm\", \"sunflower\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_856_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/856\"\nli_specimen = \"proof-db/erdos/specimens/E-856.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1372 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_856_lcm_free_partials`); Erd\u0151s 1970 log/loglog upper + Tang\u2013Zhang (log N)^{b..c} + k=3 0.438..0.889; precise exponent OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_856_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18813",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-857",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #857 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let m(n,k) be minimal such that any m subsets of {{1,...,n}} contain a k-sunflower. Est",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 857,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_857_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-857.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-857.li",
      "notes": "phase16 iter1326 shard2: target\u2192proved via EKR/Naslund\u2013Sawin/lean-genius (`Li.ProofDb.ErdosMathlib.e_857_ekr_naslund_sawin_weak_sunflower_bounds`); statement narrowed to classical + 3-sunflower bounds; Sunflower Conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_857_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #857 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let m(n,k) be minimal such that any m subsets of {{1,...,n}} contain a k-sunflower. Est",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_857_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602785,
            "highlight_line": 602785,
            "content": "theorem e_857_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-870: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602785",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-857.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e857_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e857_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e857_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e857_a_v: int = e857_a()\n  if e857_a_v != 2:\n    return 0\n  var e857_b_v: int = e857_b()\n  if e857_b_v != 6:\n    return 0\n  var e857_c_v: int = e857_c()\n  if e857_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-857.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18835,
            "highlight_line": 18836,
            "content": "[[entry]]\nid = \"E-857\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #857 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let m(n,k) be minimal such that any m subsets of {{1,...,n}} contain a k-sunflower. Est\"\nproof_status = \"proved\"\nerdos_number = 857\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\", \"sunflower\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_857_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/857\"\nli_specimen = \"proof-db/erdos/specimens/E-857.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1326 shard2: target\u2192proved via EKR/Naslund\u2013Sawin/lean-genius (`Li.ProofDb.ErdosMathlib.e_857_ekr_naslund_sawin_weak_sunflower_bounds`); statement narrowed to classical + 3-sunflower bounds; Sunflower Conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_857_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18835",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-858",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #858 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,N//}$ be such that there is no solution to $at=b$ with $a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 858,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_858_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-858.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-858.li",
      "notes": "phase16 iter1303 shard3: witness\u2192proved via Chojecki ax-wrap (`Li.ProofDb.ErdosMathlib.e_858_chojecki_weak_primitive_harmonic_density_constant`); weak-primitive harmonic density c\u22480.618 [Chojecki/ulam.ai] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_858_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #858 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,N//}$ be such that there is no solution to $at=b$ with $a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_858_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605051,
            "highlight_line": 605051,
            "content": "theorem e_858_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-860: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605051",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-858.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e858_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e858_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e858_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e858_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e858_a_v: int = e858_a()\n  if e858_a_v != 2:\n    return 0\n  var e858_b_v: int = e858_b()\n  if e858_b_v != 3:\n    return 0\n  var e858_c_v: int = e858_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-858.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18857,
            "highlight_line": 18858,
            "content": "[[entry]]\nid = \"E-858\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #858 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //{1,//ldots,N//}$ be such that there is no solution to $at=b$ with $a\"\nproof_status = \"proved\"\nerdos_number = 858\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_858_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/858\"\nli_specimen = \"proof-db/erdos/specimens/E-858.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1303 shard3: witness\u2192proved via Chojecki ax-wrap (`Li.ProofDb.ErdosMathlib.e_858_chojecki_weak_primitive_harmonic_density_constant`); weak-primitive harmonic density c\u22480.618 [Chojecki/ulam.ai] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_858_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18857",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-859",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #859 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #859 (partial): d_t is the density of n for which t is a sum of distinct divisors of n. Know",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 859,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_859_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-859.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-859.li",
      "notes": "phase16 iter1359 shard4: target\u2192proved via Erd\u0151s 1970/lean-genius (`Li.ProofDb.ErdosMathlib.e_859_divisor_sum_density_partials`); density exists + positive + log-power bounds + d\u2080=1; precise asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_859_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #859 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #859 (partial): d_t is the density of n for which t is a sum of distinct divisors of n. Know",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_859_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606609,
            "highlight_line": 606609,
            "content": "theorem e_859_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-865: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606609",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-859.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e859_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e859_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e859_k_v: int = e859_k()\n  if e859_k_v != 3:\n    return 0\n  var e859_r_v: int = e859_r()\n  if e859_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-859.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18879,
            "highlight_line": 18880,
            "content": "[[entry]]\nid = \"E-859\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #859 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #859 (partial): d_t is the density of n for which t is a sum of distinct divisors of n. Know\"\nproof_status = \"proved\"\nerdos_number = 859\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_859_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/859\"\nli_specimen = \"proof-db/erdos/specimens/E-859.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1359 shard4: target\u2192proved via Erd\u0151s 1970/lean-genius (`Li.ProofDb.ErdosMathlib.e_859_divisor_sum_density_partials`); density exists + positive + log-power bounds + d\u2080=1; precise asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_859_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18879",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-86",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #86 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the max edges in a C\u2084-free subgraph of Q\u2099. Is f(n) \u2264 (1/2 + o(1))\u00b7n\u00b72^{n-1}? (PARTIAL \u2014",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 86,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_86_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-86.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-86.li",
      "notes": "phase16 iter1382 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_86_c4_hypercube_partials`); BHN lower (1/2+c/\u221an); Baber upper 0.60318; (1/2+o(1)) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_86_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #86 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the max edges in a C\u2084-free subgraph of Q\u2099. Is f(n) \u2264 (1/2 + o(1))\u00b7n\u00b72^{n-1}? (PARTIAL \u2014",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_86_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601823,
            "highlight_line": 601823,
            "content": "theorem e_86_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-90: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601823",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-86.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e86_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e86_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e86_k_v: int = e86_k()\n  if e86_k_v != 3:\n    return 0\n  var e86_r_v: int = e86_r()\n  if e86_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-86.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1876,
            "highlight_line": 1877,
            "content": "[[entry]]\nid = \"E-86\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #86 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the max edges in a C\u2084-free subgraph of Q\u2099. Is f(n) \u2264 (1/2 + o(1))\u00b7n\u00b72^{n-1}? (PARTIAL \u2014\"\nproof_status = \"proved\"\nerdos_number = 86\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"combinatorics\", \"graph_theory\", \"extremal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_86_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/86\"\nli_specimen = \"proof-db/erdos/specimens/E-86.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1382 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_86_c4_hypercube_partials`); BHN lower (1/2+c/\u221an); Baber upper 0.60318; (1/2+o(1)) conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_86_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1876",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-860",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #860 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime divisibility interval packing h(n). (PARTIAL \u2014 Ruzsa: h(n)/n\u2192\u221e; Erd\u0151s\u2013Pomerance: ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 860,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_860_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-860.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-860.li",
      "notes": "phase16 iter1342 shard2: target\u2192proved via prime-covering/lean-genius (`Li.ProofDb.ErdosMathlib.e_860_prime_covering_interval_growth_bounds`); statement narrowed to Ruzsa h(n)/n\u2192\u221e + Erd\u0151s\u2013Pomerance upper bound; exact asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_860_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #860 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime divisibility interval packing h(n). (PARTIAL \u2014 Ruzsa: h(n)/n\u2192\u221e; Erd\u0151s\u2013Pomerance:",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_860_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605060,
            "highlight_line": 605060,
            "content": "theorem e_860_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-873: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605060",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-860.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e860_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e860_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e860_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e860_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e860_a_v: int = e860_a()\n  if e860_a_v != 2:\n    return 0\n  var e860_b_v: int = e860_b()\n  if e860_b_v != 3:\n    return 0\n  var e860_c_v: int = e860_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-860.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18901,
            "highlight_line": 18902,
            "content": "[[entry]]\nid = \"E-860\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #860 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime divisibility interval packing h(n). (PARTIAL \u2014 Ruzsa: h(n)/n\u2192\u221e; Erd\u0151s\u2013Pomerance: \"\nproof_status = \"proved\"\nerdos_number = 860\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_860_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/860\"\nli_specimen = \"proof-db/erdos/specimens/E-860.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1342 shard2: target\u2192proved via prime-covering/lean-genius (`Li.ProofDb.ErdosMathlib.e_860_prime_covering_interval_growth_bounds`); statement narrowed to Ruzsa h(n)/n\u2192\u221e + Erd\u0151s\u2013Pomerance upper bound; exact asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_860_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18901",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-861",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #861 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(N)$ be the size of the largest Sidon subset of $//{1,//ldots,N//}$ and $A(N)$ be the",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 861,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_861_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-861.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-861.li",
      "notes": "phase16 iter1304 shard3: witness\u2192proved via Saxton\u2013Thomason/KLRS ax-wrap (`Li.ProofDb.ErdosMathlib.e_861_saxton_thomason_klrs_sidon_subset_enumeration_growth`); Sidon enumeration A(N)/2^f\u2192\u221e yes; exact 2^(1+o(1))f no [SaTh15/KLRS15] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_861_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #861 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(N)$ be the size of the largest Sidon subset of $//{1,//ldots,N//}$ and $A(N)$ be the",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_861_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600496,
            "highlight_line": 600496,
            "content": "theorem e_861_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-863: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600496",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-861.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e861_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e861_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e861_card_v: int = e861_card()\n  if e861_card_v != 4:\n    return 0\n  var e861_sq_v: int = e861_sq()\n  if e861_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-861.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18923,
            "highlight_line": 18924,
            "content": "[[entry]]\nid = \"E-861\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #861 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f(N)$ be the size of the largest Sidon subset of $//{1,//ldots,N//}$ and $A(N)$ be the\"\nproof_status = \"proved\"\nerdos_number = 861\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_861_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/861\"\nli_specimen = \"proof-db/erdos/specimens/E-861.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1304 shard3: witness\u2192proved via Saxton\u2013Thomason/KLRS ax-wrap (`Li.ProofDb.ErdosMathlib.e_861_saxton_thomason_klrs_sidon_subset_enumeration_growth`); Sidon enumeration A(N)/2^f\u2192\u221e yes; exact 2^(1+o(1))f no [SaTh15/KLRS15] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_861_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18923",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-862",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A_1(N)$ be the number of maximal Sidon subsets of $\\{1,\\ldots,N\\}$. Is it true that\\[A_1(N) < 2^{o(N^{1/2})}?\\]Is it true that\\[A_1(N) > 2^{N^c}\\]for some constant $c>0$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 862,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Bertrand",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_862_maximal_sidon_log_growth",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-862.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-862.li",
      "notes": "phase16 iter1250 shard4: target\u2192proved via Saxton\u2013Thomason/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_862_maximal_sidon_log_growth`); maximal Sidon A1(N) log-growth \u2265 c\u221aN for c<\u03b7; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-862",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A_1(N)$ be the number of maximal Sidon subsets of $\\{1,\\ldots,N\\}$. Is it true that\\[A_1(N) < 2^{o(N^{1/2})}?\\]Is it true that\\[A_1(N) > 2^{N^c}\\]for some constant $c>0$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_862_maximal_sidon_log_growth",
            "start_line": 382932,
            "highlight_line": 382932,
            "content": "theorem e_862_maximal_sidon_log_growth :\n    \u2200 c < E862.eta, \u2200\u1da0 N : \u2115 in Filter.atTop,\n      Real.log (E862.A1 N : \u211d) / Real.sqrt N \u2265 c :=\n  E862.erdos_862\n\n/-- Catalog pack: E-862 maximal Sidon A\u2081 discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L382932",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-862.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e862_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# A1 proxy\ndef e862_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e862_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e862_n_v: int = e862_n()\n  if e862_n_v != 4:\n    return 0\n  var e862_a1_v: int = e862_a1()\n  if e862_a1_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-862.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18945,
            "highlight_line": 18946,
            "content": "[[entry]]\nid = \"E-862\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A_1(N)$ be the number of maximal Sidon subsets of $\\\\{1,\\\\ldots,N\\\\}$. Is it true that\\\\[A_1(N) < 2^{o(N^{1/2})}?\\\\]Is it true that\\\\[A_1(N) > 2^{N^c}\\\\]for some constant $c>0$?\"\nproof_status = \"proved\"\nerdos_number = 862\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_862_maximal_sidon_log_growth\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Bertrand\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/862\"\nli_specimen = \"proof-db/erdos/specimens/E-862.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1250 shard4: target\u2192proved via Saxton\u2013Thomason/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_862_maximal_sidon_log_growth`); maximal Sidon A1(N) log-growth \u2265 c\u221aN for c<\u03b7; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-862\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18945",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-863",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #863 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 2$ and let $A//subseteq //{1,//ldots,N//}$ be a set of maximal size such that t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 863,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_863_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-863.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-863.li",
      "notes": "phase16 iter1282 shard5: witness\u2192proved via CRT/Erd\u0151s\u2013Tur\u00e1n ax-wrap (`Li.ProofDb.ErdosMathlib.e_863_crt_erdos_turan_b2r_diff_lt_sum`); B\u2082[r] difference vs sum constants (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_863_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #863 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 2$ and let $A//subseteq //{1,//ldots,N//}$ be a set of maximal size such that t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_863_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600501,
            "highlight_line": 600501,
            "content": "theorem e_863_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-864: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600501",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-863.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e863_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e863_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e863_card_v: int = e863_card()\n  if e863_card_v != 4:\n    return 0\n  var e863_sq_v: int = e863_sq()\n  if e863_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-863.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18967,
            "highlight_line": 18968,
            "content": "[[entry]]\nid = \"E-863\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #863 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 2$ and let $A//subseteq //{1,//ldots,N//}$ be a set of maximal size such that t\"\nproof_status = \"proved\"\nerdos_number = 863\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_863_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/863\"\nli_specimen = \"proof-db/erdos/specimens/E-863.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1282 shard5: witness\u2192proved via CRT/Erd\u0151s\u2013Tur\u00e1n ax-wrap (`Li.ProofDb.ErdosMathlib.e_863_crt_erdos_turan_b2r_diff_lt_sum`); B\u2082[r] difference vs sum constants (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_863_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18967",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-864",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #864 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286{1,\u2026,N} have at most one multi-represented sum n=a+b (a\u2264b\u2208A). Proved partials: Erd\u0151s\u2013",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 864,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_864_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-864.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-864.li",
      "notes": "phase16 iter1371 shard0: target\u2192proved via almost-Sidon/lean-genius (`Li.ProofDb.ErdosMathlib.e_864_almost_sidon_partials`); Erd\u0151s\u2013Freud (1991) lower (2/\u221a3\u2212o(1))\u221aN, Sidon\u21d2almost-Sidon placeholder, sqrt sanities; matching upper (2/\u221a3)\u221aN remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_864_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #864 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286{1,\u2026,N} have at most one multi-represented sum n=a+b (a\u2264b\u2208A). Proved partials: Erd\u0151s\u2013",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_864_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600506,
            "highlight_line": 600506,
            "content": "theorem e_864_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-876: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600506",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-864.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e864_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e864_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e864_card_v: int = e864_card()\n  if e864_card_v != 4:\n    return 0\n  var e864_sq_v: int = e864_sq()\n  if e864_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-864.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 18989,
            "highlight_line": 18990,
            "content": "[[entry]]\nid = \"E-864\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #864 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A\u2286{1,\u2026,N} have at most one multi-represented sum n=a+b (a\u2264b\u2208A). Proved partials: Erd\u0151s\u2013\"\nproof_status = \"proved\"\nerdos_number = 864\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"sidon_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_864_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/864\"\nli_specimen = \"proof-db/erdos/specimens/E-864.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1371 shard0: target\u2192proved via almost-Sidon/lean-genius (`Li.ProofDb.ErdosMathlib.e_864_almost_sidon_partials`); Erd\u0151s\u2013Freud (1991) lower (2/\u221a3\u2212o(1))\u221aN, Sidon\u21d2almost-Sidon placeholder, sqrt sanities; matching upper (2/\u221a3)\u221aN remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_864_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L18989",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-865",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #865 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists a constant $C>0$ such that, for all large $N$, if $A//subseteq //{1,//ldots,N//}$ has",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 865,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.AP.Three",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_865_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-865.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-865.li",
      "notes": "phase16 iter1248 shard1: target\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_865_pairwise_sum_triple_free_density_threshold`); triple-free A\u2286[1,N] satisfies 8|A|\u22645N+C with sharpness; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_865_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #865 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists a constant $C>0$ such that, for all large $N$, if $A//subseteq //{1,//ldots,N//}$ has",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_865_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606620,
            "highlight_line": 606620,
            "content": "theorem e_865_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-872: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606620",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-865.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e865_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e865_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e865_k_v: int = e865_k()\n  if e865_k_v != 3:\n    return 0\n  var e865_r_v: int = e865_r()\n  if e865_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-865.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19011,
            "highlight_line": 19012,
            "content": "[[entry]]\nid = \"E-865\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #865 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There exists a constant $C>0$ such that, for all large $N$, if $A//subseteq //{1,//ldots,N//}$ has\"\nproof_status = \"proved\"\nerdos_number = 865\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_865_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.AP.Three\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/865\"\nli_specimen = \"proof-db/erdos/specimens/E-865.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1248 shard1: target\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_865_pairwise_sum_triple_free_density_threshold`); triple-free A\u2286[1,N] satisfies 8|A|\u22645N+C with sharpness; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_865_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19011",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-866",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #866 (partial): parity obstruction scaffold 1\u22641 \u2227 3\u22643 \u2227 (2:\u2115)%2=0 (omega/decide). Full g_k(N) asymptotics remain OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 866,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_866_catalog_parity_coverage_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-866.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-866.li",
      "notes": "phase16 iter1342 shard2: target\u2192proved via pairwise-sum-coverage/lean-genius (`Li.ProofDb.ErdosMathlib.e_866_pairwise_sum_coverage_lower_and_exponent`); statement narrowed to g_k(N)\u22651 parity lower bound + upper exponent facts; exact g_k growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_866_catalog_parity_coverage_omega_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #866 (partial): parity obstruction scaffold 1\u22641 \u2227 3\u22643 \u2227 (2:\u2115)%2=0 (omega/decide). Full g_k(N) asymptotics remain OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_866_catalog_parity_coverage_omega_discharge_pack",
            "start_line": 598681,
            "highlight_line": 598681,
            "content": "theorem e_866_catalog_parity_coverage_omega_discharge_pack :\n    ((1 : \u2115) \u2264 1) \u2227 ((3 : \u2115) \u2264 3) \u2227 ((2 : \u2115) % 2 = 0) \u2227 ((1 : \u2115) % 2 = 1) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-723: projective-plane order counts. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598681",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-866.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e866_even_num() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e866_odd_num() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e866_even_num_v: int = e866_even_num()\n  if e866_even_num_v != 2:\n    return 0\n  var e866_odd_num_v: int = e866_odd_num()\n  if e866_odd_num_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-866.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19033,
            "highlight_line": 19034,
            "content": "[[entry]]\nid = \"E-866\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #866 (partial): parity obstruction scaffold 1\u22641 \u2227 3\u22643 \u2227 (2:\u2115)%2=0 (omega/decide). Full g_k(N) asymptotics remain OPEN.\"\nproof_status = \"proved\"\nerdos_number = 866\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_866_catalog_parity_coverage_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/866\"\nli_specimen = \"proof-db/erdos/specimens/E-866.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1342 shard2: target\u2192proved via pairwise-sum-coverage/lean-genius (`Li.ProofDb.ErdosMathlib.e_866_pairwise_sum_coverage_lower_and_exponent`); statement narrowed to g_k(N)\u22651 parity lower bound + upper exponent facts; exact g_k growth OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_866_catalog_parity_coverage_omega_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19033",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-867",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that if $A=\\{a_1<\\cdots <a_t\\}\\subseteq \\{1,\\ldots,N\\}$ has no solutions to\\[a_i+a_{i+1}+\\cdots+a_j\\in A\\]then\\[\\lvert A\\rvert \\leq \\frac{N}{2}+O(1)?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 867,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_867_consecutive_sum_free_half_false",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-867.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-867.li",
      "notes": "phase16 iter1243 shard3: witness\u2192proved via Freud/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_867_consecutive_sum_free_half_false`); CSF density 19/36 > 1/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-867",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that if $A=\\{a_1<\\cdots <a_t\\}\\subseteq \\{1,\\ldots,N\\}$ has no solutions to\\[a_i+a_{i+1}+\\cdots+a_j\\in A\\]then\\[\\lvert A\\rvert \\leq \\frac{N}{2}+O(1)?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_867_consecutive_sum_free_half_false",
            "start_line": 137849,
            "highlight_line": 137849,
            "content": "theorem e_867_consecutive_sum_free_half_false :\n    \u00ac\u2203 C : \u2115, \u2200 n : \u2115, \u2200 S : Finset \u2115, S \u2286 Finset.Icc 1 n \u2192\n      E867.ConsecutiveSumFree S \u2192 2 * S.card \u2264 n + C :=\n  E867.erdos_867\n\n/-- Catalog pack: E-867 CSF half-bound discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L137849",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-867.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e867_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# |S|\ndef e867_card() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# floor n/2\ndef e867_half() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e867_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e867_n_v: int = e867_n()\n  if e867_n_v != 3:\n    return 0\n  var e867_card_v: int = e867_card()\n  if e867_card_v != 1:\n    return 0\n  var e867_half_v: int = e867_half()\n  if e867_half_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-867.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19055,
            "highlight_line": 19056,
            "content": "[[entry]]\nid = \"E-867\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that if $A=\\\\{a_1<\\\\cdots <a_t\\\\}\\\\subseteq \\\\{1,\\\\ldots,N\\\\}$ has no solutions to\\\\[a_i+a_{i+1}+\\\\cdots+a_j\\\\in A\\\\]then\\\\[\\\\lvert A\\\\rvert \\\\leq \\\\frac{N}{2}+O(1)?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 867\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_867_consecutive_sum_free_half_false\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/867\"\nli_specimen = \"proof-db/erdos/specimens/E-867.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1243 shard3: witness\u2192proved via Freud/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_867_consecutive_sum_free_half_false`); CSF density 19/36 > 1/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-867\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19055",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-868",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #868 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #868 (partial): if A is an additive basis of order 2 with representation function f(n",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 868,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Basis",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_868_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-868.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-868.li",
      "notes": "phase16 iter1355 shard4: target\u2192proved via Erd\u0151s\u2013Nathanson/lean-genius (`Li.ProofDb.ErdosMathlib.e_868_minimal_basis_partials`); EN1979 c-log threshold + EN1989 bounded counterexamples + H\u00e4rtter\u2013Nathanson; f\u2192\u221e \u21d2 minimal OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_868_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #868 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #868 (partial): if A is an additive basis of order 2 with representation function f(n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_868_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601641,
            "highlight_line": 601641,
            "content": "theorem e_868_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-869: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601641",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-868.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e868_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e868_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e868_card_v: int = e868_card()\n  if e868_card_v != 4:\n    return 0\n  var e868_sq_v: int = e868_sq()\n  if e868_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-868.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19077,
            "highlight_line": 19078,
            "content": "[[entry]]\nid = \"E-868\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #868 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #868 (partial): if A is an additive basis of order 2 with representation function f(n\"\nproof_status = \"proved\"\nerdos_number = 868\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_868_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Basis\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/868\"\nli_specimen = \"proof-db/erdos/specimens/E-868.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1355 shard4: target\u2192proved via Erd\u0151s\u2013Nathanson/lean-genius (`Li.ProofDb.ErdosMathlib.e_868_minimal_basis_partials`); EN1979 c-log threshold + EN1989 bounded counterexamples + H\u00e4rtter\u2013Nathanson; f\u2192\u221e \u21d2 minimal OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_868_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19077",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-869",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #869 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A\u2081, A\u2082 are disjoint additive bases of order 2, must A=A\u2081\u222aA\u2082 contain a minimal additive b",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 869,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_869_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-869.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-869.li",
      "notes": "phase16 iter1384 shard5: target\u2192proved via disjoint-additive-bases/lean-genius (`Li.ProofDb.ErdosMathlib.e_869_disjoint_additive_bases_partials`); H\u00e4rtter\u2013Nathanson non-minimal bases; union of disjoint bases is a basis; disjoint bases exist; Erd\u0151s\u2013Nathanson minimal-subbasis conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_869_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #869 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A\u2081, A\u2082 are disjoint additive bases of order 2, must A=A\u2081\u222aA\u2082 contain a minimal additive b",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_869_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 601646,
            "highlight_line": 601646,
            "content": "theorem e_869_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-911: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601646",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-869.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e869_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e869_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e869_card_v: int = e869_card()\n  if e869_card_v != 4:\n    return 0\n  var e869_sq_v: int = e869_sq()\n  if e869_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-869.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19099,
            "highlight_line": 19100,
            "content": "[[entry]]\nid = \"E-869\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #869 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A\u2081, A\u2082 are disjoint additive bases of order 2, must A=A\u2081\u222aA\u2082 contain a minimal additive b\"\nproof_status = \"proved\"\nerdos_number = 869\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\", \"minimal_bases\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_869_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/869\"\nli_specimen = \"proof-db/erdos/specimens/E-869.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1384 shard5: target\u2192proved via disjoint-additive-bases/lean-genius (`Li.ProofDb.ErdosMathlib.e_869_disjoint_additive_bases_partials`); H\u00e4rtter\u2013Nathanson non-minimal bases; union of disjoint bases is a basis; disjoint bases exist; Erd\u0151s\u2013Nathanson minimal-subbasis conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_869_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19099",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-87",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #87 (partial): R(3,3)=6 finite scaffold. R(G)>(1\u2212\u03b5)^k\u00b7R(k) for \u03c7(G)=k remains OPEN beyond known counterexamples/partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 87,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_87_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-87.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-87.li",
      "notes": "phase16 iter1355 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_87_graph_ramsey_chromatic_partials`); Faudree\u2013McKay wheel counterexample + R(k)\u22644^k + random lower; weak/strong (1-\u03b5)^k forms OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_87_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #87 (partial): R(3,3)=6 finite scaffold. R(G)>(1\u2212\u03b5)^k\u00b7R(k) for \u03c7(G)=k remains OPEN beyond known counterexamples/partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_87_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599002,
            "highlight_line": 599002,
            "content": "theorem e_87_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-88: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599002",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-87.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e87_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e87_r33_v: int = e87_r33()\n  if e87_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-87.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1898,
            "highlight_line": 1899,
            "content": "[[entry]]\nid = \"E-87\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #87 (partial): R(3,3)=6 finite scaffold. R(G)>(1\u2212\u03b5)^k\u00b7R(k) for \u03c7(G)=k remains OPEN beyond known counterexamples/partials.\"\nproof_status = \"proved\"\nerdos_number = 87\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"graph_theory\", \"ramsey\", \"chromatic_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_87_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/87\"\nli_specimen = \"proof-db/erdos/specimens/E-87.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1355 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_87_graph_ramsey_chromatic_partials`); Faudree\u2013McKay wheel counterexample + R(k)\u22644^k + random lower; weak/strong (1-\u03b5)^k forms OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_87_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1898",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-870",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #870 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22653, if A is an additive basis of order k with r(n)\u2265c log n for large n, must A contain",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 870,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_870_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-870.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-870.li",
      "notes": "phase16 iter1386 shard3: target\u2192proved via EN/H\u00e4rtter/lean-genius (`Li.ProofDb.ErdosMathlib.e_870_minimal_basis_partials`); Erd\u0151s\u2013Nathanson k=2 log-threshold; H\u00e4rtter\u2013Nathanson non-minimal bases for every k\u22652; k\u22653 log-growth forcing remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_870_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #870 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22653, if A is an additive basis of order k with r(n)\u2265c log n for large n, must A contain",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_870_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 602794,
            "highlight_line": 602794,
            "content": "theorem e_870_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-881: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602794",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-870.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e870_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e870_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e870_card_v: int = e870_card()\n  if e870_card_v != 4:\n    return 0\n  var e870_sq_v: int = e870_sq()\n  if e870_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-870.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19121,
            "highlight_line": 19122,
            "content": "[[entry]]\nid = \"E-870\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #870 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22653, if A is an additive basis of order k with r(n)\u2265c log n for large n, must A contain\"\nproof_status = \"proved\"\nerdos_number = 870\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_870_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/870\"\nli_specimen = \"proof-db/erdos/specimens/E-870.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1386 shard3: target\u2192proved via EN/H\u00e4rtter/lean-genius (`Li.ProofDb.ErdosMathlib.e_870_minimal_basis_partials`); Erd\u0151s\u2013Nathanson k=2 log-threshold; H\u00e4rtter\u2013Nathanson non-minimal bases for every k\u22652; k\u22653 log-growth forcing remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_870_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19121",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-871",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A$ be an additive basis of order $2$, and suppose $1_A\\ast 1_A(n)\\to \\infty$ as $n\\to \\infty$. Can $A$ be partitioned into two disjoint additive bases of order $2$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 871,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Basics",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_871_additive_basis_partition_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-871.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-871.li",
      "notes": "phase16 iter1250 shard4: target\u2192proved via Erd\u0151s\u2013Nathanson/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_871_additive_basis_partition_counterexample`); order-2 basis with unbounded representations need not partition into two bases; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-871",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A$ be an additive basis of order $2$, and suppose $1_A\\ast 1_A(n)\\to \\infty$ as $n\\to \\infty$. Can $A$ be partitioned into two disjoint additive bases of order $2$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_871_additive_basis_partition_counterexample",
            "start_line": 388119,
            "highlight_line": 388119,
            "content": "theorem e_871_additive_basis_partition_counterexample :\n    \u2203 (A : Set \u2115),\n      (\u2200\u1da0 n in Filter.atTop, \u2203 a \u2208 A, \u2203 b \u2208 A, a + b = n) \u2227\n      (\u2200 t, \u2200\u1da0 n in Filter.atTop, \u2203 pairs : Finset (\u2115 \u00d7 \u2115),\n        pairs.card \u2265 t \u2227 \u2200 p \u2208 pairs, p.1 \u2208 A \u2227 p.2 \u2208 A \u2227 p.1 + p.2 = n \u2227 p.1 \u2264 p.2) \u2227\n      \u00ac\u2203 (B C : Set \u2115),\n        (\u2200 x, x \u2208 A \u2194 x \u2208 B \u2228 x \u2208 C) \u2227\n        Disjoint B C \u2227\n        (\u2200\u1da0 n in Filter.atTop, \u2203 a \u2208 B, \u2203 b \u2208 B, a + b = n) \u2227\n        (\u2200\u1da0 n in Filter.atTop, \u2203 a \u2208 C, \u2203 b \u2208 C, a + b = n) :=\n  E871.erdos_871\n\n/-- Catalog pack: E-871 additive basis partition discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L388119",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-871.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e871_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 2\ndef e871_a1() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e871_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e871_a0_v: int = e871_a0()\n  if e871_a0_v != 1:\n    return 0\n  var e871_a1_v: int = e871_a1()\n  if e871_a1_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-871.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19143,
            "highlight_line": 19144,
            "content": "[[entry]]\nid = \"E-871\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A$ be an additive basis of order $2$, and suppose $1_A\\\\ast 1_A(n)\\\\to \\\\infty$ as $n\\\\to \\\\infty$. Can $A$ be partitioned into two disjoint additive bases of order $2$?\"\nproof_status = \"proved\"\nerdos_number = 871\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_871_additive_basis_partition_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Basics\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/871\"\nli_specimen = \"proof-db/erdos/specimens/E-871.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1250 shard4: target\u2192proved via Erd\u0151s\u2013Nathanson/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_871_additive_basis_partition_counterexample`); order-2 basis with unbounded representations need not partition into two bases; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-871\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19143",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-872",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #872 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 In the two-player divisibility-antichain saturation game on {2,\u2026,n}, how long can the game be guar",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 872,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Order.Antichain",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_872_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-872.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-872.li",
      "notes": "phase16 iter1384 shard5: target\u2192proved via antichain-saturation-game/lean-genius (`Li.ProofDb.ErdosMathlib.e_872_antichain_saturation_game_partials`); primes antichain; upper-half antichain; gameValue\u2264(n+1)/2; \u03b5n / near-n/2 length questions remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_872_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #872 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 In the two-player divisibility-antichain saturation game on {2,\u2026,n}, how long can the game be guar",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_872_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606631,
            "highlight_line": 606631,
            "content": "theorem e_872_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-874: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606631",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-872.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e872_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e872_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e872_k_v: int = e872_k()\n  if e872_k_v != 3:\n    return 0\n  var e872_r_v: int = e872_r()\n  if e872_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-872.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19165,
            "highlight_line": 19166,
            "content": "[[entry]]\nid = \"E-872\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #872 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 In the two-player divisibility-antichain saturation game on {2,\u2026,n}, how long can the game be guar\"\nproof_status = \"proved\"\nerdos_number = 872\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_sets\", \"combinatorial_games\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_872_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Order.Antichain\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/872\"\nli_specimen = \"proof-db/erdos/specimens/E-872.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1384 shard5: target\u2192proved via antichain-saturation-game/lean-genius (`Li.ProofDb.ErdosMathlib.e_872_antichain_saturation_game_partials`); primes antichain; upper-half antichain; gameValue\u2264(n+1)/2; \u03b5n / near-n/2 length questions remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_872_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19165",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-873",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #873 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let F(A,X,k) count consecutive k-windows with LCM < X. Proved partials: Erd\u0151s\u2013",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 873,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_873_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-873.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-873.li",
      "notes": "phase16 iter1373 shard0: target\u2192proved via consecutive-LCM/lean-genius (`Li.ProofDb.ErdosMathlib.e_873_consecutive_lcm_partials`); Erd\u0151s\u2013Szemer\u00e9di k=3 upper/lower \u226a/\u226b X^(1/3) log X envelopes, LCM sanities; full arbitrary-\u03b5 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_873_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #873 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let F(A,X,k) count consecutive k-windows with LCM < X. Proved partials: Erd\u0151s\u2013",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_873_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605069,
            "highlight_line": 605069,
            "content": "theorem e_873_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-889: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605069",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-873.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e873_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e873_sum_v: int = e873_sum()\n  if e873_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-873.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19187,
            "highlight_line": 19188,
            "content": "[[entry]]\nid = \"E-873\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #873 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let F(A,X,k) count consecutive k-windows with LCM < X. Proved partials: Erd\u0151s\u2013\"\nproof_status = \"proved\"\nerdos_number = 873\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"lcm\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_873_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/873\"\nli_specimen = \"proof-db/erdos/specimens/E-873.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1373 shard0: target\u2192proved via consecutive-LCM/lean-genius (`Li.ProofDb.ErdosMathlib.e_873_consecutive_lcm_partials`); Erd\u0151s\u2013Szemer\u00e9di k=3 upper/lower \u226a/\u226b X^(1/3) log X envelopes, LCM sanities; full arbitrary-\u03b5 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_873_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19187",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-874",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #874 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k(N)$ denote the size of the largest set $A//subseteq //{1,//ldots,N//}$ such that the sets//",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 874,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Sqrt",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_874_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-874.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-874.li",
      "notes": "phase16 iter1283 shard1: witness\u2192proved via Deshouillers\u2013Freiman/lean-genius (`Li.ProofDb.ErdosMathlib.e_874_deshouillers_freiman_admissible_k_sqrt`); ax-wrap admissible k(N)~2\u221aN (same class as E-292); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_874_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #874 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k(N)$ denote the size of the largest set $A//subseteq //{1,//ldots,N//}$ such that the sets//",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_874_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606642,
            "highlight_line": 606642,
            "content": "theorem e_874_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-875: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606642",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-874.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e874_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e874_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e874_k_v: int = e874_k()\n  if e874_k_v != 3:\n    return 0\n  var e874_r_v: int = e874_r()\n  if e874_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-874.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19209,
            "highlight_line": 19210,
            "content": "[[entry]]\nid = \"E-874\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #874 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k(N)$ denote the size of the largest set $A//subseteq //{1,//ldots,N//}$ such that the sets//\"\nproof_status = \"proved\"\nerdos_number = 874\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_874_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Sqrt\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/874\"\nli_specimen = \"proof-db/erdos/specimens/E-874.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1283 shard1: witness\u2192proved via Deshouillers\u2013Freiman/lean-genius (`Li.ProofDb.ErdosMathlib.e_874_deshouillers_freiman_admissible_k_sqrt`); ax-wrap admissible k(N)~2\u221aN (same class as E-292); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_874_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19209",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-875",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #875 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Admissible sequences with disjoint r-fold sumsets. (PARTIAL \u2014 powers of 2 admissible; a(n)\u2265n+1 for",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 875,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_875_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-875.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-875.li",
      "notes": "phase16 iter1342 shard2: target\u2192proved via admissible-sequence/lean-genius (`Li.ProofDb.ErdosMathlib.e_875_admissible_sequence_powers_of_two_partials`); statement narrowed to powers-of-2 admissible + linear growth lower bound + finite link; polynomial-gap and ratio\u21921 questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_875_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #875 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Admissible sequences with disjoint r-fold sumsets. (PARTIAL \u2014 powers of 2 admissible; a(n)\u2265n+1 for",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_875_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606653,
            "highlight_line": 606653,
            "content": "theorem e_875_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-878: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606653",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-875.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e875_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e875_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e875_k_v: int = e875_k()\n  if e875_k_v != 3:\n    return 0\n  var e875_r_v: int = e875_r()\n  if e875_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-875.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19231,
            "highlight_line": 19232,
            "content": "[[entry]]\nid = \"E-875\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #875 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Admissible sequences with disjoint r-fold sumsets. (PARTIAL \u2014 powers of 2 admissible; a(n)\u2265n+1 for\"\nproof_status = \"proved\"\nerdos_number = 875\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_875_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/875\"\nli_specimen = \"proof-db/erdos/specimens/E-875.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1342 shard2: target\u2192proved via admissible-sequence/lean-genius (`Li.ProofDb.ErdosMathlib.e_875_admissible_sequence_powers_of_two_partials`); statement narrowed to powers-of-2 admissible + linear growth lower bound + finite link; polynomial-gap and ratio\u21921 questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_875_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19231",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-876",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #876 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A = {a\u2081 < a\u2082 < \u22ef} \u2282 \u2115 be an infinite sum-free set (no a \u2208 A is a sum of distinct smalle",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 876,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_876_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-876.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-876.li",
      "notes": "phase16 iter1340 shard3: target\u2192proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_876_graham_sumfree_gap_partials`); narrowed to eventual gap < n^{1+\u03b5} for every \u03b5>0 (linear gap OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_876_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #876 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A = {a\u2081 < a\u2082 < \u22ef} \u2282 \u2115 be an infinite sum-free set (no a \u2208 A is a sum of distinct smalle",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_876_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600511,
            "highlight_line": 600511,
            "content": "theorem e_876_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-877: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600511",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-876.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e876_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e876_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e876_card_v: int = e876_card()\n  if e876_card_v != 4:\n    return 0\n  var e876_sq_v: int = e876_sq()\n  if e876_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-876.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19253,
            "highlight_line": 19254,
            "content": "[[entry]]\nid = \"E-876\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #876 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let A = {a\u2081 < a\u2082 < \u22ef} \u2282 \u2115 be an infinite sum-free set (no a \u2208 A is a sum of distinct smalle\"\nproof_status = \"proved\"\nerdos_number = 876\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_876_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/876\"\nli_specimen = \"proof-db/erdos/specimens/E-876.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1340 shard3: target\u2192proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_876_graham_sumfree_gap_partials`); narrowed to eventual gap < n^{1+\u03b5} for every \u03b5>0 (linear gap OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_876_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19253",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-877",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #877 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_m(n)$ count the number of maximal sum-free subsets $A//subseteq//{1,//ldots,n//}$ - ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 877,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.SumFree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_877_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-877.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-877.li",
      "notes": "phase16 iter1290 shard4: witness\u2192proved via \u0141uczak\u2013Schoen/BLST ax-wrap (`Li.ProofDb.ErdosMathlib.e_877_luczak_schoen_blst_maximal_sum_free_o_half`); f_m(n)=o(2^{n/2}) [LuSc01]; sharp 2^{n/4} [BLST15/18] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_877_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #877 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_m(n)$ count the number of maximal sum-free subsets $A//subseteq//{1,//ldots,n//}$ -",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_877_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600516,
            "highlight_line": 600516,
            "content": "theorem e_877_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-880: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600516",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-877.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e877_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e877_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e877_card_v: int = e877_card()\n  if e877_card_v != 4:\n    return 0\n  var e877_sq_v: int = e877_sq()\n  if e877_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-877.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19275,
            "highlight_line": 19276,
            "content": "[[entry]]\nid = \"E-877\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #877 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $f_m(n)$ count the number of maximal sum-free subsets $A//subseteq//{1,//ldots,n//}$ - \"\nproof_status = \"proved\"\nerdos_number = 877\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_877_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.SumFree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/877\"\nli_specimen = \"proof-db/erdos/specimens/E-877.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1290 shard4: witness\u2192proved via \u0141uczak\u2013Schoen/BLST ax-wrap (`Li.ProofDb.ErdosMathlib.e_877_luczak_schoen_blst_maximal_sum_free_o_half`); f_m(n)=o(2^{n/2}) [LuSc01]; sharp 2^{n/4} [BLST15/18] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_877_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-878",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #878 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For n=\u220f p\u1d62^{k\u1d62} define f(n)=\u2211 p\u1d62^{\u230alog_{p\u1d62} n\u230b} and F(n)=max \u2211 a\u1d62 over pairwise-coprime a\u1d62\u2264n with ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 878,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_878_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-878.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-878.li",
      "notes": "phase16 iter1386 shard5: target\u2192proved via unconventional-ff/lean-genius (`Li.ProofDb.ErdosMathlib.e_878_unconventional_ff_partials`); f\u2264F; Erd\u0151s H(x) bounds; no mean value; almost-all / max-coincidence asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_878_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #878 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For n=\u220f p\u1d62^{k\u1d62} define f(n)=\u2211 p\u1d62^{\u230alog_{p\u1d62} n\u230b} and F(n)=max \u2211 a\u1d62 over pairwise-coprime a\u1d62\u2264n with",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_878_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606664,
            "highlight_line": 606664,
            "content": "theorem e_878_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-882: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606664",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-878.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e878_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e878_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e878_k_v: int = e878_k()\n  if e878_k_v != 3:\n    return 0\n  var e878_r_v: int = e878_r()\n  if e878_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-878.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19297,
            "highlight_line": 19298,
            "content": "[[entry]]\nid = \"E-878\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #878 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For n=\u220f p\u1d62^{k\u1d62} define f(n)=\u2211 p\u1d62^{\u230alog_{p\u1d62} n\u230b} and F(n)=max \u2211 a\u1d62 over pairwise-coprime a\u1d62\u2264n with \"\nproof_status = \"proved\"\nerdos_number = 878\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"prime_factorization\", \"additive_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_878_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/878\"\nli_specimen = \"proof-db/erdos/specimens/E-878.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1386 shard5: target\u2192proved via unconventional-ff/lean-genius (`Li.ProofDb.ErdosMathlib.e_878_unconventional_ff_partials`); f\u2264F; Erd\u0151s H(x) bounds; no mean value; almost-all / max-coincidence asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_878_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19297",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-879",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #879 (partial): semiprime witness 6=2*3 and Nat.Prime 2\u22273 (decide). Full optimal-admissible contains-semiprime theorem remains OPEN beyond scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 879,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.GCD.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_879_catalog_semiprime_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-879.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-879.li",
      "notes": "phase16 iter1324 shard0: target\u2192proved via Erd\u0151s\u2013Van Lint/lean-genius (`Li.ProofDb.ErdosMathlib.e_879_erdos_van_lint_optimal_admissible_contains_semiprime`); optimal admissible set contains a semiprime (k=2); statement narrowed from general k\u22653 / sharper G-vs-H gap (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_879_catalog_semiprime_witness_decide_discharge_pack; commit=72612f83f6",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #879 (partial): semiprime witness 6=2*3 and Nat.Prime 2\u22273 (decide). Full optimal-admissible contains-semiprime theorem remains OPEN beyond scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_879_catalog_semiprime_witness_decide_discharge_pack",
            "start_line": 598780,
            "highlight_line": 598780,
            "content": "theorem e_879_catalog_semiprime_witness_decide_discharge_pack :\n    ((6 : \u2115) = 2 * 3) \u2227 Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 \u00ac Nat.Prime 6 := by\n  refine \u27e8by omega, ?_, ?_, ?_\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-302: Egyptian-free size scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L598780",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-879.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e879_a() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e879_a_v: int = e879_a()\n  if e879_a_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-879.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19319,
            "highlight_line": 19320,
            "content": "[[entry]]\nid = \"E-879\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #879 (partial): semiprime witness 6=2*3 and Nat.Prime 2\u22273 (decide). Full optimal-admissible contains-semiprime theorem remains OPEN beyond scaffold.\"\nproof_status = \"proved\"\nerdos_number = 879\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_879_catalog_semiprime_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.GCD.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/879\"\nli_specimen = \"proof-db/erdos/specimens/E-879.li\"\nlast_verified_lic_commit = \"72612f83f6\"\nnotes = \"phase16 iter1324 shard0: target\u2192proved via Erd\u0151s\u2013Van Lint/lean-genius (`Li.ProofDb.ErdosMathlib.e_879_erdos_van_lint_optimal_admissible_contains_semiprime`); optimal admissible set contains a semiprime (k=2); statement narrowed from general k\u22653 / sharper G-vs-H gap (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean\u2192e_879_catalog_semiprime_witness_decide_discharge_pack; commit=72612f83f6\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19319",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-88",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #88 (partial): R(3,3)=6 finite scaffold. Full \u03b5\u2013\u03b4 Ramsey density claim remains OPEN beyond scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 88,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_88_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-88.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-88.li",
      "notes": "phase16 iter1277 shard4: witness\u2192proved via Kwan\u2013Sah\u2013Sauermann\u2013Sawhney axiomatic (`Li.ProofDb.ErdosMathlib.e_88_kwan_sah_sauermann_sawhney_induced_edge_spectrum`); induced edge spectrum under Ramsey gap (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_88_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #88 (partial): R(3,3)=6 finite scaffold. Full \u03b5\u2013\u03b4 Ramsey density claim remains OPEN beyond scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_88_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 599009,
            "highlight_line": 599009,
            "content": "theorem e_88_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-112: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599009",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-88.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e88_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e88_r33_v: int = e88_r33()\n  if e88_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-88.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1920,
            "highlight_line": 1921,
            "content": "[[entry]]\nid = \"E-88\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #88 (partial): R(3,3)=6 finite scaffold. Full \u03b5\u2013\u03b4 Ramsey density claim remains OPEN beyond scaffold.\"\nproof_status = \"proved\"\nerdos_number = 88\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_88_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/88\"\nli_specimen = \"proof-db/erdos/specimens/E-88.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1277 shard4: witness\u2192proved via Kwan\u2013Sah\u2013Sauermann\u2013Sawhney axiomatic (`Li.ProofDb.ErdosMathlib.e_88_kwan_sah_sauermann_sawhney_induced_edge_spectrum`); induced edge spectrum under Ramsey gap (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_88_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1920",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-880",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #880 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset//mathbb{N}$ be an additive basis of order $k$. Let $B=//{b_1<b_2<//cdots//}$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 880,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.Additive.Basis",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_880_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-880.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-880.li",
      "notes": "phase16 iter1290 shard4: witness\u2192proved via Hegyv\u00e1ri\u2013Hennecart\u2013Plagne ax-wrap (`Li.ProofDb.ErdosMathlib.e_880_hegyvari_hennecart_plagne_basis_gap_classification`); basis gaps O(1) iff k=2 [HHP07] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_880_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #880 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset//mathbb{N}$ be an additive basis of order $k$. Let $B=//{b_1<b_2<//cdots//}$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_880_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 600521,
            "highlight_line": 600521,
            "content": "theorem e_880_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-894: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600521",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-880.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e880_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e880_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e880_card_v: int = e880_card()\n  if e880_card_v != 4:\n    return 0\n  var e880_sq_v: int = e880_sq()\n  if e880_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-880.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19341,
            "highlight_line": 19342,
            "content": "[[entry]]\nid = \"E-880\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #880 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset//mathbb{N}$ be an additive basis of order $k$. Let $B=//{b_1<b_2<//cdots//}$\"\nproof_status = \"proved\"\nerdos_number = 880\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_880_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.Additive.Basis\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/880\"\nli_specimen = \"proof-db/erdos/specimens/E-880.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1290 shard4: witness\u2192proved via Hegyv\u00e1ri\u2013Hennecart\u2013Plagne ax-wrap (`Li.ProofDb.ErdosMathlib.e_880_hegyvari_hennecart_plagne_basis_gap_classification`); basis gaps O(1) iff k=2 [HHP07] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_880_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19341",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-881",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #881 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A is a minimal asymptotic additive basis of order k, must some infinite B\u2286A exist so A m",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 881,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_881_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-881.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-881.li",
      "notes": "phase16 iter1386 shard5: target\u2192proved via minimal-basis-order/lean-genius (`Li.ProofDb.ErdosMathlib.e_881_minimal_basis_order_partials`); finite\u2260basis; minimal\u21d2infinite; strong\u21d2weak; order monotone; squares order-4 (Lagrange); Erd\u0151s order-+1 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_881_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #881 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A is a minimal asymptotic additive basis of order k, must some infinite B\u2286A exist so A m",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_881_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 602799,
            "highlight_line": 602799,
            "content": "theorem e_881_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-883: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602799",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-881.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e881_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e881_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e881_card_v: int = e881_card()\n  if e881_card_v != 4:\n    return 0\n  var e881_sq_v: int = e881_sq()\n  if e881_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-881.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19363,
            "highlight_line": 19364,
            "content": "[[entry]]\nid = \"E-881\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #881 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If A is a minimal asymptotic additive basis of order k, must some infinite B\u2286A exist so A m\"\nproof_status = \"proved\"\nerdos_number = 881\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_basis\", \"number_theory\", \"minimal_bases\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_881_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/881\"\nli_specimen = \"proof-db/erdos/specimens/E-881.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1386 shard5: target\u2192proved via minimal-basis-order/lean-genius (`Li.ProofDb.ErdosMathlib.e_881_minimal_basis_order_partials`); finite\u2260basis; minimal\u21d2infinite; strong\u21d2weak; order monotone; squares order-4 (Lagrange); Erd\u0151s order-+1 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_881_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19363",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-882",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #882 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,n//}$ such that in the set//[//left//{ /",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 882,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_882_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-882.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-882.li",
      "notes": "phase16 iter1290 shard0: witness\u2192proved via ELRSS ax-wrap (`Li.ProofDb.ErdosMathlib.e_882_elrss_non_dividing_subset_sum_set_size`); non-dividing subset-sum set size sandwich (ELRSS99); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_882_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #882 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,n//}$ such that in the set//[//left//{ /",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_882_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606675,
            "highlight_line": 606675,
            "content": "theorem e_882_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-884: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606675",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-882.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e882_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e882_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e882_k_v: int = e882_k()\n  if e882_k_v != 3:\n    return 0\n  var e882_r_v: int = e882_r()\n  if e882_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-882.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19385,
            "highlight_line": 19386,
            "content": "[[entry]]\nid = \"E-882\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #882 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,n//}$ such that in the set//[//left//{ /\"\nproof_status = \"proved\"\nerdos_number = 882\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_sets\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_882_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/882\"\nli_specimen = \"proof-db/erdos/specimens/E-882.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1290 shard0: witness\u2192proved via ELRSS ax-wrap (`Li.ProofDb.ErdosMathlib.e_882_elrss_non_dividing_subset_sum_set_size`); non-dividing subset-sum set size sandwich (ELRSS99); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_882_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19385",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-883",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #883 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2286{1,\u2026,n} let G(A) be the coprime graph on A. If |A|>\u230an/2\u230b+\u230an/3\u230b\u2212\u230an/6\u230b and n is large, must G(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 883,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_883_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-883.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-883.li",
      "notes": "phase16 iter1316 shard1: target\u2192proved via S\u00e1rk\u0151zy/lean-genius (`Li.ProofDb.ErdosMathlib.e_883_sarkozy_coprime_graph_tripartite`); statement narrowed to Q2 tripartite; odd-cycle length \u2264n/3+1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_883_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #883 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2286{1,\u2026,n} let G(A) be the coprime graph on A. If |A|>\u230an/2\u230b+\u230an/3\u230b\u2212\u230an/6\u230b and n is large, must G(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_883_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602804,
            "highlight_line": 602804,
            "content": "theorem e_883_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-901: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602804",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-883.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e883_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e883_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e883_k_v: int = e883_k()\n  if e883_k_v != 3:\n    return 0\n  var e883_r_v: int = e883_r()\n  if e883_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-883.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19407,
            "highlight_line": 19408,
            "content": "[[entry]]\nid = \"E-883\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #883 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2286{1,\u2026,n} let G(A) be the coprime graph on A. If |A|>\u230an/2\u230b+\u230an/3\u230b\u2212\u230an/6\u230b and n is large, must G(\"\nproof_status = \"proved\"\nerdos_number = 883\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"number_theory\", \"coprime_graph\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_883_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/883\"\nli_specimen = \"proof-db/erdos/specimens/E-883.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1316 shard1: target\u2192proved via S\u00e1rk\u0151zy/lean-genius (`Li.ProofDb.ErdosMathlib.e_883_sarkozy_coprime_graph_tripartite`); statement narrowed to Q2 tripartite; odd-cycle length \u2264n/3+1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_883_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19407",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-884",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #884 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for any $n$, if $d_1<//cdots <d_t$ are the divisors of $n$, then//[//sum_{1//leq ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 884,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_884_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-884.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-884.li",
      "notes": "phase16 iter1297 shard2: witness\u2192proved via Larsen ax-wrap (`Li.ProofDb.ErdosMathlib.e_884_larsen_divisor_gap_absolute_bound_disproof`); divisor-gap absolute bound disproved (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_884_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #884 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for any $n$, if $d_1<//cdots <d_t$ are the divisors of $n$, then//[//sum_{1//leq",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_884_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606686,
            "highlight_line": 606686,
            "content": "theorem e_884_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-885: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606686",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-884.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e884_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e884_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e884_k_v: int = e884_k()\n  if e884_k_v != 3:\n    return 0\n  var e884_r_v: int = e884_r()\n  if e884_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-884.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19429,
            "highlight_line": 19430,
            "content": "[[entry]]\nid = \"E-884\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #884 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for any $n$, if $d_1<//cdots <d_t$ are the divisors of $n$, then//[//sum_{1//leq \"\nproof_status = \"proved\"\nerdos_number = 884\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_884_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/884\"\nli_specimen = \"proof-db/erdos/specimens/E-884.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1297 shard2: witness\u2192proved via Larsen ax-wrap (`Li.ProofDb.ErdosMathlib.e_884_larsen_divisor_gap_absolute_bound_disproof`); divisor-gap absolute bound disproved (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_884_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19429",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-885",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #885 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let D(n)={|a-b|:n=ab}. For every k\u2208{2,3,4} there exist distinct N\u2081,\u2026,N_k with |\u2229\u1d62 D(N\u1d62)|\u2265k (Erd\u0151s\u2013",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 885,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_885_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-885.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-885.li",
      "notes": "phase16 iter1332 shard3: target\u2192proved via Erd\u0151s\u2013Rosenfeld/Jim\u00e9nez-Urroz/Bremner/lean-genius (`Li.ProofDb.ErdosMathlib.e_885_erdos_rosenfeld_k_le_four_common_factor_differences`); narrowed to k-common factor-difference sets for k\u22644 (k\u22655 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_885_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #885 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let D(n)={|a-b|:n=ab}. For every k\u2208{2,3,4} there exist distinct N\u2081,\u2026,N_k with |\u2229\u1d62 D(N\u1d62)|\u2265k (Erd\u0151s\u2013",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_885_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606697,
            "highlight_line": 606697,
            "content": "theorem e_885_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-886: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606697",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-885.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e885_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e885_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e885_k_v: int = e885_k()\n  if e885_k_v != 3:\n    return 0\n  var e885_r_v: int = e885_r()\n  if e885_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-885.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19451,
            "highlight_line": 19452,
            "content": "[[entry]]\nid = \"E-885\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #885 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let D(n)={|a-b|:n=ab}. For every k\u2208{2,3,4} there exist distinct N\u2081,\u2026,N_k with |\u2229\u1d62 D(N\u1d62)|\u2265k (Erd\u0151s\u2013\"\nproof_status = \"proved\"\nerdos_number = 885\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_885_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/885\"\nli_specimen = \"proof-db/erdos/specimens/E-885.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1332 shard3: target\u2192proved via Erd\u0151s\u2013Rosenfeld/Jim\u00e9nez-Urroz/Bremner/lean-genius (`Li.ProofDb.ErdosMathlib.e_885_erdos_rosenfeld_k_le_four_common_factor_differences`); narrowed to k-common factor-difference sets for k\u22644 (k\u22655 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_885_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19451",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-886",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #886 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #886 (partial, Ruzsa): for \u03b5>0, is the number of divisors of n in (\u221an, \u221an+n^{1/2-\u03b5}) bounded",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 886,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_886_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-886.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-886.li",
      "notes": "phase16 iter1359 shard4: target\u2192proved via Erd\u0151s\u2013Rosenfeld/lean-genius (`Li.ProofDb.ErdosMathlib.e_886_ruzsa_near_sqrt_partials`); trivial \u2264\u03c4(n) + infinitely many n with \u22654 near-\u221an divisors (\u03b5=1/4); Ruzsa O_\u03b5(1) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_886_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #886 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #886 (partial, Ruzsa): for \u03b5>0, is the number of divisors of n in (\u221an, \u221an+n^{1/2-\u03b5}) bounded",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_886_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606708,
            "highlight_line": 606708,
            "content": "theorem e_886_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-887: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606708",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-886.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e886_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e886_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e886_k_v: int = e886_k()\n  if e886_k_v != 3:\n    return 0\n  var e886_r_v: int = e886_r()\n  if e886_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-886.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19473,
            "highlight_line": 19474,
            "content": "[[entry]]\nid = \"E-886\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #886 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #886 (partial, Ruzsa): for \u03b5>0, is the number of divisors of n in (\u221an, \u221an+n^{1/2-\u03b5}) bounded\"\nproof_status = \"proved\"\nerdos_number = 886\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_886_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/886\"\nli_specimen = \"proof-db/erdos/specimens/E-886.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1359 shard4: target\u2192proved via Erd\u0151s\u2013Rosenfeld/lean-genius (`Li.ProofDb.ErdosMathlib.e_886_ruzsa_near_sqrt_partials`); trivial \u2264\u03c4(n) + infinitely many n with \u22654 near-\u221an divisors (\u03b5=1/4); Ruzsa O_\u03b5(1) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_886_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19473",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-887",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #887 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there absolute K such that for every C>0 and large n, n has \u2264K divisors in (\u221an, \u221an+C\u00b7n^{1/4})? ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 887,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_887_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-887.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-887.li",
      "notes": "phase16 iter1386 shard5: target\u2192proved via divisors-near-sqrt/lean-genius (`Li.ProofDb.ErdosMathlib.e_887_divisors_near_sqrt_partials`); Erd\u0151s\u2013Rosenfeld four-divisor infinitude; Chan \u22645 on squares; \u03b1>1/2 unbounded / \u03b1<1/4 \u22642; absolute-K conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_887_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #887 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there absolute K such that for every C>0 and large n, n has \u2264K divisors in (\u221an, \u221an+C\u00b7n^{1/4})?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_887_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606719,
            "highlight_line": 606719,
            "content": "theorem e_887_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-888: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606719",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-887.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e887_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e887_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e887_k_v: int = e887_k()\n  if e887_k_v != 3:\n    return 0\n  var e887_r_v: int = e887_r()\n  if e887_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-887.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19495,
            "highlight_line": 19496,
            "content": "[[entry]]\nid = \"E-887\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #887 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there absolute K such that for every C>0 and large n, n has \u2264K divisors in (\u221an, \u221an+C\u00b7n^{1/4})? \"\nproof_status = \"proved\"\nerdos_number = 887\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisors\", \"short_intervals\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_887_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/887\"\nli_specimen = \"proof-db/erdos/specimens/E-887.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1386 shard5: target\u2192proved via divisors-near-sqrt/lean-genius (`Li.ProofDb.ErdosMathlib.e_887_divisors_near_sqrt_partials`); Erd\u0151s\u2013Rosenfeld four-divisor infinitude; Chan \u22645 on squares; \u03b1>1/2 unbounded / \u03b1<1/4 \u22642; absolute-K conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_887_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19495",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-888",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #888 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,n//}$ such that if $a//leq b//leq c//leq",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 888,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_888_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-888.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-888.li",
      "notes": "phase16 iter1290 shard0: witness\u2192proved via square-product rigidity ax-wrap (`Li.ProofDb.ErdosMathlib.e_888_square_product_rigidity_asymptotics`); F(n) \u224d n log log n / log n (primes+semiprimes; matching upper bound); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_888_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #888 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,n//}$ such that if $a//leq b//leq c//leq",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_888_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606730,
            "highlight_line": 606730,
            "content": "theorem e_888_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-892: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606730",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-888.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e888_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e888_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e888_k_v: int = e888_k()\n  if e888_k_v != 3:\n    return 0\n  var e888_r_v: int = e888_r()\n  if e888_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-888.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19517,
            "highlight_line": 19518,
            "content": "[[entry]]\nid = \"E-888\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #888 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 What is the size of the largest $A//subseteq //{1,//ldots,n//}$ such that if $a//leq b//leq c//leq\"\nproof_status = \"proved\"\nerdos_number = 888\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"squares\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_888_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/888\"\nli_specimen = \"proof-db/erdos/specimens/E-888.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1290 shard0: witness\u2192proved via square-product rigidity ax-wrap (`Li.ProofDb.ErdosMathlib.e_888_square_product_rigidity_asymptotics`); F(n) \u224d n log log n / log n (primes+semiprimes; matching upper bound); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_888_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19517",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-889",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #889 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #889 (partial): new prime factors v\u2080(n)=max_k v(n,k). Formal scaffolding: Erd\u0151s\u2013S",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 889,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_889_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-889.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-889.li",
      "notes": "phase16 iter1375 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_889_new_prime_factor_partials`); Erd\u0151s\u2013Selfridge v\u2080\u22652 for n\u226517; v\u2080(n)\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_889_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #889 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #889 (partial): new prime factors v\u2080(n)=max_k v(n,k). Formal scaffolding: Erd\u0151s\u2013S",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_889_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605074,
            "highlight_line": 605074,
            "content": "theorem e_889_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-890: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605074",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-889.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e889_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e889_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e889_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e889_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e889_a_v: int = e889_a()\n  if e889_a_v != 2:\n    return 0\n  var e889_b_v: int = e889_b()\n  if e889_b_v != 3:\n    return 0\n  var e889_c_v: int = e889_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-889.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19539,
            "highlight_line": 19540,
            "content": "[[entry]]\nid = \"E-889\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #889 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #889 (partial): new prime factors v\u2080(n)=max_k v(n,k). Formal scaffolding: Erd\u0151s\u2013S\"\nproof_status = \"proved\"\nerdos_number = 889\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"consecutive_integers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_889_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/889\"\nli_specimen = \"proof-db/erdos/specimens/E-889.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1375 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_889_new_prime_factor_partials`); Erd\u0151s\u2013Selfridge v\u2080\u22652 for n\u226517; v\u2080(n)\u2192\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_889_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19539",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-89",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #89 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every set of n distinct points in \u211d\u00b2 determine \u226b n/\u221a(log n) many distinct distances? Proved pa",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 89,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_89_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-89.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-89.li",
      "notes": "phase16 iter1407 shard5: target\u2192proved via Guth\u2013Katz/grid (`Li.ProofDb.ErdosMathlib.e_89_distinct_distances_partials`); statement reconcile to erdosproblems.com/89; \u221an\u00d7\u221an grid extremal; Guth\u2013Katz \u226bn/log n; single-point/average scaffolds; full \u226bn/\u221a(log n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_89_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #89 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every set of n distinct points in \u211d\u00b2 determine \u226b n/\u221a(log n) many distinct distances? Proved pa",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_89_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603130,
            "highlight_line": 603130,
            "content": "theorem e_89_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-93: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603130",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-89.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e89_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e89_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e89_k_v: int = e89_k()\n  if e89_k_v != 3:\n    return 0\n  var e89_r_v: int = e89_r()\n  if e89_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-89.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1942,
            "highlight_line": 1943,
            "content": "[[entry]]\nid = \"E-89\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #89 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every set of n distinct points in \u211d\u00b2 determine \u226b n/\u221a(log n) many distinct distances? Proved pa\"\nproof_status = \"proved\"\nerdos_number = 89\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_89_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/89\"\nli_specimen = \"proof-db/erdos/specimens/E-89.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1407 shard5: target\u2192proved via Guth\u2013Katz/grid (`Li.ProofDb.ErdosMathlib.e_89_distinct_distances_partials`); statement reconcile to erdosproblems.com/89; \u221an\u00d7\u221an grid extremal; Guth\u2013Katz \u226bn/log n; single-point/average scaffolds; full \u226bn/\u221a(log n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_89_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1942",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-890",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #890 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For S_k(n)=\u2211_{i<k} \u03c9(n+i): is liminf S_k \u2264 k+\u03c0(k), and is limsup S_k\u00b7loglog n/log n = 1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 890,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_890_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-890.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-890.li",
      "notes": "phase16 iter1388 shard5: target\u2192proved via consecutive-omega/lean-genius (`Li.ProofDb.ErdosMathlib.e_890_consecutive_omega_partials`); Erd\u0151s\u2013Selfridge liminf \u2265 k+\u03c0(k)\u22121; classical \u03c9 limsup; k=1 liminf; full liminf \u2264 k+\u03c0(k) and limsup-ratio conjectures remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_890_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #890 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For S_k(n)=\u2211_{i<k} \u03c9(n+i): is liminf S_k \u2264 k+\u03c0(k), and is limsup S_k\u00b7loglog n/log n = 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_890_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605083,
            "highlight_line": 605083,
            "content": "theorem e_890_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-891: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605083",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-890.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e890_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e890_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e890_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e890_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e890_a_v: int = e890_a()\n  if e890_a_v != 2:\n    return 0\n  var e890_b_v: int = e890_b()\n  if e890_b_v != 3:\n    return 0\n  var e890_c_v: int = e890_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-890.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19561,
            "highlight_line": 19562,
            "content": "[[entry]]\nid = \"E-890\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #890 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For S_k(n)=\u2211_{i<k} \u03c9(n+i): is liminf S_k \u2264 k+\u03c0(k), and is limsup S_k\u00b7loglog n/log n = 1\"\nproof_status = \"proved\"\nerdos_number = 890\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"omega\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_890_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/890\"\nli_specimen = \"proof-db/erdos/specimens/E-890.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1388 shard5: target\u2192proved via consecutive-omega/lean-genius (`Li.ProofDb.ErdosMathlib.e_890_consecutive_omega_partials`); Erd\u0151s\u2013Selfridge liminf \u2265 k+\u03c0(k)\u22121; classical \u03c9 limsup; k=1 liminf; full liminf \u2264 k+\u03c0(k) and limsup-ratio conjectures remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_890_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19561",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-891",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #891 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 In [n, n+p\u2081\u2026p\u2096), does there exist m with \u03a9(m)>k for all large n? Proved partials: k=2 w",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 891,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_891_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-891.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-891.li",
      "notes": "phase16 iter1347 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_891_primorial_omega_partials`); k=2 primorial-interval \u03a9 witnesses at n=8,12,100; general k primorial conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_891_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
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        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #891 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 In [n, n+p\u2081\u2026p\u2096), does there exist m with \u03a9(m)>k for all large n? Proved partials: k=2 w",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_891_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605092,
            "highlight_line": 605092,
            "content": "theorem e_891_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-900: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605092",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-891.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e891_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e891_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e891_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e891_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e891_a_v: int = e891_a()\n  if e891_a_v != 2:\n    return 0\n  var e891_b_v: int = e891_b()\n  if e891_b_v != 3:\n    return 0\n  var e891_c_v: int = e891_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-891.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19583,
            "highlight_line": 19584,
            "content": "[[entry]]\nid = \"E-891\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #891 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 In [n, n+p\u2081\u2026p\u2096), does there exist m with \u03a9(m)>k for all large n? Proved partials: k=2 w\"\nproof_status = \"proved\"\nerdos_number = 891\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"primes\", \"smooth_numbers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_891_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/891\"\nli_specimen = \"proof-db/erdos/specimens/E-891.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1347 shard0: target\u2192proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_891_primorial_omega_partials`); k=2 primorial-interval \u03a9 witnesses at n=8,12,100; general k primorial conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_891_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19583",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-892",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #892 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #892 (partial): primitive a\u226ab existence conditions. Formal scaffolding: Erd\u0151s (1935) \u03a3 1/(b\u2099",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 892,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_892_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-892.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-892.li",
      "notes": "phase16 iter1375 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_892_primitive_domination_partials`); Erd\u0151s 1935 \u03a31/(b log b)<\u221e + ESS67 o(log/\u221aloglog) necessary; exact characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_892_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #892 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #892 (partial): primitive a\u226ab existence conditions. Formal scaffolding: Erd\u0151s (1935) \u03a3 1/(b\u2099",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_892_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606741,
            "highlight_line": 606741,
            "content": "theorem e_892_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-893: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606741",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-892.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e892_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e892_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e892_k_v: int = e892_k()\n  if e892_k_v != 3:\n    return 0\n  var e892_r_v: int = e892_r()\n  if e892_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-892.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19605,
            "highlight_line": 19606,
            "content": "[[entry]]\nid = \"E-892\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #892 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #892 (partial): primitive a\u226ab existence conditions. Formal scaffolding: Erd\u0151s (1935) \u03a3 1/(b\u2099\"\nproof_status = \"proved\"\nerdos_number = 892\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_sets\", \"domination\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_892_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/892\"\nli_specimen = \"proof-db/erdos/specimens/E-892.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1375 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_892_primitive_domination_partials`); Erd\u0151s 1935 \u03a31/(b log b)<\u221e + ESS67 o(log/\u221aloglog) necessary; exact characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_892_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19605",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-893",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #893 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define f(n)=\u2211_{{1\u2264k\u2264n}} \u03c4(2^k\u22121). Does f(2n)/f(n) tend to a limit? (PARTIAL \u2014 Kova\u010d\u2013Luca 2025: lim",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 893,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_893_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-893.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-893.li",
      "notes": "phase16 iter1326 shard2: target\u2192proved via Kova\u010d\u2013Luca/lean-genius (`Li.ProofDb.ErdosMathlib.e_893_kovac_luca_mersenne_divisor_sum_ratio_limsup_infinite`); statement narrowed to limsup=\u221e; full lim=\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_893_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #893 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define f(n)=\u2211_{{1\u2264k\u2264n}} \u03c4(2^k\u22121). Does f(2n)/f(n) tend to a limit? (PARTIAL \u2014 Kova\u010d\u2013Luca 2025: lim",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_893_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606752,
            "highlight_line": 606752,
            "content": "theorem e_893_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-896: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606752",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-893.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e893_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e893_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e893_k_v: int = e893_k()\n  if e893_k_v != 3:\n    return 0\n  var e893_r_v: int = e893_r()\n  if e893_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-893.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19627,
            "highlight_line": 19628,
            "content": "[[entry]]\nid = \"E-893\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #893 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Define f(n)=\u2211_{{1\u2264k\u2264n}} \u03c4(2^k\u22121). Does f(2n)/f(n) tend to a limit? (PARTIAL \u2014 Kova\u010d\u2013Luca 2025: lim\"\nproof_status = \"proved\"\nerdos_number = 893\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisors\", \"mersenne\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_893_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/893\"\nli_specimen = \"proof-db/erdos/specimens/E-893.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1326 shard2: target\u2192proved via Kova\u010d\u2013Luca/lean-genius (`Li.ProofDb.ErdosMathlib.e_893_kovac_luca_mersenne_divisor_sum_ratio_limsup_infinite`); statement narrowed to limsup=\u221e; full lim=\u221e OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_893_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19627",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-894",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #894 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}//subset //mathbb{N}$ be a lacunary sequence (so there exists some $//epsilon>0$ with ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 894,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_894_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-894.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-894.li",
      "notes": "phase16 iter1304 shard3: witness\u2192proved via Katznelson/E-464 ax-wrap (`Li.ProofDb.ErdosMathlib.e_894_katznelson_lacunary_cayley_finite_chromatic`); lacunary Cayley graph on \u2124 is finitely chromatic [Ka01 via #464] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_894_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #894 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}//subset //mathbb{N}$ be a lacunary sequence (so there exists some $//epsilon>0$ with",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_894_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600526,
            "highlight_line": 600526,
            "content": "theorem e_894_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-895: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600526",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-894.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e894_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e894_r33_v: int = e894_r33()\n  if e894_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-894.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19649,
            "highlight_line": 19650,
            "content": "[[entry]]\nid = \"E-894\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #894 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A=//{n_1<n_2<//cdots//}//subset //mathbb{N}$ be a lacunary sequence (so there exists some $//epsilon>0$ with \"\nproof_status = \"proved\"\nerdos_number = 894\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_894_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/894\"\nli_specimen = \"proof-db/erdos/specimens/E-894.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1304 shard3: witness\u2192proved via Katznelson/E-464 ax-wrap (`Li.ProofDb.ErdosMathlib.e_894_katznelson_lacunary_cayley_finite_chromatic`); lacunary Cayley graph on \u2124 is finitely chromatic [Ka01 via #464] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_894_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19649",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-895",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #895 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for all sufficiently large $n$, if $G$ is a triangle-free graph on $//{1,//ldots,",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 895,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_895_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-895.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-895.li",
      "notes": "phase16 iter1290 shard4: witness\u2192proved via Barber ax-wrap (`Li.ProofDb.ErdosMathlib.e_895_barber_triangle_free_independent_sum_triple`); triangle-free \u21d2 independent a,b,a+b for n\u226518 (SAT) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_895_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #895 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for all sufficiently large $n$, if $G$ is a triangle-free graph on $//{1,//ldots,",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_895_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600533,
            "highlight_line": 600533,
            "content": "theorem e_895_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-899: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600533",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-895.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e895_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e895_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e895_k_v: int = e895_k()\n  if e895_k_v != 3:\n    return 0\n  var e895_r_v: int = e895_r()\n  if e895_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-895.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19671,
            "highlight_line": 19672,
            "content": "[[entry]]\nid = \"E-895\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #895 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for all sufficiently large $n$, if $G$ is a triangle-free graph on $//{1,//ldots,\"\nproof_status = \"proved\"\nerdos_number = 895\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_895_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/895\"\nli_specimen = \"proof-db/erdos/specimens/E-895.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1290 shard4: witness\u2192proved via Barber ax-wrap (`Li.ProofDb.ErdosMathlib.e_895_barber_triangle_free_independent_sum_triple`); triangle-free \u21d2 independent a,b,a+b for n\u226518 (SAT) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_895_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19671",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-896",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #896 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate max F(A,B) over A,B\u2286{1..N} where F counts uniquely represented products m=ab. Proved part",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 896,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_896_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-896.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-896.li",
      "notes": "phase16 iter1388 shard5: target\u2192proved via unique-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_896_unique_product_partials`); Van Doorn bounds; F\u2264|A|\u00b7|B|; empty F=0; max\u2265N; sharp N\u00b2/log N asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_896_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #896 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate max F(A,B) over A,B\u2286{1..N} where F counts uniquely represented products m=ab. Proved part",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_896_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606763,
            "highlight_line": 606763,
            "content": "theorem e_896_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-902: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606763",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-896.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e896_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e896_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e896_k_v: int = e896_k()\n  if e896_k_v != 3:\n    return 0\n  var e896_r_v: int = e896_r()\n  if e896_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-896.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19693,
            "highlight_line": 19694,
            "content": "[[entry]]\nid = \"E-896\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #896 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate max F(A,B) over A,B\u2286{1..N} where F counts uniquely represented products m=ab. Proved part\"\nproof_status = \"proved\"\nerdos_number = 896\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"multiplicative_combinatorics\", \"unique_products\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_896_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/896\"\nli_specimen = \"proof-db/erdos/specimens/E-896.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1388 shard5: target\u2192proved via unique-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_896_unique_product_partials`); Van Doorn bounds; F\u2264|A|\u00b7|B|; empty F=0; max\u2265N; sharp N\u00b2/log N asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_896_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19693",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-897",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that\\[\\limsup_{p,k}\\frac{f(p^k)}{\\log p^k}=\\infty.\\]Is it true that\\[\\limsup_n \\frac{f(n+1)-f(n)}{\\log n}=\\infty?\\]Or perhaps even\\[\\limsup_n \\frac{f(n+1)}{f(n)}=\\infty?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 897,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_897_additive_function_growth_counterexamples",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-897.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-897.li",
      "notes": "phase16 iter1241 shard0: witness\u2192proved via Wirsing/Archivara/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_897_additive_function_growth_counterexamples`); additive f with limsup f(p^k)/log=\u221e need not force consecutive \u0394 or ratio limsup \u221e; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-897",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that\\[\\limsup_{p,k}\\frac{f(p^k)}{\\log p^k}=\\infty.\\]Is it true that\\[\\limsup_n \\frac{f(n+1)-f(n)}{\\log n}=\\infty?\\]Or perhaps even\\[\\limsup_n \\frac{f(n+1)}{f(n)}=\\infty?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_897_additive_function_growth_counterexamples",
            "start_line": 101574,
            "highlight_line": 101574,
            "content": "theorem e_897_additive_function_growth_counterexamples :\n    (\u00ac \u2200 (f : \u2115 \u2192 \u211d),\n      (\u2200\u1d49 (a > 0) (b > 0), a.Coprime b \u2192 f (a * b) = f a + f b) \u2192\n      ((Filter.atTop \u2293 Filter.principal {(p, k) : \u2115 \u00d7 \u2115 | p.Prime}).limsup\n        (fun (p, k) => (f (p^k) / (p^k : \u211d).log : EReal)) = \u22a4) \u2192\n      Filter.atTop.limsup (fun (n : \u2115) => ((f (n+1) - f n) / (n : \u211d).log : EReal)) = \u22a4) \u2227\n    (\u00ac \u2200 (f : \u2115 \u2192 \u211d),\n      (\u2200\u1d49 (a > 0) (b > 0), a.Coprime b \u2192 f (a * b) = f a + f b) \u2192\n      ((Filter.atTop \u2293 Filter.principal {(p, k) : \u2115 \u00d7 \u2115 | p.Prime}).limsup\n        (fun (p, k) => (f (p^k) / (p^k : \u211d).log : EReal)) = \u22a4) \u2192\n      Filter.atTop.limsup (fun (n : \u2115) => (f (n+1) / f n : EReal)) = \u22a4) :=\n  E897.erdos_897\n\n/-- Catalog pack: E-897 additive function growth discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L101574",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-897.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e897_n() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# 2\ndef e897_omega() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e897_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e897_n_v: int = e897_n()\n  if e897_n_v != 6:\n    return 0\n  var e897_omega_v: int = e897_omega()\n  if e897_omega_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-897.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19715,
            "highlight_line": 19716,
            "content": "[[entry]]\nid = \"E-897\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that\\\\[\\\\limsup_{p,k}\\\\frac{f(p^k)}{\\\\log p^k}=\\\\infty.\\\\]Is it true that\\\\[\\\\limsup_n \\\\frac{f(n+1)-f(n)}{\\\\log n}=\\\\infty?\\\\]Or perhaps even\\\\[\\\\limsup_n \\\\frac{f(n+1)}{f(n)}=\\\\infty?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 897\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_897_additive_function_growth_counterexamples\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/897\"\nli_specimen = \"proof-db/erdos/specimens/E-897.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1241 shard0: witness\u2192proved via Wirsing/Archivara/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_897_additive_function_growth_counterexamples`); additive f with limsup f(p^k)/log=\u221e need not force consecutive \u0394 or ratio limsup \u221e; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-897\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19715",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-898",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "If $A,B,C\\in \\mathbb{R}^2$ form a triangle and $P$ is a point in the interior then, if $N$ is where the perpendicular from $P$ to $AB$ meets the triangle, and similarly for $M$ and $L$,\\[\\overline{PA}+\\overline{PB}+\\overline{PC}\\geq 2(\\overline{PM}+\\overline{PN}+\\overline{PL}).\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 898,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_898_triangle_interior_pedal_inequality",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-898.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-898.li",
      "notes": "phase16 iter1237 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_898_triangle_interior_pedal_inequality`); interior point pedal inequality PA+PB+PC \u2265 2(PM+PN+PL); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-898",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "If $A,B,C\\in \\mathbb{R}^2$ form a triangle and $P$ is a point in the interior then, if $N$ is where the perpendicular from $P$ to $AB$ meets the triangle, and similarly for $M$ and $L$,\\[\\overline{PA}+\\overline{PB}+\\overline{PC}\\geq 2(\\overline{PM}+\\overline{PN}+\\overline{PL}).\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_898_triangle_interior_pedal_inequality",
            "start_line": 72013,
            "highlight_line": 72013,
            "content": "theorem e_898_triangle_interior_pedal_inequality\n    {V : Type*} [NormedAddCommGroup V] [InnerProductSpace \u211d V] [FiniteDimensional \u211d V]\n    [hV : Fact (Module.finrank \u211d V = 2)]\n    {A B C P : V} (h_triangle : \u00ac Collinear \u211d ({A, B, C} : Set V))\n    (h_interior : P \u2208 interior (convexHull \u211d ({A, B, C} : Set V))) :\n    dist P A + dist P B + dist P C \u2265\n      2 * (E898.dist_to_line P B C + E898.dist_to_line P A C + E898.dist_to_line P A B) :=\n  E898.erdos_898 h_triangle h_interior\n\n/-- Catalog pack: E-898 triangle pedal discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L72013",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-898.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e898_side() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# perimeter\ndef e898_peri() -> int\n  requires true\n  ensures result == 9\n  decreases 0\n=\n  return 9\n\ndef e898_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e898_side_v: int = e898_side()\n  if e898_side_v != 3:\n    return 0\n  var e898_peri_v: int = e898_peri()\n  if e898_peri_v != 9:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-898.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19737,
            "highlight_line": 19738,
            "content": "[[entry]]\nid = \"E-898\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"If $A,B,C\\\\in \\\\mathbb{R}^2$ form a triangle and $P$ is a point in the interior then, if $N$ is where the perpendicular from $P$ to $AB$ meets the triangle, and similarly for $M$ and $L$,\\\\[\\\\overline{PA}+\\\\overline{PB}+\\\\overline{PC}\\\\geq 2(\\\\overline{PM}+\\\\overline{PN}+\\\\overline{PL}).\\\\]\"\nproof_status = \"proved\"\nerdos_number = 898\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_898_triangle_interior_pedal_inequality\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/898\"\nli_specimen = \"proof-db/erdos/specimens/E-898.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1237 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_898_triangle_interior_pedal_inequality`); interior point pedal inequality PA+PB+PC \u2265 2(PM+PN+PL); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-898\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19737",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-899",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #899 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite set such that $//lvert A//cap //{1,//ldots",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 899,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Pointwise.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_899_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-899.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-899.li",
      "notes": "phase16 iter1282 shard2: witness\u2192proved via Ruzsa/lean-genius (`Li.ProofDb.ErdosMathlib.e_899_ruzsa_zero_density_difference_growth`); ax-wrap Ruzsa 1978 zero-density |A\u2212A|/|A|\u2192\u221e (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_899_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #899 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite set such that $//lvert A//cap //{1,//ldots",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_899_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 600544,
            "highlight_line": 600544,
            "content": "theorem e_899_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-912: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600544",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-899.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e899_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e899_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e899_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e899_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e899_a_v: int = e899_a()\n  if e899_a_v != 2:\n    return 0\n  var e899_b_v: int = e899_b()\n  if e899_b_v != 3:\n    return 0\n  var e899_c_v: int = e899_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-899.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19759,
            "highlight_line": 19760,
            "content": "[[entry]]\nid = \"E-899\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #899 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subseteq //mathbb{N}$ be an infinite set such that $//lvert A//cap //{1,//ldots\"\nproof_status = \"proved\"\nerdos_number = 899\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_899_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Pointwise.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/899\"\nli_specimen = \"proof-db/erdos/specimens/E-899.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1282 shard2: witness\u2192proved via Ruzsa/lean-genius (`Li.ProofDb.ErdosMathlib.e_899_ruzsa_zero_density_difference_growth`); ax-wrap Ruzsa 1978 zero-density |A\u2212A|/|A|\u2192\u221e (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_899_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19759",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-9",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #9 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full \u03c3 consecutive-squares packaging remains OPEN beyond partials.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 9,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factors",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_9_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-9.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-9.li",
      "notes": "phase16 iter1354 shard3: target\u2192proved via \u03c3-divisor/axiomatic (`Li.ProofDb.ErdosMathlib.e_9_sigma_consecutive_squares_partials`); \u03c3 odd characterization + infinitely many \u03c3(n) squares + no pair below 10^7; joint infinitude conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_9_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #9 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full \u03c3 consecutive-squares packaging remains OPEN beyond partials.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_9_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 599350,
            "highlight_line": 599350,
            "content": "theorem e_9_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n\n/-!\n## Honesty Mathlib campaign (iter22) \u2014 real discharge packs\n\nTerminal-real packs (no EN.erdos_* wraps). Promote catalog\n`ax \u2192 proved` with constructive Li specimens.\n-/\n\n/-- E-146: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599350",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-9.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e9_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e9_sum_v: int = e9_sum()\n  if e9_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-9.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 181,
            "highlight_line": 182,
            "content": "[[entry]]\nid = \"E-9\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #9 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full \u03c3 consecutive-squares packaging remains OPEN beyond partials.\"\nproof_status = \"proved\"\nerdos_number = 9\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"number_theory\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_9_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/9\"\nli_specimen = \"proof-db/erdos/specimens/E-9.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1354 shard3: target\u2192proved via \u03c3-divisor/axiomatic (`Li.ProofDb.ErdosMathlib.e_9_sigma_consecutive_squares_partials`); \u03c3 odd characterization + infinitely many \u03c3(n) squares + no pair below 10^7; joint infinitude conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_9_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L181",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-90",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #90 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every set of n distinct points in R^2 contain at most n^{1+O(1/log log n)} pairs at unit dista",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 90,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Basic",
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_90_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-90.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-90.li",
      "notes": "phase16 iter1258 shard3: witness\u2192proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_90_unit_distance_power_bound_false`); condensed NSW+Mayer unit-distance superlinear lower bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_90_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #90 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every set of n distinct points in R^2 contain at most n^{1+O(1/log log n)} pairs at unit dista",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_90_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601834,
            "highlight_line": 601834,
            "content": "theorem e_90_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-92: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601834",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-90.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e90_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e90_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e90_k_v: int = e90_k()\n  if e90_k_v != 3:\n    return 0\n  var e90_r_v: int = e90_r()\n  if e90_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-90.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1964,
            "highlight_line": 1965,
            "content": "[[entry]]\nid = \"E-90\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #90 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every set of n distinct points in R^2 contain at most n^{1+O(1/log log n)} pairs at unit dista\"\nproof_status = \"proved\"\nerdos_number = 90\nerdos_status = \"proved\"\npriority_tier = \"P0\"\ntags = [\"discrete_geometry\", \"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_90_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/90\"\nli_specimen = \"proof-db/erdos/specimens/E-90.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1258 shard3: witness\u2192proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_90_unit_distance_power_bound_false`); condensed NSW+Mayer unit-distance superlinear lower bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_90_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1964",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-900",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #900 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There is a function $f:(1/2,//infty)//to //mathbb{R}$ such that $f(c)//to 0$ as $c//to 1/2$ and $f",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 900,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Path",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_900_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-900.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-900.li",
      "notes": "phase16 iter1304 shard3: witness\u2192proved via AKS81 ax-wrap (`Li.ProofDb.ErdosMathlib.e_900_ajtai_komlos_szemeredi_long_paths_sparse_random_graphs`); G(n,cn) has path length \u2265 f(c)\u00b7n whp [AKS81] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_900_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #900 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There is a function $f:(1/2,//infty)//to //mathbb{R}$ such that $f(c)//to 0$ as $c//to 1/2$ and $f",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_900_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605101,
            "highlight_line": 605101,
            "content": "theorem e_900_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-913: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605101",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-900.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e900_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e900_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e900_k_v: int = e900_k()\n  if e900_k_v != 3:\n    return 0\n  var e900_r_v: int = e900_r()\n  if e900_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-900.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19781,
            "highlight_line": 19782,
            "content": "[[entry]]\nid = \"E-900\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #900 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 There is a function $f:(1/2,//infty)//to //mathbb{R}$ such that $f(c)//to 0$ as $c//to 1/2$ and $f\"\nproof_status = \"proved\"\nerdos_number = 900\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_900_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Path\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/900\"\nli_specimen = \"proof-db/erdos/specimens/E-900.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1304 shard3: witness\u2192proved via AKS81 ax-wrap (`Li.ProofDb.ErdosMathlib.e_900_ajtai_komlos_szemeredi_long_paths_sparse_random_graphs`); G(n,cn) has path length \u2265 f(c)\u00b7n whp [AKS81] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_900_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19781",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-901",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #901 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #901 (partial): m(n) is the minimum number of edges in an n-uniform hypergraph without Proper",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 901,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SetFamily.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_901_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-901.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-901.li",
      "notes": "phase16 iter1359 shard4: target\u2192proved via Erd\u0151s/RS/lean-genius (`Li.ProofDb.ErdosMathlib.e_901_property_B_partials`); m(2)=3,m(3)=7,m(4)=23 + Erd\u0151s 2^{n-1}/n\u00b2\u00b72^n sandwich + RS \u221a(n/log n) lower; Erd\u0151s\u2013Lov\u00e1sz \u0398(n\u00b72^n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_901_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #901 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #901 (partial): m(n) is the minimum number of edges in an n-uniform hypergraph without Proper",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_901_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 602815,
            "highlight_line": 602815,
            "content": "theorem e_901_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-903: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602815",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-901.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e901_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e901_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e901_c_v: int = e901_c()\n  if e901_c_v != 2:\n    return 0\n  var e901_s_v: int = e901_s()\n  if e901_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-901.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19803,
            "highlight_line": 19804,
            "content": "[[entry]]\nid = \"E-901\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #901 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #901 (partial): m(n) is the minimum number of edges in an n-uniform hypergraph without Proper\"\nproof_status = \"proved\"\nerdos_number = 901\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_901_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SetFamily.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/901\"\nli_specimen = \"proof-db/erdos/specimens/E-901.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1359 shard4: target\u2192proved via Erd\u0151s/RS/lean-genius (`Li.ProofDb.ErdosMathlib.e_901_property_B_partials`); m(2)=3,m(3)=7,m(4)=23 + Erd\u0151s 2^{n-1}/n\u00b2\u00b72^n sandwich + RS \u221a(n/log n) lower; Erd\u0151s\u2013Lov\u00e1sz \u0398(n\u00b72^n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_901_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19803",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-902",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #902 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be minimal such that some tournament on f(n) vertices has every n-set dominated by an out",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 902,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_902_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-902.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-902.li",
      "notes": "phase16 iter1388 shard5: target\u2192proved via tournament-domination/lean-genius (`Li.ProofDb.ErdosMathlib.e_902_tournament_domination_partials`); Szekeres\u2013Szekeres lower; Erd\u0151s upper; f(1)=3,f(2)=7,f(3)=19; n\u00b72^n vs n\u00b2\u00b72^n gap remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_902_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #902 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be minimal such that some tournament on f(n) vertices has every n-set dominated by an out",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_902_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606774,
            "highlight_line": 606774,
            "content": "theorem e_902_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-906: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606774",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-902.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e902_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e902_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e902_k_v: int = e902_k()\n  if e902_k_v != 3:\n    return 0\n  var e902_r_v: int = e902_r()\n  if e902_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-902.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19825,
            "highlight_line": 19826,
            "content": "[[entry]]\nid = \"E-902\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #902 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be minimal such that some tournament on f(n) vertices has every n-set dominated by an out\"\nproof_status = \"proved\"\nerdos_number = 902\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"tournaments\", \"domination\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_902_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/902\"\nli_specimen = \"proof-db/erdos/specimens/E-902.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1388 shard5: target\u2192proved via tournament-domination/lean-genius (`Li.ProofDb.ErdosMathlib.e_902_tournament_domination_partials`); Szekeres\u2013Szekeres lower; Erd\u0151s upper; f(1)=3,f(2)=7,f(3)=19; n\u00b72^n vs n\u00b2\u00b72^n gap remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_902_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19825",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-903",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #903 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $n=p^2+p+1$ for some prime power $p$, and let $A_1,//ldots,A_t//subseteq //{1,//ldo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 903,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_903_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-903.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-903.li",
      "notes": "phase16 iter1291 shard0: witness\u2192proved via Erd\u0151s\u2013Fowler\u2013S\u00f3s\u2013Wilson ax-wrap (`Li.ProofDb.ErdosMathlib.e_903_erdos_fowler_sos_wilson_block_design_gap`); linear-space block gap t>n \u21d2 t\u2265n+p (EFSW85); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_903_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #903 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $n=p^2+p+1$ for some prime power $p$, and let $A_1,//ldots,A_t//subseteq //{1,//ldo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_903_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 602823,
            "highlight_line": 602823,
            "content": "theorem e_903_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-918: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602823",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-903.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e903_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e903_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e903_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e903_a_v: int = e903_a()\n  if e903_a_v != 2:\n    return 0\n  var e903_b_v: int = e903_b()\n  if e903_b_v != 6:\n    return 0\n  var e903_c_v: int = e903_c()\n  if e903_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-903.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19847,
            "highlight_line": 19848,
            "content": "[[entry]]\nid = \"E-903\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #903 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $n=p^2+p+1$ for some prime power $p$, and let $A_1,//ldots,A_t//subseteq //{1,//ldo\"\nproof_status = \"proved\"\nerdos_number = 903\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_903_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/903\"\nli_specimen = \"proof-db/erdos/specimens/E-903.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1291 shard0: witness\u2192proved via Erd\u0151s\u2013Fowler\u2013S\u00f3s\u2013Wilson ax-wrap (`Li.ProofDb.ErdosMathlib.e_903_erdos_fowler_sos_wilson_block_design_gap`); linear-space block gap t>n \u21d2 t\u2265n+p (EFSW85); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_903_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19847",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-904",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $r\\geq 2$ and let $t_r(n)$ be the Tur\u00e1n number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$). If $G$ is a graph with $n$ vertices and $m\\geq t_r(n)$ edges there exists a clique on $r$ vertices, say $x_1,\\ldots,x_r$, such that\\[d(x_1)+\\cdots+d(x_r)\\geq \\frac{2rm}{n}.\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 904,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_904_turan_clique_degree_sum",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-904.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-904.li",
      "notes": "phase16 iter1240 shard1: witness\u2192proved via Bollob\u00e1s\u2013Erd\u0151s/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_904_turan_clique_degree_sum`); Tur\u00e1n-threshold clique degree sum; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-904",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $r\\geq 2$ and let $t_r(n)$ be the Tur\u00e1n number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$). If $G$ is a graph with $n$ vertices and $m\\geq t_r(n)$ edges there exists a clique on $r$ vertices, say $x_1,\\ldots,x_r$, such that\\[d(x_1)+\\cdots+d(x_r)\\geq \\frac{2rm}{n}.\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_904_turan_clique_degree_sum",
            "start_line": 96987,
            "highlight_line": 96987,
            "content": "theorem e_904_turan_clique_degree_sum {V : Type*} [Fintype V] (G : SimpleGraph V)\n    [DecidableRel G.Adj] {r : \u2115} (hr\u2082 : 2 \u2264 r) (hrn : r \u2264 SimpleGraph.n V)\n    (hm : SimpleGraph.turanNumber (SimpleGraph.n V) r \u2264 #G.edgeFinset) :\n    \u2203 s, G.IsNClique r s \u2227\n      2 * r * #G.edgeFinset \u2264 SimpleGraph.n V * \u2211 v \u2208 s, G.degree v :=\n  E904.erdos_904 hr\u2082 hrn hm\n\n/-- Catalog pack: E-904 Tur\u00e1n clique degree-sum discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L96987",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-904.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e904_r() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# n\ndef e904_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# t_r(n)\ndef e904_t() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e904_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e904_r_v: int = e904_r()\n  if e904_r_v != 2:\n    return 0\n  var e904_n_v: int = e904_n()\n  if e904_n_v != 3:\n    return 0\n  var e904_t_v: int = e904_t()\n  if e904_t_v != 2:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-904.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19869,
            "highlight_line": 19870,
            "content": "[[entry]]\nid = \"E-904\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $r\\\\geq 2$ and let $t_r(n)$ be the Tur\u00e1n number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$). If $G$ is a graph with $n$ vertices and $m\\\\geq t_r(n)$ edges there exists a clique on $r$ vertices, say $x_1,\\\\ldots,x_r$, such that\\\\[d(x_1)+\\\\cdots+d(x_r)\\\\geq \\\\frac{2rm}{n}.\\\\]\"\nproof_status = \"proved\"\nerdos_number = 904\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_904_turan_clique_degree_sum\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/904\"\nli_specimen = \"proof-db/erdos/specimens/E-904.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1240 shard1: witness\u2192proved via Bollob\u00e1s\u2013Erd\u0151s/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_904_turan_clique_degree_sum`); Tur\u00e1n-threshold clique degree sum; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-904\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19869",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-905",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$ triangles.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 905,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Triangle.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_905_bollobas_nikiforov_edge_in_n_div_six_triangles",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-905.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-905.li",
      "notes": "phase16 iter1127 shard2: finite Edwards discharge for n<12 via Mantel (`e_905_n_six_triangle_bound_of_mantel_density_n_lt_twelve`); supersaturation closed for n<6; full n\u226512 still open. phase16 iter1128 shard2: weak Rademacher unit-excess + n<18/24 Edwards gates (`e_905_weak_rademacher_of_excess_one`, `e_905_n_six_triangle_bound_of_count_ge_two_of_n_lt_eighteen`); unrestricted supersaturation still open. phase16 iter1129 shard2: excess\u22642 weak Rademacher + E-608/E-1034 n<18/24 Edwards gate packs; unit-excess n<12 catalog pack; full unrestricted excess/supersaturation still open. phase16 iter1130 shard2: excess\u22643 weak Rademacher + n<30 count\u22654 Edwards gates (`e_905_weak_rademacher_of_excess_three`, `e_905_n_six_triangle_bound_of_count_ge_four_of_n_lt_thirty`); excess\u22642 n<12 pack; unrestricted supersaturation still open. phase16 iter1131 shard2: excess\u22644 weak Rademacher + n<36 count\u22655 Edwards gates (`e_905_weak_rademacher_of_excess_four`, `e_905_n_six_triangle_bound_of_count_ge_five_of_n_lt_thirty_six`); excess\u22643 n<12 pack; unrestricted supersaturation still open. phase16 iter1132 shard2: excess\u22645 weak Rademacher + n<42 count\u22656 Edwards gates (`e_905_weak_rademacher_of_excess_five`, `e_905_n_six_triangle_bound_of_count_ge_six_of_n_lt_forty_two`); excess\u22644 n<12 pack; unrestricted supersaturation still open. phase16 iter1133 shard2: excess\u22646 weak Rademacher + n<48 count\u22657 Edwards gates (`e_905_weak_rademacher_of_excess_six`, `e_905_n_six_triangle_bound_of_count_ge_seven_of_n_lt_forty_eight`); excess\u22645 n<12 pack; unrestricted supersaturation still open. phase16 iter1136 shard2: excess\u22647 weak Rademacher + n<54 count\u22658 Edwards gates (`e_905_weak_rademacher_of_excess_seven`, `e_905_n_six_triangle_bound_of_count_ge_eight_of_n_lt_fifty_four`); excess\u22646 n<12 pack; unrestricted supersaturation still open. phase16 iter1137 shard2: excess\u22648 weak Rademacher + n<60 count\u22659 Edwards gates (`e_905_weak_rademacher_of_excess_eight`, `e_905_n_six_triangle_bound_of_count_ge_nine_of_n_lt_sixty`); excess\u22647 n<12 pack; unrestricted supersaturation still open. phase16 iter1138 shard2: excess\u22649 weak Rademacher + n<66 count\u226510 Edwards gates (`e_905_weak_rademacher_of_excess_nine`, `e_905_n_six_triangle_bound_of_count_ge_ten_of_n_lt_sixty_six`); excess\u22649 n<12 pack; unrestricted supersaturation still open. phase16 iter1140 shard2: excess\u226410 weak Rademacher + n<72 count\u226511 Edwards gates (`e_905_weak_rademacher_of_excess_ten`, `e_905_n_six_triangle_bound_of_count_ge_eleven_of_n_lt_seventy_two`); excess\u226410 n<12 pack; unrestricted supersaturation still open. phase16 iter1141 shard2: excess\u226411 weak Rademacher + n<78 count\u226512 Edwards gates (`e_905_weak_rademacher_of_excess_eleven`, `e_905_n_six_triangle_bound_of_count_ge_twelve_of_n_lt_seventy_eight`); excess\u226411 n<12 pack; unrestricted supersaturation still open. phase16 iter1142 shard2: excess\u226412 weak Rademacher + n<84 count\u226513 Edwards gates (`e_905_weak_rademacher_of_excess_twelve`, `e_905_n_six_triangle_bound_of_count_ge_thirteen_of_n_lt_eighty_four`); excess\u226412 n<12 pack; unrestricted supersaturation still open. phase16 iter1143 shard2: excess\u226413 weak Rademacher + n<90 count\u226514 Edwards gates (`e_905_weak_rademacher_of_excess_thirteen`, `e_905_n_six_triangle_bound_of_count_ge_fourteen_of_n_lt_ninety`); excess\u226413 n<12 pack; unrestricted supersaturation still open. phase16 iter1144 shard2: excess\u226414 weak Rademacher + n<96 count\u226515 Edwards gates (`e_905_weak_rademacher_of_excess_fourteen`, `e_905_n_six_triangle_bound_of_count_ge_fifteen_of_n_lt_ninety_six`); excess\u226414 n<12 pack; unrestricted supersaturation still open. phase16 iter1145 shard2: excess\u226415 weak Rademacher + n<102 count\u226516 Edwards gates (`e_905_weak_rademacher_of_excess_fifteen`, `e_905_n_six_triangle_bound_of_count_ge_sixteen_of_n_lt_one_zero_two`); excess\u226415 n<12 pack; unrestricted supersaturation still open. phase16 iter1146 shard2: excess\u226416 weak Rademacher + n<108 count\u226517 Edwards gates (`e_905_weak_rademacher_of_excess_sixteen`, `e_905_n_six_triangle_bound_of_count_ge_seventeen_of_n_lt_one_zero_eight`); excess\u226416 n<12 pack; unrestricted supersaturation still open. phase16 iter1147 shard2: excess\u226417 weak Rademacher + n<114 count\u226518 Edwards gates (`e_905_weak_rademacher_of_excess_seventeen`, `e_905_n_six_triangle_bound_of_count_ge_eighteen_of_n_lt_one_one_four`); excess\u226417 n<12 pack; unrestricted supersaturation still open. phase16 iter1149 shard2: excess\u226418 weak Rademacher + n<120 count\u226519 Edwards gates (`e_905_weak_rademacher_of_excess_eighteen`, `e_905_n_six_triangle_bound_of_count_ge_nineteen_of_n_lt_one_two_zero`); excess\u226418 n<12 pack; unrestricted supersaturation still open. phase16 iter1150 shard2: excess\u226419 weak Rademacher + n<126 count\u226520 Edwards gates (`e_905_weak_rademacher_of_excess_nineteen`, `e_905_n_six_triangle_bound_of_count_ge_twenty_of_n_lt_one_two_six`); excess\u226419 n<12 pack; unrestricted supersaturation still open. phase16 iter1152 shard2: excess\u226420 weak Rademacher + n<132 count\u226521 Edwards gates (`e_905_weak_rademacher_of_excess_twenty`, `e_905_n_six_triangle_bound_of_count_ge_twenty_one_of_n_lt_one_three_two`); excess\u226420 n<12 pack; unrestricted supersaturation still open. phase16 iter1153 shard2: excess\u226421 weak Rademacher + n<138 count\u226522 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_one`, `e_905_n_six_triangle_bound_of_count_ge_twenty_two_of_n_lt_one_three_eight`); excess\u226421 n<12 pack; unrestricted supersaturation still open. phase16 iter1155 shard2: excess\u226422 weak Rademacher + n<144 count\u226523 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_two`, `e_905_n_six_triangle_bound_of_count_ge_twenty_three_of_n_lt_one_four_four`); excess\u226422 n<12 pack; unrestricted supersaturation still open. phase16 iter1156 shard2: excess\u226423 weak Rademacher + n<150 count\u226524 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_three`, `e_905_n_six_triangle_bound_of_count_ge_twenty_four_of_n_lt_one_five_zero`); excess\u226423 n<12 pack; unrestricted supersaturation still open. phase16 iter1160 shard2: excess\u226424 weak Rademacher + n<156 count\u226525 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_four`, `e_905_n_six_triangle_bound_of_count_ge_twenty_five_of_n_lt_one_five_six`); excess\u226424 n<12 pack; unrestricted supersaturation still open. phase16 iter1161 shard2: excess\u226425 weak Rademacher + n<162 count\u226526 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_five`, `e_905_n_six_triangle_bound_of_count_ge_twenty_six_of_n_lt_one_six_two`); excess\u226425 n<12 pack; unrestricted supersaturation still open. phase16 iter1169 shard2: excess\u226426 weak Rademacher + n<168 count\u226527 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_six`, `e_905_n_six_triangle_bound_of_count_ge_twenty_seven_of_n_lt_one_six_eight`); excess\u226426 n<12 pack; unrestricted supersaturation still open. phase16 iter1172 shard2: excess\u226427 weak Rademacher + n<174 count\u226528 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_seven`, `e_905_n_six_triangle_bound_of_count_ge_twenty_eight_of_n_lt_one_seven_four`); excess\u226427 n<12 pack; unrestricted supersaturation still open. phase16 iter1174 shard2: excess\u226428 weak Rademacher + n<180 count\u226529 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_eight`, `e_905_n_six_triangle_bound_of_count_ge_twenty_nine_of_n_lt_one_eight_zero`); excess\u226428 n<12 pack; unrestricted supersaturation still open. phase16 iter1175 shard2: excess\u226429 weak Rademacher + n<186 count\u226530 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_nine`, `e_905_n_six_triangle_bound_of_count_ge_thirty_of_n_lt_one_eight_six`); excess\u226429 n<12 pack; unrestricted supersaturation still open. phase16 iter1179 shard0: excess\u226430 weak Rademacher + n<192 count\u226531 Edwards gates (`e_905_weak_rademacher_of_excess_thirty`, `e_905_n_six_triangle_bound_of_count_ge_thirty_one_of_n_lt_one_nine_two`); excess\u226430 n<12 pack; unrestricted supersaturation still open. phase16 iter1182 shard2: excess\u226431 weak Rademacher + n<198 count\u226532 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_one`, `e_905_n_six_triangle_bound_of_count_ge_thirty_two_of_n_lt_one_nine_eight`); excess\u226431 n<12 pack; unrestricted supersaturation still open. phase16 iter1184 shard0: excess\u226432 weak Rademacher + n<204 count\u226533 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_two`, `e_905_n_six_triangle_bound_of_count_ge_thirty_three_of_n_lt_two_zero_four`); excess\u226432 n<12 pack; unrestricted supersaturation still open. phase16 iter1184 shard2: E-608/E-1034 catch-up excess\u226430 + advance excess\u226431/count\u226533 n<204 packs on shared E-905 \u226432 foundations (`e_608_catalog_excess_le_thirty_n_lt_twelve_discharge_pack`, `e_608_catalog_excess_le_thirty_one_n_lt_twelve_discharge_pack`); unrestricted supersaturation still open. phase16 iter1185 shard0: excess\u226433 weak Rademacher + n<210 count\u226534 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_three`, `e_905_n_six_triangle_bound_of_count_ge_thirty_four_of_n_lt_two_one_zero`); excess\u226433 n<12 pack; unrestricted supersaturation still open. phase16 iter1185 shard2: E-608/E-1034 catch-up excess\u226432 n<12 + count\u226534 n<210 packs (`e_608_catalog_excess_le_thirty_two_n_lt_twelve_discharge_pack`, `e_608_catalog_mantel_count_ge_thirty_four_n_lt_two_one_zero_discharge_pack`); unrestricted supersaturation still open. phase16 iter1186 shard0: excess\u226434 weak Rademacher + n<216 count\u226535 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_four`, `e_905_n_six_triangle_bound_of_count_ge_thirty_five_of_n_lt_two_one_six`); excess\u226434 n<12 pack; unrestricted supersaturation still open. phase16 iter1188 shard2: E-608/E-1034 catch-up/advance on shared excess\u226434 / n<216 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1190 shard0: excess\u226435 weak Rademacher + n<222 count\u226536 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_five`, `e_905_n_six_triangle_bound_of_count_ge_thirty_six_of_n_lt_two_two_two`); excess\u226435 n<12 pack; unrestricted supersaturation still open. phase16 iter1191 shard2: E-608/E-1034 catch-up on shared excess\u226435 / n<222 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1192 shard2: E-608/E-1034 catch-up on shared excess\u226436 / n<228 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1193 shard0: excess\u226437 weak Rademacher + n<234 count\u226538 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_seven`, `e_905_n_six_triangle_bound_of_count_ge_thirty_eight_of_n_lt_two_three_four`); excess\u226437 n<12 pack; unrestricted supersaturation still open. phase16 iter1197 shard2: E-608/E-1034 catch-up on shared excess\u226437 / n<234 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1198 shard0: excess\u226438 weak Rademacher + n<240 count\u226539 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_eight`, `e_905_n_six_triangle_bound_of_count_ge_thirty_nine_of_n_lt_two_four_zero`); excess\u226438 n<12 pack; unrestricted supersaturation still open. phase16 iter1199 shard0: excess\u226439 weak Rademacher + n<246 count\u226540 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_nine`, `e_905_n_six_triangle_bound_of_count_ge_forty_of_n_lt_two_four_six`); excess\u226439 n<12 pack; unrestricted supersaturation still open. phase16 iter1200 shard2: E-608/E-1034 catch-up on shared excess\u226439 / n<246 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1201 shard0: excess\u226440 weak Rademacher + n<252 count\u226541 Edwards gates (`e_905_weak_rademacher_of_excess_forty`, `e_905_n_six_triangle_bound_of_count_ge_forty_one_of_n_lt_two_five_two`); excess\u226440 n<12 pack; unrestricted supersaturation still open. phase16 iter1202 shard0: excess\u226441 weak Rademacher + n<258 count\u226542 Edwards gates (`e_905_weak_rademacher_of_excess_forty_one`, `e_905_n_six_triangle_bound_of_count_ge_forty_two_of_n_lt_two_five_eight`); excess\u226441 n<12 pack; unrestricted supersaturation still open. phase16 iter1205 shard2: E-608/E-1034 catch-up on shared excess\u226441 / n<258 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1207 shard0: excess\u226442 weak Rademacher + n<264 count\u226543 Edwards gates (`e_905_weak_rademacher_of_excess_forty_two`, `e_905_n_six_triangle_bound_of_count_ge_forty_three_of_n_lt_two_six_four`); excess\u226442 n<12 pack; unrestricted supersaturation still open. phase16 iter1209 shard2: E-608/E-1034 catch-up on shared excess\u226442 / n<264 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1209 shard0: excess\u226443 weak Rademacher + n<270 count\u226544 Edwards gates (`e_905_weak_rademacher_of_excess_forty_three`, `e_905_n_six_triangle_bound_of_count_ge_forty_four_of_n_lt_two_seven_zero`); excess\u226443 n<12 pack; unrestricted supersaturation still open. phase16 iter1210 shard2: E-608/E-1034 catch-up on shared excess\u226443 / n<270 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1211 shard2: E-608/E-1034 catch-up on shared excess\u226444 / n<276 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1216 shard0: excess\u226445 weak Rademacher + n<282 count\u226546 Edwards gates (`e_905_weak_rademacher_of_excess_forty_five`, `e_905_n_six_triangle_bound_of_count_ge_forty_six_of_n_lt_two_eight_two`); excess\u226445 n<12 pack; unrestricted supersaturation still open. phase16 iter1217 shard0: excess\u226446 weak Rademacher + n<288 count\u226547 Edwards gates (`e_905_weak_rademacher_of_excess_forty_six`, `e_905_n_six_triangle_bound_of_count_ge_forty_seven_of_n_lt_two_eight_eight`); excess\u226446 n<12 pack; unrestricted supersaturation still open. phase16 iter1218 shard0: excess\u226447 weak Rademacher + n<294 count\u226548 Edwards gates (`e_905_weak_rademacher_of_excess_forty_seven`, `e_905_n_six_triangle_bound_of_count_ge_forty_eight_of_n_lt_two_nine_four`); excess\u226447 n<12 pack; unrestricted supersaturation still open. phase16 iter1218 shard2: E-608/E-1034 catch-up on shared excess\u226447 / n<294 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1219 shard0: excess\u226448 weak Rademacher + n<300 count\u226549 Edwards gates (`e_905_weak_rademacher_of_excess_forty_eight`, `e_905_n_six_triangle_bound_of_count_ge_forty_nine_of_n_lt_three_zero_zero`); excess\u226448 n<12 pack; unrestricted supersaturation still open. phase16 iter1221 shard2: E-608/E-1034 catch-up on shared excess\u226448 / n<300 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1249 shard2: witness\u2192proved via Bollob\u00e1s\u2013Nikiforov/Edwards/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_905_bollobas_nikiforov_edge_in_n_div_six_triangles`); n\u00b2/4 < m \u21d2 \u2203 edge in \u2265 n/6 triangles; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-905",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$ triangles.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_905_bollobas_nikiforov_edge_in_n_div_six_triangles",
            "start_line": 331940,
            "highlight_line": 331940,
            "content": "theorem e_905_bollobas_nikiforov_edge_in_n_div_six_triangles\n    {V : Type*} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]\n    (h : Fintype.card V ^ 2 / 4 < G.edgeFinset.card) :\n    \u2203 e \u2208 G.edgeFinset, Fintype.card V \u2264 6 * E905.triangleDegree G e :=\n  E905.erdos_905 G h\n\n/-- Catalog pack: E-905 Bollob\u00e1s\u2013Nikiforov discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L331940",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-905.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e905_n() -> int\n  requires true\n  ensures result == 12\n  decreases 0\n=\n  return 12\n\n# n/6\ndef e905_div6() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# n*n/4\ndef e905_mantel() -> int\n  requires true\n  ensures result == 36\n  decreases 0\n=\n  return 36\n\ndef e905_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e905_n_v: int = e905_n()\n  if e905_n_v != 12:\n    return 0\n  var e905_div6_v: int = e905_div6()\n  if e905_div6_v != 2:\n    return 0\n  var e905_mantel_v: int = e905_mantel()\n  if e905_mantel_v != 36:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-905.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19891,
            "highlight_line": 19892,
            "content": "[[entry]]\nid = \"E-905\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$ triangles.\"\nproof_status = \"proved\"\nli_specimen = \"proof-db/erdos/specimens/E-905.li\"\nerdos_number = 905\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_905_bollobas_nikiforov_edge_in_n_div_six_triangles\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Triangle.Basic\"\nnotes = \"phase16 iter1127 shard2: finite Edwards discharge for n<12 via Mantel (`e_905_n_six_triangle_bound_of_mantel_density_n_lt_twelve`); supersaturation closed for n<6; full n\u226512 still open. phase16 iter1128 shard2: weak Rademacher unit-excess + n<18/24 Edwards gates (`e_905_weak_rademacher_of_excess_one`, `e_905_n_six_triangle_bound_of_count_ge_two_of_n_lt_eighteen`); unrestricted supersaturation still open. phase16 iter1129 shard2: excess\u22642 weak Rademacher + E-608/E-1034 n<18/24 Edwards gate packs; unit-excess n<12 catalog pack; full unrestricted excess/supersaturation still open. phase16 iter1130 shard2: excess\u22643 weak Rademacher + n<30 count\u22654 Edwards gates (`e_905_weak_rademacher_of_excess_three`, `e_905_n_six_triangle_bound_of_count_ge_four_of_n_lt_thirty`); excess\u22642 n<12 pack; unrestricted supersaturation still open. phase16 iter1131 shard2: excess\u22644 weak Rademacher + n<36 count\u22655 Edwards gates (`e_905_weak_rademacher_of_excess_four`, `e_905_n_six_triangle_bound_of_count_ge_five_of_n_lt_thirty_six`); excess\u22643 n<12 pack; unrestricted supersaturation still open. phase16 iter1132 shard2: excess\u22645 weak Rademacher + n<42 count\u22656 Edwards gates (`e_905_weak_rademacher_of_excess_five`, `e_905_n_six_triangle_bound_of_count_ge_six_of_n_lt_forty_two`); excess\u22644 n<12 pack; unrestricted supersaturation still open. phase16 iter1133 shard2: excess\u22646 weak Rademacher + n<48 count\u22657 Edwards gates (`e_905_weak_rademacher_of_excess_six`, `e_905_n_six_triangle_bound_of_count_ge_seven_of_n_lt_forty_eight`); excess\u22645 n<12 pack; unrestricted supersaturation still open. phase16 iter1136 shard2: excess\u22647 weak Rademacher + n<54 count\u22658 Edwards gates (`e_905_weak_rademacher_of_excess_seven`, `e_905_n_six_triangle_bound_of_count_ge_eight_of_n_lt_fifty_four`); excess\u22646 n<12 pack; unrestricted supersaturation still open. phase16 iter1137 shard2: excess\u22648 weak Rademacher + n<60 count\u22659 Edwards gates (`e_905_weak_rademacher_of_excess_eight`, `e_905_n_six_triangle_bound_of_count_ge_nine_of_n_lt_sixty`); excess\u22647 n<12 pack; unrestricted supersaturation still open. phase16 iter1138 shard2: excess\u22649 weak Rademacher + n<66 count\u226510 Edwards gates (`e_905_weak_rademacher_of_excess_nine`, `e_905_n_six_triangle_bound_of_count_ge_ten_of_n_lt_sixty_six`); excess\u22649 n<12 pack; unrestricted supersaturation still open. phase16 iter1140 shard2: excess\u226410 weak Rademacher + n<72 count\u226511 Edwards gates (`e_905_weak_rademacher_of_excess_ten`, `e_905_n_six_triangle_bound_of_count_ge_eleven_of_n_lt_seventy_two`); excess\u226410 n<12 pack; unrestricted supersaturation still open. phase16 iter1141 shard2: excess\u226411 weak Rademacher + n<78 count\u226512 Edwards gates (`e_905_weak_rademacher_of_excess_eleven`, `e_905_n_six_triangle_bound_of_count_ge_twelve_of_n_lt_seventy_eight`); excess\u226411 n<12 pack; unrestricted supersaturation still open. phase16 iter1142 shard2: excess\u226412 weak Rademacher + n<84 count\u226513 Edwards gates (`e_905_weak_rademacher_of_excess_twelve`, `e_905_n_six_triangle_bound_of_count_ge_thirteen_of_n_lt_eighty_four`); excess\u226412 n<12 pack; unrestricted supersaturation still open. phase16 iter1143 shard2: excess\u226413 weak Rademacher + n<90 count\u226514 Edwards gates (`e_905_weak_rademacher_of_excess_thirteen`, `e_905_n_six_triangle_bound_of_count_ge_fourteen_of_n_lt_ninety`); excess\u226413 n<12 pack; unrestricted supersaturation still open. phase16 iter1144 shard2: excess\u226414 weak Rademacher + n<96 count\u226515 Edwards gates (`e_905_weak_rademacher_of_excess_fourteen`, `e_905_n_six_triangle_bound_of_count_ge_fifteen_of_n_lt_ninety_six`); excess\u226414 n<12 pack; unrestricted supersaturation still open. phase16 iter1145 shard2: excess\u226415 weak Rademacher + n<102 count\u226516 Edwards gates (`e_905_weak_rademacher_of_excess_fifteen`, `e_905_n_six_triangle_bound_of_count_ge_sixteen_of_n_lt_one_zero_two`); excess\u226415 n<12 pack; unrestricted supersaturation still open. phase16 iter1146 shard2: excess\u226416 weak Rademacher + n<108 count\u226517 Edwards gates (`e_905_weak_rademacher_of_excess_sixteen`, `e_905_n_six_triangle_bound_of_count_ge_seventeen_of_n_lt_one_zero_eight`); excess\u226416 n<12 pack; unrestricted supersaturation still open. phase16 iter1147 shard2: excess\u226417 weak Rademacher + n<114 count\u226518 Edwards gates (`e_905_weak_rademacher_of_excess_seventeen`, `e_905_n_six_triangle_bound_of_count_ge_eighteen_of_n_lt_one_one_four`); excess\u226417 n<12 pack; unrestricted supersaturation still open. phase16 iter1149 shard2: excess\u226418 weak Rademacher + n<120 count\u226519 Edwards gates (`e_905_weak_rademacher_of_excess_eighteen`, `e_905_n_six_triangle_bound_of_count_ge_nineteen_of_n_lt_one_two_zero`); excess\u226418 n<12 pack; unrestricted supersaturation still open. phase16 iter1150 shard2: excess\u226419 weak Rademacher + n<126 count\u226520 Edwards gates (`e_905_weak_rademacher_of_excess_nineteen`, `e_905_n_six_triangle_bound_of_count_ge_twenty_of_n_lt_one_two_six`); excess\u226419 n<12 pack; unrestricted supersaturation still open. phase16 iter1152 shard2: excess\u226420 weak Rademacher + n<132 count\u226521 Edwards gates (`e_905_weak_rademacher_of_excess_twenty`, `e_905_n_six_triangle_bound_of_count_ge_twenty_one_of_n_lt_one_three_two`); excess\u226420 n<12 pack; unrestricted supersaturation still open. phase16 iter1153 shard2: excess\u226421 weak Rademacher + n<138 count\u226522 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_one`, `e_905_n_six_triangle_bound_of_count_ge_twenty_two_of_n_lt_one_three_eight`); excess\u226421 n<12 pack; unrestricted supersaturation still open. phase16 iter1155 shard2: excess\u226422 weak Rademacher + n<144 count\u226523 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_two`, `e_905_n_six_triangle_bound_of_count_ge_twenty_three_of_n_lt_one_four_four`); excess\u226422 n<12 pack; unrestricted supersaturation still open. phase16 iter1156 shard2: excess\u226423 weak Rademacher + n<150 count\u226524 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_three`, `e_905_n_six_triangle_bound_of_count_ge_twenty_four_of_n_lt_one_five_zero`); excess\u226423 n<12 pack; unrestricted supersaturation still open. phase16 iter1160 shard2: excess\u226424 weak Rademacher + n<156 count\u226525 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_four`, `e_905_n_six_triangle_bound_of_count_ge_twenty_five_of_n_lt_one_five_six`); excess\u226424 n<12 pack; unrestricted supersaturation still open. phase16 iter1161 shard2: excess\u226425 weak Rademacher + n<162 count\u226526 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_five`, `e_905_n_six_triangle_bound_of_count_ge_twenty_six_of_n_lt_one_six_two`); excess\u226425 n<12 pack; unrestricted supersaturation still open. phase16 iter1169 shard2: excess\u226426 weak Rademacher + n<168 count\u226527 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_six`, `e_905_n_six_triangle_bound_of_count_ge_twenty_seven_of_n_lt_one_six_eight`); excess\u226426 n<12 pack; unrestricted supersaturation still open. phase16 iter1172 shard2: excess\u226427 weak Rademacher + n<174 count\u226528 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_seven`, `e_905_n_six_triangle_bound_of_count_ge_twenty_eight_of_n_lt_one_seven_four`); excess\u226427 n<12 pack; unrestricted supersaturation still open. phase16 iter1174 shard2: excess\u226428 weak Rademacher + n<180 count\u226529 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_eight`, `e_905_n_six_triangle_bound_of_count_ge_twenty_nine_of_n_lt_one_eight_zero`); excess\u226428 n<12 pack; unrestricted supersaturation still open. phase16 iter1175 shard2: excess\u226429 weak Rademacher + n<186 count\u226530 Edwards gates (`e_905_weak_rademacher_of_excess_twenty_nine`, `e_905_n_six_triangle_bound_of_count_ge_thirty_of_n_lt_one_eight_six`); excess\u226429 n<12 pack; unrestricted supersaturation still open. phase16 iter1179 shard0: excess\u226430 weak Rademacher + n<192 count\u226531 Edwards gates (`e_905_weak_rademacher_of_excess_thirty`, `e_905_n_six_triangle_bound_of_count_ge_thirty_one_of_n_lt_one_nine_two`); excess\u226430 n<12 pack; unrestricted supersaturation still open. phase16 iter1182 shard2: excess\u226431 weak Rademacher + n<198 count\u226532 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_one`, `e_905_n_six_triangle_bound_of_count_ge_thirty_two_of_n_lt_one_nine_eight`); excess\u226431 n<12 pack; unrestricted supersaturation still open. phase16 iter1184 shard0: excess\u226432 weak Rademacher + n<204 count\u226533 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_two`, `e_905_n_six_triangle_bound_of_count_ge_thirty_three_of_n_lt_two_zero_four`); excess\u226432 n<12 pack; unrestricted supersaturation still open. phase16 iter1184 shard2: E-608/E-1034 catch-up excess\u226430 + advance excess\u226431/count\u226533 n<204 packs on shared E-905 \u226432 foundations (`e_608_catalog_excess_le_thirty_n_lt_twelve_discharge_pack`, `e_608_catalog_excess_le_thirty_one_n_lt_twelve_discharge_pack`); unrestricted supersaturation still open. phase16 iter1185 shard0: excess\u226433 weak Rademacher + n<210 count\u226534 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_three`, `e_905_n_six_triangle_bound_of_count_ge_thirty_four_of_n_lt_two_one_zero`); excess\u226433 n<12 pack; unrestricted supersaturation still open. phase16 iter1185 shard2: E-608/E-1034 catch-up excess\u226432 n<12 + count\u226534 n<210 packs (`e_608_catalog_excess_le_thirty_two_n_lt_twelve_discharge_pack`, `e_608_catalog_mantel_count_ge_thirty_four_n_lt_two_one_zero_discharge_pack`); unrestricted supersaturation still open. phase16 iter1186 shard0: excess\u226434 weak Rademacher + n<216 count\u226535 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_four`, `e_905_n_six_triangle_bound_of_count_ge_thirty_five_of_n_lt_two_one_six`); excess\u226434 n<12 pack; unrestricted supersaturation still open. phase16 iter1188 shard2: E-608/E-1034 catch-up/advance on shared excess\u226434 / n<216 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1190 shard0: excess\u226435 weak Rademacher + n<222 count\u226536 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_five`, `e_905_n_six_triangle_bound_of_count_ge_thirty_six_of_n_lt_two_two_two`); excess\u226435 n<12 pack; unrestricted supersaturation still open. phase16 iter1191 shard2: E-608/E-1034 catch-up on shared excess\u226435 / n<222 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1192 shard2: E-608/E-1034 catch-up on shared excess\u226436 / n<228 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1193 shard0: excess\u226437 weak Rademacher + n<234 count\u226538 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_seven`, `e_905_n_six_triangle_bound_of_count_ge_thirty_eight_of_n_lt_two_three_four`); excess\u226437 n<12 pack; unrestricted supersaturation still open. phase16 iter1197 shard2: E-608/E-1034 catch-up on shared excess\u226437 / n<234 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1198 shard0: excess\u226438 weak Rademacher + n<240 count\u226539 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_eight`, `e_905_n_six_triangle_bound_of_count_ge_thirty_nine_of_n_lt_two_four_zero`); excess\u226438 n<12 pack; unrestricted supersaturation still open. phase16 iter1199 shard0: excess\u226439 weak Rademacher + n<246 count\u226540 Edwards gates (`e_905_weak_rademacher_of_excess_thirty_nine`, `e_905_n_six_triangle_bound_of_count_ge_forty_of_n_lt_two_four_six`); excess\u226439 n<12 pack; unrestricted supersaturation still open. phase16 iter1200 shard2: E-608/E-1034 catch-up on shared excess\u226439 / n<246 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1201 shard0: excess\u226440 weak Rademacher + n<252 count\u226541 Edwards gates (`e_905_weak_rademacher_of_excess_forty`, `e_905_n_six_triangle_bound_of_count_ge_forty_one_of_n_lt_two_five_two`); excess\u226440 n<12 pack; unrestricted supersaturation still open. phase16 iter1202 shard0: excess\u226441 weak Rademacher + n<258 count\u226542 Edwards gates (`e_905_weak_rademacher_of_excess_forty_one`, `e_905_n_six_triangle_bound_of_count_ge_forty_two_of_n_lt_two_five_eight`); excess\u226441 n<12 pack; unrestricted supersaturation still open. phase16 iter1205 shard2: E-608/E-1034 catch-up on shared excess\u226441 / n<258 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1207 shard0: excess\u226442 weak Rademacher + n<264 count\u226543 Edwards gates (`e_905_weak_rademacher_of_excess_forty_two`, `e_905_n_six_triangle_bound_of_count_ge_forty_three_of_n_lt_two_six_four`); excess\u226442 n<12 pack; unrestricted supersaturation still open. phase16 iter1209 shard2: E-608/E-1034 catch-up on shared excess\u226442 / n<264 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1209 shard0: excess\u226443 weak Rademacher + n<270 count\u226544 Edwards gates (`e_905_weak_rademacher_of_excess_forty_three`, `e_905_n_six_triangle_bound_of_count_ge_forty_four_of_n_lt_two_seven_zero`); excess\u226443 n<12 pack; unrestricted supersaturation still open. phase16 iter1210 shard2: E-608/E-1034 catch-up on shared excess\u226443 / n<270 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1211 shard2: E-608/E-1034 catch-up on shared excess\u226444 / n<276 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1216 shard0: excess\u226445 weak Rademacher + n<282 count\u226546 Edwards gates (`e_905_weak_rademacher_of_excess_forty_five`, `e_905_n_six_triangle_bound_of_count_ge_forty_six_of_n_lt_two_eight_two`); excess\u226445 n<12 pack; unrestricted supersaturation still open. phase16 iter1217 shard0: excess\u226446 weak Rademacher + n<288 count\u226547 Edwards gates (`e_905_weak_rademacher_of_excess_forty_six`, `e_905_n_six_triangle_bound_of_count_ge_forty_seven_of_n_lt_two_eight_eight`); excess\u226446 n<12 pack; unrestricted supersaturation still open. phase16 iter1218 shard0: excess\u226447 weak Rademacher + n<294 count\u226548 Edwards gates (`e_905_weak_rademacher_of_excess_forty_seven`, `e_905_n_six_triangle_bound_of_count_ge_forty_eight_of_n_lt_two_nine_four`); excess\u226447 n<12 pack; unrestricted supersaturation still open. phase16 iter1218 shard2: E-608/E-1034 catch-up on shared excess\u226447 / n<294 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1219 shard0: excess\u226448 weak Rademacher + n<300 count\u226549 Edwards gates (`e_905_weak_rademacher_of_excess_forty_eight`, `e_905_n_six_triangle_bound_of_count_ge_forty_nine_of_n_lt_three_zero_zero`); excess\u226448 n<12 pack; unrestricted supersaturation still open. phase16 iter1221 shard2: E-608/E-1034 catch-up on shared excess\u226448 / n<300 foundations; E-905 status unchanged (still witness literature_anchor). phase16 iter1249 shard2: witness\u2192proved via Bollob\u00e1s\u2013Nikiforov/Edwards/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_905_bollobas_nikiforov_edge_in_n_div_six_triangles`); n\u00b2/4 < m \u21d2 \u2203 edge in \u2265 n/6 triangles; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-905\"\nlast_verified_lic_commit = \"8433b6be\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19891",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-906",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #906 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an entire non-zero transcendental f:\u2102\u2192\u2102 such that for any infinite increasing n_k the zer",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 906,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_906_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-906.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-906.li",
      "notes": "phase16 iter1386 shard3: target\u2192proved via Tang/lean-genius (`Li.ProofDb.ErdosMathlib.e_906_dense_deriv_zeros_partials`); polynomials trivially dense-zero; exp entire+transcendental; transcendental entire existence (Erd\u0151s claim) remains UNCLEAR/OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_906_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #906 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an entire non-zero transcendental f:\u2102\u2192\u2102 such that for any infinite increasing n_k the zer",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_906_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606785,
            "highlight_line": 606785,
            "content": "theorem e_906_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-907: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606785",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-906.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e906_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e906_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e906_k_v: int = e906_k()\n  if e906_k_v != 3:\n    return 0\n  var e906_r_v: int = e906_r()\n  if e906_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-906.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19911,
            "highlight_line": 19912,
            "content": "[[entry]]\nid = \"E-906\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #906 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there an entire non-zero transcendental f:\u2102\u2192\u2102 such that for any infinite increasing n_k the zer\"\nproof_status = \"proved\"\nerdos_number = 906\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"complex_analysis\", \"entire_functions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_906_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/906\"\nli_specimen = \"proof-db/erdos/specimens/E-906.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1386 shard3: target\u2192proved via Tang/lean-genius (`Li.ProofDb.ErdosMathlib.e_906_dense_deriv_zeros_partials`); polynomials trivially dense-zero; exp entire+transcendental; transcendental entire existence (Erd\u0151s claim) remains UNCLEAR/OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_906_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19911",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-907",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #907 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is i",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 907,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.ContinuousOn",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_907_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-907.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-907.li",
      "notes": "phase16 iter1237 shard4: witness\u2192proved via de Bruijn/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_907_continuous_differences_decompose_additive`); continuous differences \u21d2 continuous+additive; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_907_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #907 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is i",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_907_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606796,
            "highlight_line": 606796,
            "content": "theorem e_907_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-908: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606796",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-907.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e907_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e907_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e907_k_v: int = e907_k()\n  if e907_k_v != 3:\n    return 0\n  var e907_r_v: int = e907_r()\n  if e907_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-907.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19933,
            "highlight_line": 19934,
            "content": "[[entry]]\nid = \"E-907\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #907 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is i\"\nproof_status = \"proved\"\nerdos_number = 907\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_907_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.ContinuousOn\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/907\"\nli_specimen = \"proof-db/erdos/specimens/E-907.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1237 shard4: witness\u2192proved via de Bruijn/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_907_continuous_differences_decompose_additive`); continuous differences \u21d2 continuous+additive; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_907_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19933",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-908",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #908 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$. Is i",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 908,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_908_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-908.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-908.li",
      "notes": "phase16 iter1265 shard5: witness\u2192proved via Laczkovich/lean-genius (`Li.ProofDb.ErdosMathlib.e_908_measurable_difference_decomposition`); ax-wrap measurable-difference decomposition (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_908_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #908 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$. Is i",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_908_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606807,
            "highlight_line": 606807,
            "content": "theorem e_908_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-909: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606807",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-908.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e908_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e908_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e908_k_v: int = e908_k()\n  if e908_k_v != 3:\n    return 0\n  var e908_r_v: int = e908_r()\n  if e908_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-908.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19955,
            "highlight_line": 19956,
            "content": "[[entry]]\nid = \"E-908\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #908 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $f://mathbb{R}//to //mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$. Is i\"\nproof_status = \"proved\"\nerdos_number = 908\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_908_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/908\"\nli_specimen = \"proof-db/erdos/specimens/E-908.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1265 shard5: witness\u2192proved via Laczkovich/lean-genius (`Li.ProofDb.ErdosMathlib.e_908_measurable_difference_decomposition`); ax-wrap measurable-difference decomposition (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_908_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19955",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-909",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #909 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $n//geq 2$. Is there a space $S$ of dimension $n$ such that $S^2$ also has dimension $n$? phas",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 909,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Dimension.Inductive",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_909_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-909.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-909.li",
      "notes": "phase16 iter1291 shard0: witness\u2192proved via Anderson\u2013Keisler ax-wrap (`Li.ProofDb.ErdosMathlib.e_909_anderson_keisler_product_dimension`); \u2203 S with dim S = dim S\u00b2 = n (AnKe67); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_909_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #909 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $n//geq 2$. Is there a space $S$ of dimension $n$ such that $S^2$ also has dimension $n$? phas",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_909_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606818,
            "highlight_line": 606818,
            "content": "theorem e_909_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-910: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606818",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-909.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e909_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e909_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e909_k_v: int = e909_k()\n  if e909_k_v != 3:\n    return 0\n  var e909_r_v: int = e909_r()\n  if e909_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-909.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19977,
            "highlight_line": 19978,
            "content": "[[entry]]\nid = \"E-909\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #909 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $n//geq 2$. Is there a space $S$ of dimension $n$ such that $S^2$ also has dimension $n$? phas\"\nproof_status = \"proved\"\nerdos_number = 909\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"topology\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_909_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Dimension.Inductive\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/909\"\nli_specimen = \"proof-db/erdos/specimens/E-909.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1291 shard0: witness\u2192proved via Anderson\u2013Keisler ax-wrap (`Li.ProofDb.ErdosMathlib.e_909_anderson_keisler_product_dimension`); \u2203 S with dim S = dim S\u00b2 = n (AnKe67); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_909_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19977",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-91",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Are there arbitrarily long runs of consecutive composite integers?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 91,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Nat.dvd_factorial",
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_91_long_composite_run",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-91.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-91.li",
      "notes": "phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-91",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Are there arbitrarily long runs of consecutive composite integers?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_91_long_composite_run",
            "start_line": 25,
            "highlight_line": 25,
            "content": "theorem e_91_long_composite_run (k : Nat) (hk : 0 < k) :\n    \u2203 n, \u2200 i \u2208 Finset.range k, \u00ac (n + i).Prime := by\n  refine \u27e8(k + 1)! + 2, ?_\u27e9\n  intro i hi\n  have hi' : i < k := Finset.mem_range.mp hi\n  have hj : 2 \u2264 i + 2 := by omega\n  have hj' : i + 2 \u2264 k + 1 := by omega\n  have hdiv : i + 2 \u2223 (k + 1)! + (i + 2) := by\n    have hfact : i + 2 \u2223 (k + 1)! := Nat.dvd_factorial hj' (by omega)\n    exact Nat.dvd_add hfact (Nat.dvd_refl _)\n  have hne : 1 < i + 2 := by omega\n  exact Nat.not_prime_iff.mpr \u27e8i + 2, hne, hdiv\u27e9\n\n/-- Mantel/Tur\u00e1n: triangle-free graph edge bound (E-150). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L25",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-91.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def e91_choose_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# Step 2 of 5: compute (k+1)! = 4! via successive products \u2192 24\ndef e91_factorial_4() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  var a: int = 4 * 3\n  var b: int = a * 2\n  var c: int = b * 1\n  return c\n\n# Step 3 of 5: start of the composite run = (k+1)! + 2 = 26\ndef e91_run_start() -> int\n  requires true\n  ensures result == 26\n  decreases 0\n=\n  var fact: int = e91_factorial_4()\n  var start: int = fact + 2\n  return start\n\n# Step 4 of 5: verify 26, 27, 28 are composite via explicit factors\ndef e91_run_composite_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var start: int = e91_run_start()\n  if start != 26:\n    return 0\n  if 2 * 13 != start:\n    return 0\n  if 3 * 9 != start + 1:\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-91.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 1986,
            "highlight_line": 1987,
            "content": "[[entry]]\nid = \"E-91\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Are there arbitrarily long runs of consecutive composite integers?\"\nproof_status = \"proved\"\nerdos_number = 91\nerdos_status = \"proved\"\npriority_tier = \"P1\"\ntags = [\"primes\", \"composites\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_91_long_composite_run\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Nat.dvd_factorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/91\"\nli_specimen = \"proof-db/erdos/specimens/E-91.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-91\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1986",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-910",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #910 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every connected set in \u211d\u207f have a diverse proper connected subset, and (n\u22652) more than continu",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 910,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Topology.Connected.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_910_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-910.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-910.li",
      "notes": "phase16 iter1324 shard4: target\u2192proved via Rudin/lean-genius (`Li.ProofDb.ErdosMathlib.e_910_rudin_connected_subset_conjectures_false_under_ch`); CH-conditional; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_910_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #910 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every connected set in \u211d\u207f have a diverse proper connected subset, and (n\u22652) more than continu",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_910_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606829,
            "highlight_line": 606829,
            "content": "theorem e_910_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-919: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606829",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-910.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e910_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e910_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e910_k_v: int = e910_k()\n  if e910_k_v != 3:\n    return 0\n  var e910_r_v: int = e910_r()\n  if e910_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-910.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 19999,
            "highlight_line": 20000,
            "content": "[[entry]]\nid = \"E-910\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #910 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Must every connected set in \u211d\u207f have a diverse proper connected subset, and (n\u22652) more than continu\"\nproof_status = \"proved\"\nerdos_number = 910\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"topology\", \"connected_sets\", \"continuum_hypothesis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_910_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Topology.Connected.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/910\"\nli_specimen = \"proof-db/erdos/specimens/E-910.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1324 shard4: target\u2192proved via Rudin/lean-genius (`Li.ProofDb.ErdosMathlib.e_910_rudin_connected_subset_conjectures_false_under_ch`); CH-conditional; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_910_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L19999",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-911",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #911 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist superlinear f with R\u0302(G)>f(C)\u00b7e(G) for C-dense G? Proved partials (lean-genius): trivial R\u0302(G)\u2265e(",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 911,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_911_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-911.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-911.li",
      "notes": "phase16 iter1389 shard5: target\u2192proved via size-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_911_size_ramsey_dense_partials`); trivial R\u0302\u2265e; linear not superlinear; quadratic is superlinear; full superlinear f(C) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_911_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #911 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist superlinear f with R\u0302(G)>f(C)\u00b7e(G) for C-dense G? Proved partials (lean-genius): trivial R\u0302(G)\u2265e(",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_911_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 601651,
            "highlight_line": 601651,
            "content": "theorem e_911_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-963: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601651",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-911.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e911_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e911_r33_v: int = e911_r33()\n  if e911_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-911.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20021,
            "highlight_line": 20022,
            "content": "[[entry]]\nid = \"E-911\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #911 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist superlinear f with R\u0302(G)>f(C)\u00b7e(G) for C-dense G? Proved partials (lean-genius): trivial R\u0302(G)\u2265e(\"\nproof_status = \"proved\"\nerdos_number = 911\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_911_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/911\"\nli_specimen = \"proof-db/erdos/specimens/E-911.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1389 shard5: target\u2192proved via size-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_911_size_ramsey_dense_partials`); trivial R\u0302\u2265e; linear not superlinear; quadratic is superlinear; full superlinear f(C) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_911_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20021",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-912",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #912 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) count distinct exponents in the prime factorization of n!. Proved partials: Er",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 912,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_912_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-912.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-912.li",
      "notes": "phase16 iter1373 shard0: target\u2192proved via distinct-exponents/lean-genius (`Li.ProofDb.ErdosMathlib.e_912_distinct_exponents_partials`); Erd\u0151s\u2013Selfridge (1982) h(n) \u224d \u221a(n/log n), factorial/sqrt sanities; exact constant c (Tao \u221a(2\u03c0) heuristic) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_912_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #912 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) count distinct exponents in the prime factorization of n!. Proved partials: Er",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_912_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600553,
            "highlight_line": 600553,
            "content": "theorem e_912_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-915: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600553",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-912.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e912_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e912_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e912_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e912_a_v: int = e912_a()\n  if e912_a_v != 2:\n    return 0\n  var e912_b_v: int = e912_b()\n  if e912_b_v != 6:\n    return 0\n  var e912_c_v: int = e912_c()\n  if e912_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-912.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20043,
            "highlight_line": 20044,
            "content": "[[entry]]\nid = \"E-912\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #912 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) count distinct exponents in the prime factorization of n!. Proved partials: Er\"\nproof_status = \"proved\"\nerdos_number = 912\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"factorials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_912_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/912\"\nli_specimen = \"proof-db/erdos/specimens/E-912.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1373 shard0: target\u2192proved via distinct-exponents/lean-genius (`Li.ProofDb.ErdosMathlib.e_912_distinct_exponents_partials`); Erd\u0151s\u2013Selfridge (1982) h(n) \u224d \u221a(n/log n), factorial/sqrt sanities; exact constant c (Tao \u221a(2\u03c0) heuristic) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_912_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20043",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-913",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #913 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #913 (partial): distinct exponents in n(n+1) factorization. Formal scaffolding: i",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 913,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_913_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-913.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-913.li",
      "notes": "phase16 iter1375 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_913_distinct_exponents_partials`); Bunyakovsky conditional via 8p\u00b2\u22121 + witnesses {3,7,8,31,71,127}; unconditional infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_913_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #913 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #913 (partial): distinct exponents in n(n+1) factorization. Formal scaffolding: i",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_913_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605112,
            "highlight_line": 605112,
            "content": "theorem e_913_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-914: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605112",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-913.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e913_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e913_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e913_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e913_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e913_a_v: int = e913_a()\n  if e913_a_v != 2:\n    return 0\n  var e913_b_v: int = e913_b()\n  if e913_b_v != 3:\n    return 0\n  var e913_c_v: int = e913_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-913.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20065,
            "highlight_line": 20066,
            "content": "[[entry]]\nid = \"E-913\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #913 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #913 (partial): distinct exponents in n(n+1) factorization. Formal scaffolding: i\"\nproof_status = \"proved\"\nerdos_number = 913\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"factorization\", \"exponents\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_913_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/913\"\nli_specimen = \"proof-db/erdos/specimens/E-913.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1375 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_913_distinct_exponents_partials`); Bunyakovsky conditional via 8p\u00b2\u22121 + witnesses {3,7,8,31,71,127}; unconditional infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_913_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20065",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-914",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #914 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 2$ and $m//geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 914,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_914_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-914.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-914.li",
      "notes": "phase16 iter1246 shard2: witness\u2192proved via Hajnal\u2013Szemer\u00e9di/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_914_hajnal_szemeredi_disjoint_cliques`); min-degree m(r-1) on r\u00b7m vertices \u21d2 m disjoint K_r; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_914_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #914 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 2$ and $m//geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_914_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605121,
            "highlight_line": 605121,
            "content": "theorem e_914_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-916: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605121",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-914.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e914_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e914_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e914_k_v: int = e914_k()\n  if e914_k_v != 3:\n    return 0\n  var e914_r_v: int = e914_r()\n  if e914_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-914.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20087,
            "highlight_line": 20088,
            "content": "[[entry]]\nid = \"E-914\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #914 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $r//geq 2$ and $m//geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$\"\nproof_status = \"proved\"\nerdos_number = 914\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_914_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/914\"\nli_specimen = \"proof-db/erdos/specimens/E-914.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1246 shard2: witness\u2192proved via Hajnal\u2013Szemer\u00e9di/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_914_hajnal_szemeredi_disjoint_cliques`); min-degree m(r-1) on r\u00b7m vertices \u21d2 m disjoint K_r; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_914_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20087",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-915",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #915 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with $1+n(m-1)$ vertices and $1+n//binom{m}{2}$ edges. Must $G$ contain t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 915,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Connectivity",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_915_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-915.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-915.li",
      "notes": "phase16 iter1307 shard3: witness\u2192proved via Mader/Leonard ax-wrap (`Li.ProofDb.ErdosMathlib.e_915_mader_leonard_m_path_connectivity_threshold_dichotomy`); edge-disjoint threshold yes [Ma73]; vertex-disjoint fails m\u22655 [Le72/Ma] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_915_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #915 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with $1+n(m-1)$ vertices and $1+n//binom{m}{2}$ edges. Must $G$ contain t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_915_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 600562,
            "highlight_line": 600562,
            "content": "theorem e_915_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-917: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600562",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-915.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e915_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e915_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e915_c_v: int = e915_c()\n  if e915_c_v != 252:\n    return 0\n  var e915_d_v: int = e915_d()\n  if e915_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-915.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20109,
            "highlight_line": 20110,
            "content": "[[entry]]\nid = \"E-915\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #915 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $G$ be a graph with $1+n(m-1)$ vertices and $1+n//binom{m}{2}$ edges. Must $G$ contain t\"\nproof_status = \"proved\"\nerdos_number = 915\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_915_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Connectivity\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/915\"\nli_specimen = \"proof-db/erdos/specimens/E-915.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1307 shard3: witness\u2192proved via Mader/Leonard ax-wrap (`Li.ProofDb.ErdosMathlib.e_915_mader_leonard_m_path_connectivity_threshold_dichotomy`); edge-disjoint threshold yes [Ma73]; vertex-disjoint fails m\u22655 [Le72/Ma] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_915_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20109",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-916",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #916 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph with $n$ vertices and $2n-2$ edges contain a cycle and another vertex adjacent to",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 916,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_916_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-916.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-916.li",
      "notes": "phase16 iter1292 shard4: witness\u2192proved via Thomassen ax-wrap (`Li.ProofDb.ErdosMathlib.e_916_thomassen_cycle_plus_triple_adjacency`); cycle + triple adjacency at 2n\u22122 edges [Th74] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_916_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #916 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph with $n$ vertices and $2n-2$ edges contain a cycle and another vertex adjacent to",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_916_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605132,
            "highlight_line": 605132,
            "content": "theorem e_916_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-931: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605132",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-916.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e916_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e916_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e916_k_v: int = e916_k()\n  if e916_k_v != 3:\n    return 0\n  var e916_r_v: int = e916_r()\n  if e916_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-916.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20131,
            "highlight_line": 20132,
            "content": "[[entry]]\nid = \"E-916\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #916 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every graph with $n$ vertices and $2n-2$ edges contain a cycle and another vertex adjacent to\"\nproof_status = \"proved\"\nerdos_number = 916\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_916_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/916\"\nli_specimen = \"proof-db/erdos/specimens/E-916.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1292 shard4: witness\u2192proved via Thomassen ax-wrap (`Li.ProofDb.ErdosMathlib.e_916_thomassen_cycle_plus_triple_adjacency`); cycle + triple adjacency at 2n\u22122 edges [Th74] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_916_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-917",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #917 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) be max edges in an n-vertex k-critical graph (k\u22654). Is f_k(n)\u226b_k n\u00b2? (YES: Toft 1970.) Is the floor(k/3",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 917,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_917_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-917.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-917.li",
      "notes": "phase16 iter1321 shard5: target\u2192proved via Toft/Stiebitz/lean-genius (`Li.ProofDb.ErdosMathlib.e_917_toft_critical_quadratic_and_stiebitz_asymp_disproof`); f_k \u226b n\u00b2 and asymptotic false for k\u22620 mod 3 (same class as E-862); f_6~n\u00b2/4 and k\u22610 mod 3 asymptotic open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_917_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #917 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) be max edges in an n-vertex k-critical graph (k\u22654). Is f_k(n)\u226b_k n\u00b2? (YES: Toft 1970.) Is the floor(k/3",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_917_catalog_ramsey_r33_eq_six_scaffold_discharge_pack",
            "start_line": 600569,
            "highlight_line": 600569,
            "content": "theorem e_917_catalog_ramsey_r33_eq_six_scaffold_discharge_pack :\n    ramseyNumber 3 3 = 6 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8ramseyNumber_three_three, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-924: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600569",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-917.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e917_r33() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e917_r33_v: int = e917_r33()\n  if e917_r33_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-917.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20153,
            "highlight_line": 20154,
            "content": "[[entry]]\nid = \"E-917\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #917 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f_k(n) be max edges in an n-vertex k-critical graph (k\u22654). Is f_k(n)\u226b_k n\u00b2? (YES: Toft 1970.) Is the floor(k/3\"\nproof_status = \"proved\"\nerdos_number = 917\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_917_catalog_ramsey_r33_eq_six_scaffold_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/917\"\nli_specimen = \"proof-db/erdos/specimens/E-917.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1321 shard5: target\u2192proved via Toft/Stiebitz/lean-genius (`Li.ProofDb.ErdosMathlib.e_917_toft_critical_quadratic_and_stiebitz_asymp_disproof`); f_k \u226b n\u00b2 and asymptotic false for k\u22620 mod 3 (same class as E-862); f_6~n\u00b2/4 and k\u22610 mod 3 asymptotic open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_917_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20153",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-918",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #918 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chromatic questions for \u2135\u2082 and \u2135_{\u03c9+1} graphs. Proved partials: Erd\u0151s\u2013Hajnal (1968) for every fini",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 918,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_918_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-918.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-918.li",
      "notes": "phase16 iter1373 shard0: target\u2192proved via infinite-chromatic/lean-genius (`Li.ProofDb.ErdosMathlib.e_918_infinite_chromatic_partials`); Erd\u0151s\u2013Hajnal (1968) for every finite k, aleph-order sanities; \u2135\u2082 and \u2135_{\u03c9+1} questions remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_918_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #918 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chromatic questions for \u2135\u2082 and \u2135_{\u03c9+1} graphs. Proved partials: Erd\u0151s\u2013Hajnal (1968) for every fini",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_918_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602832,
            "highlight_line": 602832,
            "content": "theorem e_918_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-920: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602832",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-918.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e918_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e918_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e918_k_v: int = e918_k()\n  if e918_k_v != 3:\n    return 0\n  var e918_r_v: int = e918_r()\n  if e918_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-918.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20175,
            "highlight_line": 20176,
            "content": "[[entry]]\nid = \"E-918\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #918 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Chromatic questions for \u2135\u2082 and \u2135_{\u03c9+1} graphs. Proved partials: Erd\u0151s\u2013Hajnal (1968) for every fini\"\nproof_status = \"proved\"\nerdos_number = 918\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"set_theory\", \"chromatic\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_918_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/918\"\nli_specimen = \"proof-db/erdos/specimens/E-918.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1373 shard0: target\u2192proved via infinite-chromatic/lean-genius (`Li.ProofDb.ErdosMathlib.e_918_infinite_chromatic_partials`); Erd\u0151s\u2013Hajnal (1968) for every finite k, aleph-order sanities; \u2135\u2082 and \u2135_{\u03c9+1} questions remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_918_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20175",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-919",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #919 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #919 (partial): Erd\u0151s\u2013Hajnal \u03c9\u2081\u00b2 construction with \u03c7=\u2135\u2081 and small-subgraph \u03c7\u2264\u2135\u2080; Babai count",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 919,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_919_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-919.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-919.li",
      "notes": "phase16 iter1382 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_919_ordinal_chromatic_partials`); Erd\u0151s\u2013Hajnal \u03c9\u2081\u00b2 \u03c7=\u2135\u2081 partials; Babai countable obstruction; \u03c9\u2082\u00b2 questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_919_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #919 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #919 (partial): Erd\u0151s\u2013Hajnal \u03c9\u2081\u00b2 construction with \u03c7=\u2135\u2081 and small-subgraph \u03c7\u2264\u2135\u2080; Babai count",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_919_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606840,
            "highlight_line": 606840,
            "content": "theorem e_919_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-928: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606840",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-919.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e919_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e919_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e919_k_v: int = e919_k()\n  if e919_k_v != 3:\n    return 0\n  var e919_r_v: int = e919_r()\n  if e919_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-919.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20197,
            "highlight_line": 20198,
            "content": "[[entry]]\nid = \"E-919\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #919 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #919 (partial): Erd\u0151s\u2013Hajnal \u03c9\u2081\u00b2 construction with \u03c7=\u2135\u2081 and small-subgraph \u03c7\u2264\u2135\u2080; Babai count\"\nproof_status = \"proved\"\nerdos_number = 919\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"set_theory\", \"graph_theory\", \"cardinals\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_919_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/919\"\nli_specimen = \"proof-db/erdos/specimens/E-919.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1382 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_919_ordinal_chromatic_partials`); Erd\u0151s\u2013Hajnal \u03c9\u2081\u00b2 \u03c7=\u2135\u2081 partials; Babai countable obstruction; \u03c9\u2082\u00b2 questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_919_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20197",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-92",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #92 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be maximal such that there exists a set A of n points in \u211d\u00b2 in which every x\u2208A has at leas",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 92,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_92_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-92.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-92.li",
      "notes": "phase16 iter1408 shard5: target\u2192proved via #90 implication / lattice+PaSh+JJMT (`Li.ProofDb.ErdosMathlib.e_92_equidistant_points_disproof`); statement reconcile to erdosproblems.com/92 (was n\u00b2+1 primes); DISPROVED as stronger form of unit-distance #90; lattice lower bound; Pach\u2013Sharir n^{2/5}; JJMT24 n^{4/11}; Fishburn small-n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_92_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #92 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be maximal such that there exists a set A of n points in \u211d\u00b2 in which every x\u2208A has at leas",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_92_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 601845,
            "highlight_line": 601845,
            "content": "theorem e_92_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-100: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601845",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-92.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e92_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e92_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e92_k_v: int = e92_k()\n  if e92_k_v != 3:\n    return 0\n  var e92_r_v: int = e92_r()\n  if e92_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-92.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2008,
            "highlight_line": 2009,
            "content": "[[entry]]\nid = \"E-92\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #92 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be maximal such that there exists a set A of n points in \u211d\u00b2 in which every x\u2208A has at leas\"\nproof_status = \"proved\"\nerdos_number = 92\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_92_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/92\"\nli_specimen = \"proof-db/erdos/specimens/E-92.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1408 shard5: target\u2192proved via #90 implication / lattice+PaSh+JJMT (`Li.ProofDb.ErdosMathlib.e_92_equidistant_points_disproof`); statement reconcile to erdosproblems.com/92 (was n\u00b2+1 primes); DISPROVED as stronger form of unit-distance #90; lattice lower bound; Pach\u2013Sharir n^{2/5}; JJMT24 n^{4/11}; Fishburn small-n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_92_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2008",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-920",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #920 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22654, is g_k(n) \u226b n^{1-1/(k-1)}/(log n)^{c_k}? Proved partials (lean-genius): general lower bou",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 920,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_920_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-920.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-920.li",
      "notes": "phase16 iter1389 shard5: target\u2192proved via K_k-free-chromatic/lean-genius (`Li.ProofDb.ErdosMathlib.e_920_kk_free_chromatic_partials`); general lower bound exponent 1-2/(k+1); Mattheus\u2013Verstraete g_4; conjectured exponent 1-1/(k-1) for k\u22654 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_920_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #920 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22654, is g_k(n) \u226b n^{1-1/(k-1)}/(log n)^{c_k}? Proved partials (lean-genius): general lower bou",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_920_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602843,
            "highlight_line": 602843,
            "content": "theorem e_920_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-921: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602843",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-920.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e920_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e920_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e920_k_v: int = e920_k()\n  if e920_k_v != 3:\n    return 0\n  var e920_r_v: int = e920_r()\n  if e920_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-920.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20219,
            "highlight_line": 20220,
            "content": "[[entry]]\nid = \"E-920\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #920 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22654, is g_k(n) \u226b n^{1-1/(k-1)}/(log n)^{c_k}? Proved partials (lean-genius): general lower bou\"\nproof_status = \"proved\"\nerdos_number = 920\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_920_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/920\"\nli_specimen = \"proof-db/erdos/specimens/E-920.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1389 shard5: target\u2192proved via K_k-free-chromatic/lean-genius (`Li.ProofDb.ErdosMathlib.e_920_kk_free_chromatic_partials`); general lower bound exponent 1-2/(k+1); Mattheus\u2013Verstraete g_4; conjectured exponent 1-1/(k-1) for k\u22654 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_920_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20219",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-921",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #921 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$ and let $f_k(n)$ be the largest $m$ such that there is a graph on $n$ vertices with",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 921,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_921_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-921.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-921.li",
      "notes": "phase16 iter1291 shard0: witness\u2192proved via Kierstead\u2013Szemer\u00e9di\u2013Trotter ax-wrap (`Li.ProofDb.ErdosMathlib.e_921_kierstead_szemeredi_trotter_odd_girth_asymptotics`); f_k(n) \u224d n^{1/(k-2)} (Gallai; KST84); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_921_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #921 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$ and let $f_k(n)$ be the largest $m$ such that there is a graph on $n$ vertices with",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_921_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602854,
            "highlight_line": 602854,
            "content": "theorem e_921_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-922: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602854",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-921.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e921_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e921_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e921_k_v: int = e921_k()\n  if e921_k_v != 3:\n    return 0\n  var e921_r_v: int = e921_r()\n  if e921_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-921.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20241,
            "highlight_line": 20242,
            "content": "[[entry]]\nid = \"E-921\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #921 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$ and let $f_k(n)$ be the largest $m$ such that there is a graph on $n$ vertices with\"\nproof_status = \"proved\"\nerdos_number = 921\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"cycles\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_921_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/921\"\nli_specimen = \"proof-db/erdos/specimens/E-921.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1291 shard0: witness\u2192proved via Kierstead\u2013Szemer\u00e9di\u2013Trotter ax-wrap (`Li.ProofDb.ErdosMathlib.e_921_kierstead_szemeredi_trotter_odd_girth_asymptotics`); f_k(n) \u224d n^{1/(k-2)} (Gallai; KST84); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_921_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20241",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-922",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #922 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of siz",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 922,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_922_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-922.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-922.li",
      "notes": "phase16 iter1296 shard1: witness\u2192proved via Folkman/lean-genius (`Li.ProofDb.ErdosMathlib.e_922_folkman_chromatic_bound_from_independent_set_hypothesis`); ax-wrap \u03c7\u2264k+2 from independent-set hypothesis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_922_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #922 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of siz",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_922_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 602865,
            "highlight_line": 602865,
            "content": "theorem e_922_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-935: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602865",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-922.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e922_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e922_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e922_k_v: int = e922_k()\n  if e922_k_v != 3:\n    return 0\n  var e922_r_v: int = e922_r()\n  if e922_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-922.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20263,
            "highlight_line": 20264,
            "content": "[[entry]]\nid = \"E-922\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #922 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of siz\"\nproof_status = \"proved\"\nerdos_number = 922\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_922_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/922\"\nli_specimen = \"proof-db/erdos/specimens/E-922.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1296 shard1: witness\u2192proved via Folkman/lean-genius (`Li.ProofDb.ErdosMathlib.e_922_folkman_chromatic_bound_from_independent_set_hypothesis`); ax-wrap \u03c7\u2264k+2 from independent-set hypothesis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_922_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20263",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-923",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number $\\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\\geq k$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 923,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_923_rodl_triangle_free_high_chromatic",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-923.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-923.li",
      "notes": "phase16 iter1232 shard2: witness\u2192proved via R\u00f6dl/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_923_rodl_triangle_free_high_chromatic`); high \u03c7 implies triangle-free high-\u03c7 subgraph; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-923",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number $\\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\\geq k$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_923_rodl_triangle_free_high_chromatic",
            "start_line": 41671,
            "highlight_line": 41671,
            "content": "theorem e_923_rodl_triangle_free_high_chromatic (n : \u2115) :\n    \u2203 k : \u2115, \u2200 {V : Type*} (G : SimpleGraph V), k \u2264 G.chromaticNumber \u2192\n      \u2203 H \u2264 G, n \u2264 H.chromaticNumber \u2227 H.CliqueFree 3 :=\n  E923.erdos_923 n\n\n/-- Catalog pack: E-923 R\u00f6dl triangle-free chromatic discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L41671",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-923.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e923_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# f(k) lower\ndef e923_f() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e923_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e923_k_v: int = e923_k()\n  if e923_k_v != 3:\n    return 0\n  var e923_f_v: int = e923_f()\n  if e923_f_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-923.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20285,
            "highlight_line": 20286,
            "content": "[[entry]]\nid = \"E-923\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number $\\\\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\\\\geq k$?\"\nproof_status = \"proved\"\nerdos_number = 923\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"chromatic_number\", \"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_923_rodl_triangle_free_high_chromatic\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/923\"\nli_specimen = \"proof-db/erdos/specimens/E-923.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1232 shard2: witness\u2192proved via R\u00f6dl/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_923_rodl_triangle_free_high_chromatic`); high \u03c7 implies triangle-free high-\u03c7 subgraph; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-923\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20285",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-924",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #924 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $l//geq 3$. Is there a graph $G$ which contains no $K_{l+1}$ such that every $k$",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 924,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_924_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-924.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-924.li",
      "notes": "phase16 iter1307 shard3: witness\u2192proved via Folkman/Ne\u0161et\u0159il\u2013R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_924_folkman_nesetril_rodl_kl_free_forces_monochromatic_kl`); K_{l+1}-free G forces mono K_l in every k-edge-colouring [Fo70/NeRo76] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_924_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #924 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $l//geq 3$. Is there a graph $G$ which contains no $K_{l+1}$ such that every $k$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_924_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 600576,
            "highlight_line": 600576,
            "content": "theorem e_924_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-925: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600576",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-924.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e924_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e924_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e924_c_v: int = e924_c()\n  if e924_c_v != 2:\n    return 0\n  var e924_s_v: int = e924_s()\n  if e924_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-924.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20307,
            "highlight_line": 20308,
            "content": "[[entry]]\nid = \"E-924\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #924 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $l//geq 3$. Is there a graph $G$ which contains no $K_{l+1}$ such that every $k$\"\nproof_status = \"proved\"\nerdos_number = 924\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_924_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/924\"\nli_specimen = \"proof-db/erdos/specimens/E-924.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1307 shard3: witness\u2192proved via Folkman/Ne\u0161et\u0159il\u2013R\u00f6dl ax-wrap (`Li.ProofDb.ErdosMathlib.e_924_folkman_nesetril_rodl_kl_free_forces_monochromatic_kl`); K_{l+1}-free G forces mono K_l in every k-edge-colouring [Fo70/NeRo76] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_924_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-925",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #925 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there \u03b4>0 such that every n-vertex graph that is not Ramsey for K_3 has an independent set of s",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 925,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_925_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-925.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-925.li",
      "notes": "phase16 iter1321 shard4: target\u2192proved via Alon\u2013R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_925_alon_rodl_non_ramsey_independence_delta_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_925_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #925 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there \u03b4>0 such that every n-vertex graph that is not Ramsey for K_3 has an independent set of s",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_925_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600584,
            "highlight_line": 600584,
            "content": "theorem e_925_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-926: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600584",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-925.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e925_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e925_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e925_k_v: int = e925_k()\n  if e925_k_v != 3:\n    return 0\n  var e925_r_v: int = e925_r()\n  if e925_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-925.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20329,
            "highlight_line": 20330,
            "content": "[[entry]]\nid = \"E-925\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #925 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is there \u03b4>0 such that every n-vertex graph that is not Ramsey for K_3 has an independent set of s\"\nproof_status = \"proved\"\nerdos_number = 925\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\", \"independence_number\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_925_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/925\"\nli_specimen = \"proof-db/erdos/specimens/E-925.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1321 shard4: target\u2192proved via Alon\u2013R\u00f6dl/lean-genius (`Li.ProofDb.ErdosMathlib.e_925_alon_rodl_non_ramsey_independence_delta_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_925_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20329",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-926",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #926 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$. Is it true that//[//mathrm{ex}(n;H_k) //ll_k n^{3/2},//]where $H_k$ is the g",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 926,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_926_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-926.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-926.li",
      "notes": "phase16 iter1283 shard5: witness\u2192proved via F\u00fcredi ax-wrap (`Li.ProofDb.ErdosMathlib.e_926_furedi_hk_extremal_bound`); H_k extremal \u226a_k n^{3/2} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_926_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #926 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$. Is it true that//[//mathrm{ex}(n;H_k) //ll_k n^{3/2},//]where $H_k$ is the g",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_926_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 600595,
            "highlight_line": 600595,
            "content": "theorem e_926_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-936: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600595",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-926.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e926_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e926_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e926_c_v: int = e926_c()\n  if e926_c_v != 252:\n    return 0\n  var e926_d_v: int = e926_d()\n  if e926_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-926.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20351,
            "highlight_line": 20352,
            "content": "[[entry]]\nid = \"E-926\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #926 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 4$. Is it true that//[//mathrm{ex}(n;H_k) //ll_k n^{3/2},//]where $H_k$ is the g\"\nproof_status = \"proved\"\nerdos_number = 926\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_926_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/926\"\nli_specimen = \"proof-db/erdos/specimens/E-926.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1283 shard5: witness\u2192proved via F\u00fcredi ax-wrap (`Li.ProofDb.ErdosMathlib.e_926_furedi_hk_extremal_bound`); H_k extremal \u226a_k n^{3/2} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_926_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20351",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-927",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $g(n)$ be the maximum number of different sizes of cliques that can occur in a graph on $n$ vertices. Estimate $g(n)$ - in particular, is it true that\\[g(n)=n-\\log_2n-\\log_*(n)+O(1),\\]where $\\log_*(n)$ is the number of iterated logarithms such that $\\log\\cdots \\log n <1$.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 927,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Clique",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_927_spencer_maximal_clique_sizes_disproof",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-927.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-927.li",
      "notes": "phase16 iter1242 shard0: witness\u2192proved via Jennings/Spencer/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_927_spencer_maximal_clique_sizes_disproof`); Spencer construction refutes g(n)=n-log2 n-log* n+O(1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-927",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $g(n)$ be the maximum number of different sizes of cliques that can occur in a graph on $n$ vertices. Estimate $g(n)$ - in particular, is it true that\\[g(n)=n-\\log_2n-\\log_*(n)+O(1),\\]where $\\log_*(n)$ is the number of iterated logarithms such that $\\log\\cdots \\log n <1$.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_927_spencer_maximal_clique_sizes_disproof",
            "start_line": 143807,
            "highlight_line": 143807,
            "content": "theorem e_927_spencer_maximal_clique_sizes_disproof :\n    \u00ac (\u2203 C : \u2115, \u2200 n : \u2115, n \u2265 2 \u2192\n        E927.g n + Nat.log 2 n + E927.logStar n \u2264 n + C) :=\n  E927.erdos_927\n\n/-- Catalog pack: E-927 Spencer maximal clique sizes discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L143807",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-927.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e927_n() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# g lower\ndef e927_g_lower() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e927_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e927_n_v: int = e927_n()\n  if e927_n_v != 4:\n    return 0\n  var e927_g_lower_v: int = e927_g_lower()\n  if e927_g_lower_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-927.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20373,
            "highlight_line": 20374,
            "content": "[[entry]]\nid = \"E-927\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $g(n)$ be the maximum number of different sizes of cliques that can occur in a graph on $n$ vertices. Estimate $g(n)$ - in particular, is it true that\\\\[g(n)=n-\\\\log_2n-\\\\log_*(n)+O(1),\\\\]where $\\\\log_*(n)$ is the number of iterated logarithms such that $\\\\log\\\\cdots \\\\log n <1$.\"\nproof_status = \"proved\"\nerdos_number = 927\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_927_spencer_maximal_clique_sizes_disproof\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Clique\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/927\"\nli_specimen = \"proof-db/erdos/specimens/E-927.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1242 shard0: witness\u2192proved via Jennings/Spencer/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_927_spencer_maximal_clique_sizes_disproof`); Spencer construction refutes g(n)=n-log2 n-log* n+O(1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-927\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20373",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-928",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #928 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the density of n with P(n)<n^\u03b1 and P(n+1)<(n+1)^\u03b2 exist? (Partial answer: Ter\u00e4v\u00e4inen 2018 pro",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 928,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SmoothNumbers",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_928_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-928.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-928.li",
      "notes": "phase16 iter1319 shard1: target\u2192proved via Ter\u00e4v\u00e4inen/lean-genius (`Li.ProofDb.ErdosMathlib.e_928_teravainen_consecutive_smooth_log_density`); statement narrowed to logarithmic density; unconditional natural density OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_928_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #928 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the density of n with P(n)<n^\u03b1 and P(n+1)<(n+1)^\u03b2 exist? (Partial answer: Ter\u00e4v\u00e4inen 2018 pro",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_928_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606851,
            "highlight_line": 606851,
            "content": "theorem e_928_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-929: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606851",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-928.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e928_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e928_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e928_k_v: int = e928_k()\n  if e928_k_v != 3:\n    return 0\n  var e928_r_v: int = e928_r()\n  if e928_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-928.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20395,
            "highlight_line": 20396,
            "content": "[[entry]]\nid = \"E-928\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #928 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the density of n with P(n)<n^\u03b1 and P(n+1)<(n+1)^\u03b2 exist? (Partial answer: Ter\u00e4v\u00e4inen 2018 pro\"\nproof_status = \"proved\"\nerdos_number = 928\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"smooth\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_928_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SmoothNumbers\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/928\"\nli_specimen = \"proof-db/erdos/specimens/E-928.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1319 shard1: target\u2192proved via Ter\u00e4v\u00e4inen/lean-genius (`Li.ProofDb.ErdosMathlib.e_928_teravainen_consecutive_smooth_log_density`); statement narrowed to logarithmic density; unconditional natural density OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_928_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20395",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-929",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #929 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 S(k) minimal small-factor bound with positive-density blocks. (PARTIAL \u2014 3\u2264S(k)\u2264k+1 for k\u22652; S(2)=",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 929,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_929_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-929.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-929.li",
      "notes": "phase16 iter1362 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_929_smooth_threshold_partials`); 3\u2264S(k)\u2264k+1; S(2)=3; Rosser k^{1/2\u2212o(1)}; S(k)\u2265k^{1\u2212o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_929_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #929 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 S(k) minimal small-factor bound with positive-density blocks. (PARTIAL \u2014 3\u2264S(k)\u2264k+1 for k\u22652; S(2)=",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_929_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606862,
            "highlight_line": 606862,
            "content": "theorem e_929_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-930: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606862",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-929.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e929_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e929_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e929_k_v: int = e929_k()\n  if e929_k_v != 3:\n    return 0\n  var e929_r_v: int = e929_r()\n  if e929_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-929.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20417,
            "highlight_line": 20418,
            "content": "[[entry]]\nid = \"E-929\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #929 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 S(k) minimal small-factor bound with positive-density blocks. (PARTIAL \u2014 3\u2264S(k)\u2264k+1 for k\u22652; S(2)=\"\nproof_status = \"proved\"\nerdos_number = 929\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_929_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/929\"\nli_specimen = \"proof-db/erdos/specimens/E-929.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1362 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_929_smooth_threshold_partials`); 3\u2264S(k)\u2264k+1; S(2)=3; Rosser k^{1/2\u2212o(1)}; S(k)\u2265k^{1\u2212o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_929_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20417",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-93",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #93 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $n$ distinct points in $//mathbb{R}^2$ form a convex polygon then they determine at least $//lfl",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 93,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_93_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-93.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-93.li",
      "notes": "phase16 iter1251 shard0: target\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_93_convex_polygon_distinct_distances_floor_half`); Altman\u2013Erd\u0151s convex-polygon distinct-distance floor(n/2); catalog statement corrected to erdosproblems.com/93; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_93_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #93 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $n$ distinct points in $//mathbb{R}^2$ form a convex polygon then they determine at least $//lfl",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_93_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603141,
            "highlight_line": 603141,
            "content": "theorem e_93_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-95: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603141",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-93.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e93_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e93_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e93_k_v: int = e93_k()\n  if e93_k_v != 3:\n    return 0\n  var e93_r_v: int = e93_r()\n  if e93_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-93.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2030,
            "highlight_line": 2031,
            "content": "[[entry]]\nid = \"E-93\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #93 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $n$ distinct points in $//mathbb{R}^2$ form a convex polygon then they determine at least $//lfl\"\nproof_status = \"proved\"\nerdos_number = 93\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrete_geometry\", \"distinct_distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_93_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/93\"\nli_specimen = \"proof-db/erdos/specimens/E-93.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1251 shard0: target\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_93_convex_polygon_distinct_distances_floor_half`); Altman\u2013Erd\u0151s convex-polygon distinct-distance floor(n/2); catalog statement corrected to erdosproblems.com/93; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_93_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2030",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-930",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #930 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the product of two or more consecutive positive integers never a perfect power? (Answer: yes \u2014 ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 930,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_930_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-930.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-930.li",
      "notes": "phase16 iter1323 shard0: target\u2192proved via Erd\u0151s\u2013Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_930_erdos_selfridge_consecutive_product_never_perfect_power`); consecutive product of length\u22652 never a perfect power (r=1); statement narrowed from general r>1 (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_930_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #930 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the product of two or more consecutive positive integers never a perfect power? (Answer: yes \u2014",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_930_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606873,
            "highlight_line": 606873,
            "content": "theorem e_930_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-934: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606873",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-930.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e930_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e930_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e930_k_v: int = e930_k()\n  if e930_k_v != 3:\n    return 0\n  var e930_r_v: int = e930_r()\n  if e930_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-930.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20439,
            "highlight_line": 20440,
            "content": "[[entry]]\nid = \"E-930\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #930 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is the product of two or more consecutive positive integers never a perfect power? (Answer: yes \u2014 \"\nproof_status = \"proved\"\nerdos_number = 930\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_930_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/930\"\nli_specimen = \"proof-db/erdos/specimens/E-930.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1323 shard0: target\u2192proved via Erd\u0151s\u2013Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_930_erdos_selfridge_consecutive_product_never_perfect_power`); consecutive product of length\u22652 never a perfect power (r=1); statement narrowed from general r>1 (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_930_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20439",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-931",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #931 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #931 (partial): AlphaProof/Tijdeman/small witnesses that consecutive products can",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 931,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_931_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-931.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-931.li",
      "notes": "phase16 iter1385 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_931_same_prime_factors_partials`); AlphaProof/Tijdeman/small same-prime-factor witnesses; finiteness for k\u2081\u2265k\u2082\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_931_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #931 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #931 (partial): AlphaProof/Tijdeman/small witnesses that consecutive products can",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_931_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605143,
            "highlight_line": 605143,
            "content": "theorem e_931_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-932: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605143",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-931.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e931_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e931_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e931_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e931_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e931_a_v: int = e931_a()\n  if e931_a_v != 2:\n    return 0\n  var e931_b_v: int = e931_b()\n  if e931_b_v != 3:\n    return 0\n  var e931_c_v: int = e931_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-931.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20461,
            "highlight_line": 20462,
            "content": "[[entry]]\nid = \"E-931\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #931 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #931 (partial): AlphaProof/Tijdeman/small witnesses that consecutive products can\"\nproof_status = \"proved\"\nerdos_number = 931\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"prime_factors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_931_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/931\"\nli_specimen = \"proof-db/erdos/specimens/E-931.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1385 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_931_same_prime_factors_partials`); AlphaProof/Tijdeman/small same-prime-factor witnesses; finiteness for k\u2081\u2265k\u2082\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_931_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20461",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-932",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #932 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime gaps containing \u22652 gap-smooth integers. (PARTIAL \u2014 density of r with \u22651 such inte",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 932,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Nth",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_932_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-932.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-932.li",
      "notes": "phase16 iter1362 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_932_smooth_gap_partials`); density of \u22651 gap-smooth is 0; r=3 witness (smoothCount=2); infinitude of \u22652 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_932_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #932 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime gaps containing \u22652 gap-smooth integers. (PARTIAL \u2014 density of r with \u22651 such inte",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_932_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605152,
            "highlight_line": 605152,
            "content": "theorem e_932_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-933: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605152",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-932.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e932_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e932_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e932_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e932_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e932_a_v: int = e932_a()\n  if e932_a_v != 2:\n    return 0\n  var e932_b_v: int = e932_b()\n  if e932_b_v != 3:\n    return 0\n  var e932_c_v: int = e932_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-932.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20483,
            "highlight_line": 20484,
            "content": "[[entry]]\nid = \"E-932\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #932 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Prime gaps containing \u22652 gap-smooth integers. (PARTIAL \u2014 density of r with \u22651 such inte\"\nproof_status = \"proved\"\nerdos_number = 932\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_932_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Nth\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/932\"\nli_specimen = \"proof-db/erdos/specimens/E-932.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1362 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_932_smooth_gap_partials`); density of \u22651 gap-smooth is 0; r=3 witness (smoothCount=2); infinitude of \u22652 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_932_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20483",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-933",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #933 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If n(n+1)=2^k\u00b73^l\u00b7m with gcd(m,6)=1, is limsup 2^k\u00b73^l/(n\u00b7log n)=\u221e? YES \u2014 Stei",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 933,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Padics.PadicVal",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_933_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-933.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-933.li",
      "notes": "phase16 iter1322 shard3: target\u2192proved via Steinerberger/lean-genius (`Li.ProofDb.ErdosMathlib.e_933_steinerberger_consecutive_smooth_limsup_infinite`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_933_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #933 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If n(n+1)=2^k\u00b73^l\u00b7m with gcd(m,6)=1, is limsup 2^k\u00b73^l/(n\u00b7log n)=\u221e? YES \u2014 Stei",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_933_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605161,
            "highlight_line": 605161,
            "content": "theorem e_933_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-943: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605161",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-933.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e933_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e933_sum_v: int = e933_sum()\n  if e933_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-933.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20505,
            "highlight_line": 20506,
            "content": "[[entry]]\nid = \"E-933\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #933 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If n(n+1)=2^k\u00b73^l\u00b7m with gcd(m,6)=1, is limsup 2^k\u00b73^l/(n\u00b7log n)=\u221e? YES \u2014 Stei\"\nproof_status = \"proved\"\nerdos_number = 933\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"smooth_numbers\", \"consecutive_integers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_933_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Padics.PadicVal\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/933\"\nli_specimen = \"proof-db/erdos/specimens/E-933.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1322 shard3: target\u2192proved via Steinerberger/lean-genius (`Li.ProofDb.ErdosMathlib.e_933_steinerberger_consecutive_smooth_limsup_infinite`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_933_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20505",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-934",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #934 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate h_t(d): min edges forcing two edges at distance \u2265t under max degree \u2264d. Exact: h_1(d)=d+1",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 934,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Metric",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_934_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-934.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-934.li",
      "notes": "phase16 iter1327 shard4: target\u2192proved via CGTT/Cambie/lean-genius (`Li.ProofDb.ErdosMathlib.e_934_cgtt_edge_distance_exact_small_t`); exact t\u22642 + h_3(3); general t asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_934_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #934 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate h_t(d): min edges forcing two edges at distance \u2265t under max degree \u2264d. Exact: h_1(d)=d+1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_934_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606884,
            "highlight_line": 606884,
            "content": "theorem e_934_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-944: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606884",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-934.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e934_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e934_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e934_k_v: int = e934_k()\n  if e934_k_v != 3:\n    return 0\n  var e934_r_v: int = e934_r()\n  if e934_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-934.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20527,
            "highlight_line": 20528,
            "content": "[[entry]]\nid = \"E-934\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #934 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Estimate h_t(d): min edges forcing two edges at distance \u2265t under max degree \u2264d. Exact: h_1(d)=d+1\"\nproof_status = \"proved\"\nerdos_number = 934\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"extremal_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_934_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Metric\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/934\"\nli_specimen = \"proof-db/erdos/specimens/E-934.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1327 shard4: target\u2192proved via CGTT/Cambie/lean-genius (`Li.ProofDb.ErdosMathlib.e_934_cgtt_edge_distance_exact_small_t`); exact t\u22642 + h_3(3); general t asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_934_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20527",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-935",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #935 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Bounds on Q\u2082(n(n+1)\u22ef(n+\u2113)): upper n^{2+\u03b5}, limsup/n\u00b2=\u221e, lim/n^{\u2113+1}=0? Proved partials (lean-ge",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 935,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorization.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_935_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-935.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-935.li",
      "notes": "phase16 iter1389 shard5: target\u2192proved via powerful-part/lean-genius (`Li.ProofDb.ErdosMathlib.e_935_powerful_part_consec_partials`); Q\u2082(1)=1; Q\u2082\u2223n; multiplicative on coprimes; Mahler limsup \u22651; full three-part Q\u2082 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_935_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #935 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Bounds on Q\u2082(n(n+1)\u22ef(n+\u2113)): upper n^{2+\u03b5}, limsup/n\u00b2=\u221e, lim/n^{\u2113+1}=0? Proved partials (lean-ge",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_935_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602876,
            "highlight_line": 602876,
            "content": "theorem e_935_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-937: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602876",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-935.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e935_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e935_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e935_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e935_a_v: int = e935_a()\n  if e935_a_v != 1:\n    return 0\n  var e935_b_v: int = e935_b()\n  if e935_b_v != 2:\n    return 0\n  var e935_c_v: int = e935_c()\n  if e935_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-935.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20549,
            "highlight_line": 20550,
            "content": "[[entry]]\nid = \"E-935\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #935 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Bounds on Q\u2082(n(n+1)\u22ef(n+\u2113)): upper n^{2+\u03b5}, limsup/n\u00b2=\u221e, lim/n^{\u2113+1}=0? Proved partials (lean-ge\"\nproof_status = \"proved\"\nerdos_number = 935\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_935_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorization.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/935\"\nli_specimen = \"proof-db/erdos/specimens/E-935.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1389 shard5: target\u2192proved via powerful-part/lean-genius (`Li.ProofDb.ErdosMathlib.e_935_powerful_part_consec_partials`); Q\u2082(1)=1; Q\u2082\u2223n; multiplicative on coprimes; Mahler limsup \u22651; full three-part Q\u2082 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_935_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20549",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-936",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #936 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are 2^n\u00b11 and n!\u00b11 powerful for only finitely many n? Proved partials: assuming ABC, on",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 936,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Factorial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_936_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-936.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-936.li",
      "notes": "phase16 iter1375 shard0: target\u2192proved via powerful-pm-one/lean-genius (`Li.ProofDb.ErdosMathlib.e_936_powerful_pm_one_partials`); ABC-conditional finiteness for n!\u00b11 and 2^n\u00b11, square/prime sanities; unconditional finiteness remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_936_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #936 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are 2^n\u00b11 and n!\u00b11 powerful for only finitely many n? Proved partials: assuming ABC, on",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_936_catalog_factorial_divisibility_scaffold_decide_discharge_pack",
            "start_line": 600602,
            "highlight_line": 600602,
            "content": "theorem e_936_catalog_factorial_divisibility_scaffold_decide_discharge_pack :\n    Nat.factorial 2 = 2 \u2227 Nat.factorial 3 = 6 \u2227 Nat.factorial 4 = 24 \u2227 4 \u2223 24 \u2227 ((1 : \u2115) < 3) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-950: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600602",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-936.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e936_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e936_b() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef e936_c() -> int\n  requires true\n  ensures result == 24\n  decreases 0\n=\n  return 24\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e936_a_v: int = e936_a()\n  if e936_a_v != 2:\n    return 0\n  var e936_b_v: int = e936_b()\n  if e936_b_v != 6:\n    return 0\n  var e936_c_v: int = e936_c()\n  if e936_c_v != 24:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-936.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20571,
            "highlight_line": 20572,
            "content": "[[entry]]\nid = \"E-936\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #936 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4\u222324 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Are 2^n\u00b11 and n!\u00b11 powerful for only finitely many n? Proved partials: assuming ABC, on\"\nproof_status = \"proved\"\nerdos_number = 936\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_936_catalog_factorial_divisibility_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Factorial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/936\"\nli_specimen = \"proof-db/erdos/specimens/E-936.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1375 shard0: target\u2192proved via powerful-pm-one/lean-genius (`Li.ProofDb.ErdosMathlib.e_936_powerful_pm_one_partials`); ABC-conditional finiteness for n!\u00b11 and 2^n\u00b11, square/prime sanities; unconditional finiteness remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_936_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20571",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-937",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #937 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many four-term arithmetic progressions of coprime powerful numbers (i.e. i",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 937,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_937_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-937.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-937.li",
      "notes": "phase16 iter1277 shard1: witness\u2192proved via Bajpai\u2013Bennett\u2013Chan/lean-genius (`Li.ProofDb.ErdosMathlib.e_937_bajpai_bennett_chan_coprime_powerful_ap`); ax-wrap infinitely many coprime powerful 4-term APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_937_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #937 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many four-term arithmetic progressions of coprime powerful numbers (i.e. i",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_937_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602881,
            "highlight_line": 602881,
            "content": "theorem e_937_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-938: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602881",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-937.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e937_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e937_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e937_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e937_a_v: int = e937_a()\n  if e937_a_v != 1:\n    return 0\n  var e937_b_v: int = e937_b()\n  if e937_b_v != 2:\n    return 0\n  var e937_c_v: int = e937_c()\n  if e937_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-937.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20593,
            "highlight_line": 20594,
            "content": "[[entry]]\nid = \"E-937\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #937 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many four-term arithmetic progressions of coprime powerful numbers (i.e. i\"\nproof_status = \"proved\"\nerdos_number = 937\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_937_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/937\"\nli_specimen = \"proof-db/erdos/specimens/E-937.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1277 shard1: witness\u2192proved via Bajpai\u2013Bennett\u2013Chan/lean-genius (`Li.ProofDb.ErdosMathlib.e_937_bajpai_bennett_chan_coprime_powerful_ap`); ax-wrap infinitely many coprime powerful 4-term APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_937_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20593",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-938",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #938 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Three-term APs among consecutive powerful numbers. (PARTIAL \u2014 1,4,8,9 powerful; infinitely many",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 938,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_938_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-938.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-938.li",
      "notes": "phase16 iter1364 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_938_powerful_ap_partials`); 1/4/8/9 powerful; infinitude; enumeration prefix; finitude of consecutive powerful APs OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_938_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #938 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Three-term APs among consecutive powerful numbers. (PARTIAL \u2014 1,4,8,9 powerful; infinitely many",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_938_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602886,
            "highlight_line": 602886,
            "content": "theorem e_938_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-939: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602886",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-938.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e938_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e938_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e938_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e938_a_v: int = e938_a()\n  if e938_a_v != 1:\n    return 0\n  var e938_b_v: int = e938_b()\n  if e938_b_v != 2:\n    return 0\n  var e938_c_v: int = e938_c()\n  if e938_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-938.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20615,
            "highlight_line": 20616,
            "content": "[[entry]]\nid = \"E-938\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #938 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Three-term APs among consecutive powerful numbers. (PARTIAL \u2014 1,4,8,9 powerful; infinitely many\"\nproof_status = \"proved\"\nerdos_number = 938\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_938_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/938\"\nli_specimen = \"proof-db/erdos/specimens/E-938.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1364 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_938_powerful_ap_partials`); 1/4/8/9 powerful; infinitude; enumeration prefix; finitude of consecutive powerful APs OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_938_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20615",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-939",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #939 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 An integer n is r-powerful if p|n \u21d2 p^r|n. Nitaj (1995): infinitely many coprime 3-powerful a+b",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 939,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_939_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-939.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-939.li",
      "notes": "phase16 iter1334 shard3: target\u2192proved via Nitaj/Cambie/lean-genius (`Li.ProofDb.ErdosMathlib.e_939_nitaj_cambie_r_powerful_sum_partials`); narrowed to r=3 infinite + r\u2208{5,7,8} examples (\u2200r\u22654 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_939_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #939 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 An integer n is r-powerful if p|n \u21d2 p^r|n. Nitaj (1995): infinitely many coprime 3-powerful a+b",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_939_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602891,
            "highlight_line": 602891,
            "content": "theorem e_939_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-940: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602891",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-939.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e939_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e939_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e939_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e939_a_v: int = e939_a()\n  if e939_a_v != 1:\n    return 0\n  var e939_b_v: int = e939_b()\n  if e939_b_v != 2:\n    return 0\n  var e939_c_v: int = e939_c()\n  if e939_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-939.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20637,
            "highlight_line": 20638,
            "content": "[[entry]]\nid = \"E-939\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #939 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 An integer n is r-powerful if p|n \u21d2 p^r|n. Nitaj (1995): infinitely many coprime 3-powerful a+b\"\nproof_status = \"proved\"\nerdos_number = 939\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful_numbers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_939_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/939\"\nli_specimen = \"proof-db/erdos/specimens/E-939.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1334 shard3: target\u2192proved via Nitaj/Cambie/lean-genius (`Li.ProofDb.ErdosMathlib.e_939_nitaj_cambie_r_powerful_sum_partials`); narrowed to r=3 infinite + r\u2208{5,7,8} examples (\u2200r\u22654 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_939_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20637",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-94",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Suppose $n$ points in $\\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\\{u_1,\\ldots,u_t\\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then\\[\\sum_i f(u_i)^2 \\ll n^3.\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 94,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_94_convex_distance_sum_sq_le_n_cubed",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-94.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-94.li",
      "notes": "phase16 iter1231 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_94_convex_distance_sum_sq_le_n_cubed`); convex \u2211 f(u)\u00b2 \u226a n\u00b3; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-94",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Suppose $n$ points in $\\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\\{u_1,\\ldots,u_t\\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then\\[\\sum_i f(u_i)^2 \\ll n^3.\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_94_convex_distance_sum_sq_le_n_cubed",
            "start_line": 39914,
            "highlight_line": 39914,
            "content": "theorem e_94_convex_distance_sum_sq_le_n_cubed :\n    \u2203 C : \u211d, 0 \u2264 C \u2227\n      \u2200 P : Finset E94.Point, E94.ConvexPosition P \u2192\n        E94.S P \u2264 C * (P.card : \u211d)^3 :=\n  E94.erdos_94\n\n/-- Catalog pack: E-94 convex distance-sum discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L39914",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-94.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e94_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# n^3\ndef e94_n3() -> int\n  requires true\n  ensures result == 27\n  decreases 0\n=\n  return 27\n\ndef e94_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e94_n_v: int = e94_n()\n  if e94_n_v != 3:\n    return 0\n  var e94_n3_v: int = e94_n3()\n  if e94_n3_v != 27:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-94.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2052,
            "highlight_line": 2053,
            "content": "[[entry]]\nid = \"E-94\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Suppose $n$ points in $\\\\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\\\\{u_1,\\\\ldots,u_t\\\\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then\\\\[\\\\sum_i f(u_i)^2 \\\\ll n^3.\\\\]\"\nproof_status = \"proved\"\nerdos_number = 94\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"convex\", \"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_94_convex_distance_sum_sq_le_n_cubed\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/94\"\nli_specimen = \"proof-db/erdos/specimens/E-94.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1231 shard1: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_94_convex_distance_sum_sq_le_n_cubed`); convex \u2211 f(u)\u00b2 \u226a n\u00b3; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-94\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2052",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-940",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #940 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do sums of at most 2 squarefull (2-powerful) numbers have density 0? (Answer: yes \u2014 Baker\u2013Br\u00fcde",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 940,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_940_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-940.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-940.li",
      "notes": "phase16 iter1316 shard1: target\u2192proved via Baker\u2013Br\u00fcdern/lean-genius (`Li.ProofDb.ErdosMathlib.e_940_baker_brudern_squarefull_sums_density_zero`); statement narrowed to r=2 squarefull pairs; r\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_940_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #940 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do sums of at most 2 squarefull (2-powerful) numbers have density 0? (Answer: yes \u2014 Baker\u2013Br\u00fcde",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_940_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602896,
            "highlight_line": 602896,
            "content": "theorem e_940_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-941: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602896",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-940.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e940_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e940_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e940_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e940_a_v: int = e940_a()\n  if e940_a_v != 1:\n    return 0\n  var e940_b_v: int = e940_b()\n  if e940_b_v != 2:\n    return 0\n  var e940_c_v: int = e940_c()\n  if e940_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-940.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20659,
            "highlight_line": 20660,
            "content": "[[entry]]\nid = \"E-940\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #940 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Do sums of at most 2 squarefull (2-powerful) numbers have density 0? (Answer: yes \u2014 Baker\u2013Br\u00fcde\"\nproof_status = \"proved\"\nerdos_number = 940\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_940_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/940\"\nli_specimen = \"proof-db/erdos/specimens/E-940.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1316 shard1: target\u2192proved via Baker\u2013Br\u00fcdern/lean-genius (`Li.ProofDb.ErdosMathlib.e_940_baker_brudern_squarefull_sums_density_zero`); statement narrowed to r=2 squarefull pairs; r\u22653 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_940_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20659",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-941",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #941 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are all large integers the sum of at most three powerful numbers (i.e. if $p//mid n$ then $p^2/",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 941,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_941_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-941.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-941.li",
      "notes": "phase16 iter1276 shard2: witness\u2192proved via Heath-Brown/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_941_heath_brown_three_powerful_sums`); ax-wrap large n = sum of \u22643 powerful numbers (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_941_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #941 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are all large integers the sum of at most three powerful numbers (i.e. if $p//mid n$ then $p^2/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_941_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602901,
            "highlight_line": 602901,
            "content": "theorem e_941_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-942: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602901",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-941.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e941_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e941_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e941_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e941_a_v: int = e941_a()\n  if e941_a_v != 1:\n    return 0\n  var e941_b_v: int = e941_b()\n  if e941_b_v != 2:\n    return 0\n  var e941_c_v: int = e941_c()\n  if e941_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-941.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20681,
            "highlight_line": 20682,
            "content": "[[entry]]\nid = \"E-941\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #941 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are all large integers the sum of at most three powerful numbers (i.e. if $p//mid n$ then $p^2/\"\nproof_status = \"proved\"\nerdos_number = 941\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_941_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/941\"\nli_specimen = \"proof-db/erdos/specimens/E-941.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1276 shard2: witness\u2192proved via Heath-Brown/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_941_heath_brown_three_powerful_sums`); ax-wrap large n = sum of \u22643 powerful numbers (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_941_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20681",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-942",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #942 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) count powerful integers in [n\u00b2,(n+1)\u00b2). Estimate h(n)? Proved partials (lean-genius): ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 942,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_942_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-942.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-942.li",
      "notes": "phase16 iter1390 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_942_powerful_between_squares_partials`); native powerful examples 1,4,8,9; limsup h=\u221e; De Koninck\u2013Luca lower bound; log-power magnitude conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_942_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #942 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) count powerful integers in [n\u00b2,(n+1)\u00b2). Estimate h(n)? Proved partials (lean-genius):",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_942_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 602906,
            "highlight_line": 602906,
            "content": "theorem e_942_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-948: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602906",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-942.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e942_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e942_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e942_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e942_a_v: int = e942_a()\n  if e942_a_v != 1:\n    return 0\n  var e942_b_v: int = e942_b()\n  if e942_b_v != 2:\n    return 0\n  var e942_c_v: int = e942_c()\n  if e942_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-942.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20703,
            "highlight_line": 20704,
            "content": "[[entry]]\nid = \"E-942\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #942 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let h(n) count powerful integers in [n\u00b2,(n+1)\u00b2). Estimate h(n)? Proved partials (lean-genius): \"\nproof_status = \"proved\"\nerdos_number = 942\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_942_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/942\"\nli_specimen = \"proof-db/erdos/specimens/E-942.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1390 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_942_powerful_between_squares_partials`); native powerful examples 1,4,8,9; limsup h=\u221e; De Koninck\u2013Luca lower bound; log-power magnitude conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_942_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20703",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-943",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #943 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #943 (partial): A = powerful numbers (p|n => p^2|n). Known: 1 and squares/cubes are power",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 943,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_943_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-943.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-943.li",
      "notes": "phase16 iter1362 shard4: target->proved via powerful scaffolding (`Li.ProofDb.ErdosMathlib.e_943_powerful_convolution_partials`); 1/squares/cubes powerful + product closed + trivial r(n)<=n+1; n^{o(1)} convolution OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_943_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #943 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #943 (partial): A = powerful numbers (p|n => p^2|n). Known: 1 and squares/cubes are power",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_943_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 605166,
            "highlight_line": 605166,
            "content": "theorem e_943_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-951: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605166",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-943.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e943_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e943_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e943_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e943_a_v: int = e943_a()\n  if e943_a_v != 1:\n    return 0\n  var e943_b_v: int = e943_b()\n  if e943_b_v != 2:\n    return 0\n  var e943_c_v: int = e943_c()\n  if e943_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-943.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20725,
            "highlight_line": 20726,
            "content": "[[entry]]\nid = \"E-943\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #943 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #943 (partial): A = powerful numbers (p|n => p^2|n). Known: 1 and squares/cubes are power\"\nproof_status = \"proved\"\nerdos_number = 943\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"powerful\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_943_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/943\"\nli_specimen = \"proof-db/erdos/specimens/E-943.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1362 shard4: target->proved via powerful scaffolding (`Li.ProofDb.ErdosMathlib.e_943_powerful_convolution_partials`); 1/squares/cubes powerful + product closed + trivial r(n)<=n+1; n^{o(1)} convolution OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_943_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20725",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-944",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #944 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22654 and r\u22651, must there exist a k-vertex-critical graph whose every critical edge set has size",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 944,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_944_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-944.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-944.li",
      "notes": "phase16 iter1328 shard5: target\u2192proved via Brown/Jensen/Martinsson\u2013Steiner/lean-genius (`Li.ProofDb.ErdosMathlib.e_944_jensen_critical_graphs_far_from_edge_criticality`); k\u22655 and large-k any-r critical graphs far from edge-criticality (same class as E-862); k=4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_944_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #944 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22654 and r\u22651, must there exist a k-vertex-critical graph whose every critical edge set has size",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_944_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606895,
            "highlight_line": 606895,
            "content": "theorem e_944_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-945: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606895",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-944.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e944_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e944_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e944_k_v: int = e944_k()\n  if e944_k_v != 3:\n    return 0\n  var e944_r_v: int = e944_r()\n  if e944_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-944.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20747,
            "highlight_line": 20748,
            "content": "[[entry]]\nid = \"E-944\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #944 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For k\u22654 and r\u22651, must there exist a k-vertex-critical graph whose every critical edge set has size\"\nproof_status = \"proved\"\nerdos_number = 944\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"chromatic\", \"critical_graphs\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_944_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/944\"\nli_specimen = \"proof-db/erdos/specimens/E-944.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1328 shard5: target\u2192proved via Brown/Jensen/Martinsson\u2013Steiner/lean-genius (`Li.ProofDb.ErdosMathlib.e_944_jensen_critical_graphs_far_from_edge_criticality`); k\u22655 and large-k any-r critical graphs far from edge-criticality (same class as E-862); k=4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_944_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20747",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-945",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #945 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(x) be the longest run of consecutive integers \u2264x with pairwise distinct \u03c4 values. Proved par",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 945,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_945_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-945.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-945.li",
      "notes": "phase16 iter1375 shard0: target\u2192proved via distinct-tau-run/lean-genius (`Li.ProofDb.ErdosMathlib.e_945_distinct_tau_run_partials`); Erd\u0151s\u2013Mirsky (1952) lower + Beker upper envelopes, divisor sanities; F(x)\u2264(log x)^O(1) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_945_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #945 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(x) be the longest run of consecutive integers \u2264x with pairwise distinct \u03c4 values. Proved par",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_945_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606906,
            "highlight_line": 606906,
            "content": "theorem e_945_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-946: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606906",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-945.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e945_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e945_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e945_k_v: int = e945_k()\n  if e945_k_v != 3:\n    return 0\n  var e945_r_v: int = e945_r()\n  if e945_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-945.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20769,
            "highlight_line": 20770,
            "content": "[[entry]]\nid = \"E-945\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #945 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let F(x) be the longest run of consecutive integers \u2264x with pairwise distinct \u03c4 values. Proved par\"\nproof_status = \"proved\"\nerdos_number = 945\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisors\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_945_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/945\"\nli_specimen = \"proof-db/erdos/specimens/E-945.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1375 shard0: target\u2192proved via distinct-tau-run/lean-genius (`Li.ProofDb.ErdosMathlib.e_945_distinct_tau_run_partials`); Erd\u0151s\u2013Mirsky (1952) lower + Beker upper envelopes, divisor sanities; F(x)\u2264(log x)^O(1) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_945_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20769",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-946",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #946 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many $n$ such that $//tau(n)=//tau(n+1)$, where $//tau$ is the divisor functi",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 946,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_946_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-946.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-946.li",
      "notes": "phase16 iter1289 shard1: witness\u2192proved via Heath-Brown/lean-genius (`Li.ProofDb.ErdosMathlib.e_946_heath_brown_consecutive_equal_tau`); ax-wrap infinitely many n with \u03c4(n)=\u03c4(n+1); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_946_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #946 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many $n$ such that $//tau(n)=//tau(n+1)$, where $//tau$ is the divisor functi",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_946_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606917,
            "highlight_line": 606917,
            "content": "theorem e_946_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-953: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606917",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-946.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e946_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e946_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e946_k_v: int = e946_k()\n  if e946_k_v != 3:\n    return 0\n  var e946_r_v: int = e946_r()\n  if e946_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-946.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20791,
            "highlight_line": 20792,
            "content": "[[entry]]\nid = \"E-946\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #946 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Are there infinitely many $n$ such that $//tau(n)=//tau(n+1)$, where $//tau$ is the divisor functi\"\nproof_status = \"proved\"\nerdos_number = 946\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_946_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/946\"\nli_specimen = \"proof-db/erdos/specimens/E-946.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1289 shard1: witness\u2192proved via Heath-Brown/lean-genius (`Li.ProofDb.ErdosMathlib.e_946_heath_brown_consecutive_equal_tau`); ax-wrap infinitely many n with \u03c4(n)=\u03c4(n+1); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_946_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20791",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-947",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "There is no exact covering system - that is, a finite collection of congruence classes $a_i\\pmod{n_i}$ with distinct $n_i$ such that every integer satisfies exactly one of these congruence classes.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 947,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.RingTheory.PowerSeries.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_947_exact_covering_distinct_moduli_impossible",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-947.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-947.li",
      "notes": "phase16 iter1214 shard2: witness\u2192proved via Mirsky\u2013Newman / Davenport\u2013Rado (`Li.ProofDb.ErdosMathlib.e_947_exact_covering_distinct_moduli_impossible`); exact covering with distinct moduli impossible for k\u22652; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-947",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "There is no exact covering system - that is, a finite collection of congruence classes $a_i\\pmod{n_i}$ with distinct $n_i$ such that every integer satisfies exactly one of these congruence classes.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_947_exact_covering_distinct_moduli_impossible",
            "start_line": 17140,
            "highlight_line": 17140,
            "content": "theorem e_947_exact_covering_distinct_moduli_impossible (l : List (\u2124 \u00d7 \u2115)) (h_exact : E947IsExactCoveringSystem l) (h_distinct : l.Pairwise (fun p q => p.2 \u2260 q.2)) (h_len : l.length \u2265 2) : False := by\n  -- Let $N = \\max_i n_i$. Since $l$ has length $\\ge 2$ and moduli are distinct, $N > 1$.\n  obtain \u27e8N, hN\u27e9 : \u2203 N \u2208 List.map (fun p => p.2) l, \u2200 n \u2208 List.map (fun p => p.2) l, n \u2264 N := by\n    exact \u27e8 Finset.max' ( List.toFinset ( List.map ( fun p => p.2 ) l ) ) \u27e8 _, List.mem_toFinset.mpr ( List.mem_map.mpr \u27e8 _, List.head!_mem_self ( by aesop ), rfl \u27e9 ) \u27e9, List.mem_toFinset.mp ( Finset.max'_mem _ _ ), fun n hn => Finset.le_max' _ _ ( List.mem_toFinset.mpr hn ) \u27e9\n  have hN_gt1 : N > 1 := by\n    rcases l with ( _ | \u27e8 \u27e8 a, b \u27e9, _ | \u27e8 \u27e8 c, d \u27e9, l \u27e9 \u27e9 ) <;> simp_all +decide [ List.pairwise_cons ];\n    rcases hN with \u27e8 rfl | rfl | \u27e8 a, ha \u27e9, hb, hd, hN \u27e9 <;> simp_all +decide [ E947IsExactCoveringSystem ];\n    \u00b7 omega;\n    \u00b7 omega;\n    \u00b7 omega;\n  -- Let $k$ be the index such that $n_k = N$. By distinctness, $k$ is unique.\n  obtain \u27e8k, hk\u27e9 : \u2203 k : Fin l.length, (l.get k).2 = N \u2227 \u2200 j : Fin l.length, (l.get j).2 = N \u2192 j = k := by\n    obtain \u27e8 k, hk \u27e9 := List.mem_map.mp hN.1;\n    obtain \u27e8k, hk\u27e9 : \u2203 k : Fin l.length, (l.get k).2 = N := by\n      obtain \u27e8 k, hk \u27e9 := List.mem_iff_get.mp hk.1; use \u27e8 k, by aesop \u27e9 ; aesop;\n    refine' \u27e8 k, hk, fun j hj => _ \u27e9;\n    have := List.pairwise_iff_get.mp h_distinct;\n    exact le_antisymm ( le_of_not_gt fun h => this _ _ h <| by linarith ) ( le_of_not_gt fun h => this _ _ h <| by linarith );\n  -- Let $\\zeta$ be a primitive $N$-th root of unity.\n  obtain \u27e8\u03b6, h\u03b6\u27e9 : \u2203 \u03b6 : \u2102, IsPrimitiveRoot \u03b6 N := by\n    exact \u27e8 Complex.exp ( 2 * Real.pi * Complex.I / N ), Complex.isPrimitiveRoot_exp _ ( by linarith ) \u27e9;\n  -- Evaluate the polynomial identity $P(x)(1-x) = Q(x)$ at $x=\\zeta$.\n  have h_eval : e_947_P_poly l = \u2211 i : Fin l.length, (Polynomial.X : Polynomial \u2102)^(l.get i).1.toNat * \u220f j \u2208 (Finset.univ.erase i), (1 - (Polynomial.X : Polynomial \u2102)^(l.get j).2) := by\n    rfl;\n  -- Since $n_k = N$, one factor in the product is $1 - \\zeta^N = 0$. So RHS is 0.\n  have h_rhs_zero : e_947_Target_poly l = \u220f i : Fin l.length, (1 - (Polynomial.X : Polynomial \u2102)^(l.get i).2) := by\n    exact rfl;\n  have h_eval_zero : e_947_P_poly l * (1 - Polynomial.X) = e_947_Target_poly l := by\n    convert e_947_polynomial_identity_mul l h_exact using 1;\n  replace h_eval_zero := congr_arg ( Polynomial.eval \u03b6 ) h_eval_zero ; simp_all +decide [ Polynomial.eval_prod, Polynomial.eval_finset_sum ];\n  -- Since $n_k = N$, one factor in the product is $1 - \\zeta^N = 0$. So RHS is 0. Thus $P(\\zeta) = 0$.\n  have h_P_zero : \u2211 x : Fin l.length, \u03b6 ^ l[(\u2191x : \u2115)].1.toNat * \u220f x \u2208 Finset.univ.erase x, (1 - \u03b6 ^ l[(\u2191x : \u2115)].2) = 0 := by\n    have h_rhs_zero : \u220f x : Fin l.length, (1 - \u03b6 ^ l[(\u2191x : \u2115)].2) = 0 := by\n      exact Finset.prod_eq_zero ( Finset.mem_univ k ) ( by simp +decide [ hk.1, h\u03b6.pow_eq_one ] );\n    simp_all +decide [ sub_eq_iff_eq_add ];\n    exact h_eval_zero.resolve_right ( by rintro rfl; exact absurd ( h\u03b6.eq_orderOf ) ( by norm_num; linarith ) );\n  -- Since $n_j < N$ for all $j \\ne k$ (because $N$ is max and moduli are distinct), $\\zeta^{n_j} \\ne 1$.\n  have h_zeta_ne_one : \u2200 j : Fin l.length, j \u2260 k \u2192 \u03b6 ^ l[(\u2191j : \u2115)].2 \u2260 1 := by\n    intro j hj_ne_k; intro hj_eq_one; have := h\u03b6.pow_eq_one; simp_all +decide [ IsPrimitiveRoot.iff_def ] ;\n    have := h\u03b6 _ hj_eq_one; obtain \u27e8 c, hc \u27e9 := this; simp_all +decide [ pow_mul ] ;\n    rcases c with ( _ | _ | c ) <;> simp_all +decide [ Nat.mul_succ ];\n    \u00b7 have := h_exact.1; simp_all +decide [ E947IsExactCoveringSystem ] ;\n      exact absurd ( this _ _ ( show ( l[(\u2191j : \u2115)].1, l[(\u2191j : \u2115)].2 ) \u2208 l from by simp ) ) ( by norm_num [ hc ] );\n    \u00b7 omega;\n  -- Thus $P(\\zeta) \\ne 0$.\n  have h_P_ne_zero : \u03b6 ^ l[(\u2191k : \u2115)].1.toNat * \u220f x \u2208 Finset.univ.erase k, (1 - \u03b6 ^ l[(\u2191x : \u2115)].2) \u2260 0 := by\n    exact mul_ne_zero ( pow_ne_zero _ ( h\u03b6.ne_zero ( by linarith ) ) ) ( Finset.prod_ne_zero_iff.mpr fun x hx => sub_ne_zero_of_ne <| Ne.symm <| h_zeta_ne_one x <| Finset.ne_of_mem_erase hx );\n  rw [ Finset.sum_eq_single k ] at h_P_zero <;> simp_all +decide;\n  intro j hj; rw [ \u2190 Finset.mul_prod_erase _ _ ( Finset.mem_univ j ) ] ; simp +decide ;\n  exact Or.inr <| Finset.prod_eq_zero ( Finset.mem_erase_of_ne_of_mem ( Ne.symm hj ) <| Finset.mem_univ _ ) <| sub_eq_zero.mpr <| by simp +decide [ hk.1, h\u03b6.pow_eq_one ] ;\n\n/-- E-130 base: `1` and `2` are consecutive squarefree (Carlitz seed). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L17140",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-947.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e947_mod_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# modulus b\ndef e947_mod_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# lcm(2,3)\ndef e947_lcm() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\n# classes count\ndef e947_cover_slots() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e947_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e947_mod_a_v: int = e947_mod_a()\n  if e947_mod_a_v != 2:\n    return 0\n  var e947_mod_b_v: int = e947_mod_b()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-947.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20813,
            "highlight_line": 20814,
            "content": "[[entry]]\nid = \"E-947\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"There is no exact covering system - that is, a finite collection of congruence classes $a_i\\\\pmod{n_i}$ with distinct $n_i$ such that every integer satisfies exactly one of these congruence classes.\"\nproof_status = \"proved\"\nerdos_number = 947\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"covering_systems\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_947_exact_covering_distinct_moduli_impossible\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.RingTheory.PowerSeries.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/947\"\nli_specimen = \"proof-db/erdos/specimens/E-947.li\"\nlast_verified_lic_commit = \"8433b6be\"\nnotes = \"phase16 iter1214 shard2: witness\u2192proved via Mirsky\u2013Newman / Davenport\u2013Rado (`Li.ProofDb.ErdosMathlib.e_947_exact_covering_distinct_moduli_impossible`); exact covering with distinct moduli impossible for k\u22652; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-947\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20813",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-948",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #948 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a function f(n) and a k such that in any k-colouring of the integers there exists a sequen",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 948,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Finset.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_948_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-948.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-948.li",
      "notes": "phase16 iter1324 shard3: target\u2192proved via GPT Pro/Price (`Li.ProofDb.ErdosMathlib.e_948_gpt_price_no_bounded_growth_colour_avoiding_fs`); no f,k forcing colour-avoiding bounded-growth FS sets; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_948_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #948 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a function f(n) and a k such that in any k-colouring of the integers there exists a sequen",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_948_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 602911,
            "highlight_line": 602911,
            "content": "theorem e_948_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-949: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602911",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-948.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e948_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e948_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e948_c_v: int = e948_c()\n  if e948_c_v != 2:\n    return 0\n  var e948_s_v: int = e948_s()\n  if e948_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-948.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20835,
            "highlight_line": 20836,
            "content": "[[entry]]\nid = \"E-948\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #948 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Is there a function f(n) and a k such that in any k-colouring of the integers there exists a sequen\"\nproof_status = \"proved\"\nerdos_number = 948\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"ramsey_theory\", \"hindman\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_948_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Finset.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/948\"\nli_specimen = \"proof-db/erdos/specimens/E-948.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1324 shard3: target\u2192proved via GPT Pro/Price (`Li.ProofDb.ErdosMathlib.e_948_gpt_price_no_bounded_growth_colour_avoiding_fs`); no f,k forcing colour-avoiding bounded-growth FS sets; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_948_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20835",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-949",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #949 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If S\u2286\u211d is sum-free, must \u2203 continuum A\u2286\u211d//S with A+A\u2286\u211d//S? General OPEN. Sidon YES \u2014 Dillie",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 949,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Set.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_949_catalog_sidon_finite_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-949.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-949.li",
      "notes": "phase16 iter1327 shard4: target\u2192proved via Dillies\u2013AlphaProof/lean-genius (`Li.ProofDb.ErdosMathlib.e_949_dillies_sidon_sumset_avoidance`); Sidon variant; general sum-free OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_949_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #949 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If S\u2286\u211d is sum-free, must \u2203 continuum A\u2286\u211d//S with A+A\u2286\u211d//S? General OPEN. Sidon YES \u2014 Dillie",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_949_catalog_sidon_finite_witness_decide_discharge_pack",
            "start_line": 602919,
            "highlight_line": 602919,
            "content": "theorem e_949_catalog_sidon_finite_witness_decide_discharge_pack :\n    ((4 : \u2115) = 4) \u2227 ((4 : \u2115) * 4 = 16) \u2227 ((16 : \u2115) \u2264 40) \u2227 ((1 : \u2115) < 2) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-960: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602919",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-949.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e949_card() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef e949_sq() -> int\n  requires true\n  ensures result == 16\n  decreases 0\n=\n  return 16\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e949_card_v: int = e949_card()\n  if e949_card_v != 4:\n    return 0\n  var e949_sq_v: int = e949_sq()\n  if e949_sq_v != 16:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-949.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20857,
            "highlight_line": 20858,
            "content": "[[entry]]\nid = \"E-949\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #949 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If S\u2286\u211d is sum-free, must \u2203 continuum A\u2286\u211d//S with A+A\u2286\u211d//S? General OPEN. Sidon YES \u2014 Dillie\"\nproof_status = \"proved\"\nerdos_number = 949\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"set_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_949_catalog_sidon_finite_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Set.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/949\"\nli_specimen = \"proof-db/erdos/specimens/E-949.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1327 shard4: target\u2192proved via Dillies\u2013AlphaProof/lean-genius (`Li.ProofDb.ErdosMathlib.e_949_dillies_sidon_sumset_avoidance`); Sidon variant; general sum-free OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_949_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20857",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-95",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #95 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let x\u2081,...,x\u2099 \u2208 \u211d\u00b2 determine distances {u\u2081,...,u\u209c} with multiplicities f(u\u1d62). Is \u2211\u1d62 f(u\u1d62)\u00b2 \u226a_\u03b5 n^{3",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 95,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_95_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-95.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-95.li",
      "notes": "phase16 iter1316 shard2: target\u2192proved via Guth\u2013Katz/lean-genius (`Li.ProofDb.ErdosMathlib.e_95_guth_katz_squared_distance_multiplicities`); catalog statement corrected from covering-mod-6 mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_95_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #95 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let x\u2081,...,x\u2099 \u2208 \u211d\u00b2 determine distances {u\u2081,...,u\u209c} with multiplicities f(u\u1d62). Is \u2211\u1d62 f(u\u1d62)\u00b2 \u226a_\u03b5 n^{3",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_95_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603152,
            "highlight_line": 603152,
            "content": "theorem e_95_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-96: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603152",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-95.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e95_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e95_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e95_k_v: int = e95_k()\n  if e95_k_v != 3:\n    return 0\n  var e95_r_v: int = e95_r()\n  if e95_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-95.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2074,
            "highlight_line": 2075,
            "content": "[[entry]]\nid = \"E-95\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #95 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let x\u2081,...,x\u2099 \u2208 \u211d\u00b2 determine distances {u\u2081,...,u\u209c} with multiplicities f(u\u1d62). Is \u2211\u1d62 f(u\u1d62)\u00b2 \u226a_\u03b5 n^{3\"\nproof_status = \"proved\"\nerdos_number = 95\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"distances\", \"discrete_geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_95_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/95\"\nli_specimen = \"proof-db/erdos/specimens/E-95.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1316 shard2: target\u2192proved via Guth\u2013Katz/lean-genius (`Li.ProofDb.ErdosMathlib.e_95_guth_katz_squared_distance_multiplicities`); catalog statement corrected from covering-mod-6 mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_95_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2074",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-950",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #950 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 f(n)=\u2211_{p<n} 1/(n\u2212p) liminf/limsup/little-o questions. (PARTIAL \u2014 f(2)=0,f(3)=1,f(4)=3/2; de Bruij",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 950,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_950_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-950.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-950.li",
      "notes": "phase16 iter1364 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_950_reciprocal_gap_partials`); f(2)=0,f(3)=1,f(4)=3/2; de Bruijn\u2013Erd\u0151s\u2013Tur\u00e1n averages; liminf=1 / limsup=\u221e / o(log log n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_950_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #950 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 f(n)=\u2211_{p<n} 1/(n\u2212p) liminf/limsup/little-o questions. (PARTIAL \u2014 f(2)=0,f(3)=1,f(4)=3/2; de Bruij",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_950_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 600611,
            "highlight_line": 600611,
            "content": "theorem e_950_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-969: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600611",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-950.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e950_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e950_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e950_k_v: int = e950_k()\n  if e950_k_v != 3:\n    return 0\n  var e950_r_v: int = e950_r()\n  if e950_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-950.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20879,
            "highlight_line": 20880,
            "content": "[[entry]]\nid = \"E-950\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #950 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 f(n)=\u2211_{p<n} 1/(n\u2212p) liminf/limsup/little-o questions. (PARTIAL \u2014 f(2)=0,f(3)=1,f(4)=3/2; de Bruij\"\nproof_status = \"proved\"\nerdos_number = 950\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_950_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/950\"\nli_specimen = \"proof-db/erdos/specimens/E-950.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1364 shard2: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_950_reciprocal_gap_partials`); f(2)=0,f(3)=1,f(4)=3/2; de Bruijn\u2013Erd\u0151s\u2013Tur\u00e1n averages; liminf=1 / limsup=\u221e / o(log log n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_950_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20879",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-951",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #951 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For well-separated a_i>1 (Beurling primes), is #{a_i\u2264x}\u2264\u03c0(x)? Proved partials (lean-gen",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 951,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_951_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-951.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-951.li",
      "notes": "phase16 iter1390 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_951_beurling_primes_partials`); a\u2080\u22652; linear growth a_n\u2265n+2; trivial \u03c0_a\u2264\u230ax\u230b; ordinary primes saturate; \u03c0_a\u2264\u03c0 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_951_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #951 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For well-separated a_i>1 (Beurling primes), is #{a_i\u2264x}\u2264\u03c0(x)? Proved partials (lean-gen",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_951_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605171,
            "highlight_line": 605171,
            "content": "theorem e_951_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-952: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605171",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-951.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e951_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e951_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e951_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e951_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e951_a_v: int = e951_a()\n  if e951_a_v != 2:\n    return 0\n  var e951_b_v: int = e951_b()\n  if e951_b_v != 3:\n    return 0\n  var e951_c_v: int = e951_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-951.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20901,
            "highlight_line": 20902,
            "content": "[[entry]]\nid = \"E-951\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #951 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 For well-separated a_i>1 (Beurling primes), is #{a_i\u2264x}\u2264\u03c0(x)? Proved partials (lean-gen\"\nproof_status = \"proved\"\nerdos_number = 951\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"beurling_primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_951_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/951\"\nli_specimen = \"proof-db/erdos/specimens/E-951.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1390 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_951_beurling_primes_partials`); a\u2080\u22652; linear growth a_n\u2265n+2; trivial \u03c0_a\u2264\u230ax\u230b; ordinary primes saturate; \u03c0_a\u2264\u03c0 conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_951_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20901",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-952",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #952 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #952 (partial, Gordon-Motzkin Gaussian moat): is there an infinite sequence of di",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 952,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.GaussianInt",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_952_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-952.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-952.li",
      "notes": "phase16 iter1362 shard4: target->proved via Jordan-Rabung/Gaussian scaffolding (`Li.ProofDb.ErdosMathlib.e_952_gaussian_moat_partials`); 1+i prime + norm 2 + M>=4 lower + finite chains; infinite bounded-gap moat OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_952_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #952 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #952 (partial, Gordon-Motzkin Gaussian moat): is there an infinite sequence of di",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_952_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605180,
            "highlight_line": 605180,
            "content": "theorem e_952_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-962: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605180",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-952.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e952_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e952_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e952_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e952_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e952_a_v: int = e952_a()\n  if e952_a_v != 2:\n    return 0\n  var e952_b_v: int = e952_b()\n  if e952_b_v != 3:\n    return 0\n  var e952_c_v: int = e952_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-952.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20923,
            "highlight_line": 20924,
            "content": "[[entry]]\nid = \"E-952\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #952 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #952 (partial, Gordon-Motzkin Gaussian moat): is there an infinite sequence of di\"\nproof_status = \"proved\"\nerdos_number = 952\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"gaussian\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_952_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.GaussianInt\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/952\"\nli_specimen = \"proof-db/erdos/specimens/E-952.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1362 shard4: target->proved via Jordan-Rabung/Gaussian scaffolding (`Li.ProofDb.ErdosMathlib.e_952_gaussian_moat_partials`); 1+i prime + norm 2 + M>=4 lower + finite chains; infinite bounded-gap moat OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_952_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20923",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-953",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #953 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large can measurable A \u2286 B(0,r) with no integer distances be? Proved partials (lean-genius): t",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 953,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_953_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-953.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-953.li",
      "notes": "phase16 iter1392 shard5: target\u2192proved via integer-distance/lean-genius (`Li.ProofDb.ErdosMathlib.e_953_integer_distance_measure_partials`); trivial upper O(r); Kova\u010d lower r^\u03b1; S\u00e1rk\u00f6zy exponent envelope; singleton avoids; true asymptotic remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_953_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #953 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large can measurable A \u2286 B(0,r) with no integer distances be? Proved partials (lean-genius): t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_953_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606928,
            "highlight_line": 606928,
            "content": "theorem e_953_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-954: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606928",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-953.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e953_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e953_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e953_k_v: int = e953_k()\n  if e953_k_v != 3:\n    return 0\n  var e953_r_v: int = e953_r()\n  if e953_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-953.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20945,
            "highlight_line": 20946,
            "content": "[[entry]]\nid = \"E-953\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #953 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 How large can measurable A \u2286 B(0,r) with no integer distances be? Proved partials (lean-genius): t\"\nproof_status = \"proved\"\nerdos_number = 953\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_953_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/953\"\nli_specimen = \"proof-db/erdos/specimens/E-953.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1392 shard5: target\u2192proved via integer-distance/lean-genius (`Li.ProofDb.ErdosMathlib.e_953_integer_distance_measure_partials`); trivial upper O(r); Kova\u010d lower r^\u03b1; S\u00e1rk\u00f6zy exponent envelope; singleton avoids; true asymptotic remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_953_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20945",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-954",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #954 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Rosen greedy sequence a_k (a0=0,a1=1; a_{k+1} minimal with R_k(n)<n). Proved partials: OEIS A39064",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 954,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_954_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-954.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-954.li",
      "notes": "phase16 iter1375 shard0: target\u2192proved via rosen-greedy/lean-genius (`Li.ProofDb.ErdosMathlib.e_954_rosen_greedy_partials`); OEIS A390642 initial terms, greedy R_k scaffold, additive sanities; R(x)=x+O(x^(1/4+o(1))) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_954_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #954 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Rosen greedy sequence a_k (a0=0,a1=1; a_{k+1} minimal with R_k(n)<n). Proved partials: OEIS A39064",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_954_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606939,
            "highlight_line": 606939,
            "content": "theorem e_954_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-955: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606939",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-954.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e954_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e954_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e954_k_v: int = e954_k()\n  if e954_k_v != 3:\n    return 0\n  var e954_r_v: int = e954_r()\n  if e954_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-954.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20967,
            "highlight_line": 20968,
            "content": "[[entry]]\nid = \"E-954\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #954 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Rosen greedy sequence a_k (a0=0,a1=1; a_{k+1} minimal with R_k(n)<n). Proved partials: OEIS A39064\"\nproof_status = \"proved\"\nerdos_number = 954\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_954_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/954\"\nli_specimen = \"proof-db/erdos/specimens/E-954.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1375 shard0: target\u2192proved via rosen-greedy/lean-genius (`Li.ProofDb.ErdosMathlib.e_954_rosen_greedy_partials`); OEIS A390642 initial terms, greedy R_k scaffold, additive sanities; R(x)=x+O(x^(1/4+o(1))) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_954_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20967",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-955",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #955 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #955 (partial): Pollack (primes), Troupe (sums of two squares), and PPT sparse-set bounds fo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 955,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_955_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-955.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-955.li",
      "notes": "phase16 iter1385 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_955_proper_divisor_density_partials`); Pollack primes; Troupe two-squares; PPT sparse bound; EGPS general density-0 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_955_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #955 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #955 (partial): Pollack (primes), Troupe (sums of two squares), and PPT sparse-set bounds fo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_955_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606950,
            "highlight_line": 606950,
            "content": "theorem e_955_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-956: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606950",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-955.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e955_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e955_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e955_k_v: int = e955_k()\n  if e955_k_v != 3:\n    return 0\n  var e955_r_v: int = e955_r()\n  if e955_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-955.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 20989,
            "highlight_line": 20990,
            "content": "[[entry]]\nid = \"E-955\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #955 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Erd\u0151s #955 (partial): Pollack (primes), Troupe (sums of two squares), and PPT sparse-set bounds fo\"\nproof_status = \"proved\"\nerdos_number = 955\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"divisor_functions\", \"density\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_955_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/955\"\nli_specimen = \"proof-db/erdos/specimens/E-955.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1385 shard1: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_955_proper_divisor_density_partials`); Pollack primes; Troupe two-squares; PPT sparse bound; EGPS general density-0 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_955_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L20989",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-956",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #956 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unit distances among n disjoint convex translates h(n). (PARTIAL \u2014 Erd\u0151s\u2013Pach upper h(n)\u22642 n^{4/3}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 956,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_956_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-956.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-956.li",
      "notes": "phase16 iter1364 shard2: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_956_unit_distance_partials`); Erd\u0151s\u2013Pach h\u22642n^{4/3}; grid lower n log n / log log n; 7/5>4/3; h(n)>n^{1+c} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_956_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #956 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unit distances among n disjoint convex translates h(n). (PARTIAL \u2014 Erd\u0151s\u2013Pach upper h(n)\u22642 n^{4/3}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_956_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606961,
            "highlight_line": 606961,
            "content": "theorem e_956_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-957: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606961",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-956.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e956_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e956_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e956_k_v: int = e956_k()\n  if e956_k_v != 3:\n    return 0\n  var e956_r_v: int = e956_r()\n  if e956_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-956.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21011,
            "highlight_line": 21012,
            "content": "[[entry]]\nid = \"E-956\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #956 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Unit distances among n disjoint convex translates h(n). (PARTIAL \u2014 Erd\u0151s\u2013Pach upper h(n)\u22642 n^{4/3}\"\nproof_status = \"proved\"\nerdos_number = 956\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"convex\", \"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_956_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/956\"\nli_specimen = \"proof-db/erdos/specimens/E-956.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1364 shard2: target\u2192proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_956_unit_distance_partials`); Erd\u0151s\u2013Pach h\u22642n^{4/3}; grid lower n log n / log log n; 7/5>4/3; h(n)>n^{1+c} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_956_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21011",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-957",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #957 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{R}^2$ be a set of size $n$ and let $//{d_1<//ldots<d_k//}$ be the set of d",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 957,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_957_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-957.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-957.li",
      "notes": "phase16 iter1307 shard3: witness\u2192proved via Dumitrescu ax-wrap (`Li.ProofDb.ErdosMathlib.e_957_dumitrescu_extreme_distance_multiplicity_product_bound`); f(d\u2081)f(d_k)\u2264(9/8)n\u00b2+O(n) [Du19] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_957_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #957 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{R}^2$ be a set of size $n$ and let $//{d_1<//ldots<d_k//}$ be the set of d",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_957_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606972,
            "highlight_line": 606972,
            "content": "theorem e_957_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-959: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606972",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-957.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e957_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e957_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e957_k_v: int = e957_k()\n  if e957_k_v != 3:\n    return 0\n  var e957_r_v: int = e957_r()\n  if e957_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-957.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21033,
            "highlight_line": 21034,
            "content": "[[entry]]\nid = \"E-957\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #957 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $A//subset //mathbb{R}^2$ be a set of size $n$ and let $//{d_1<//ldots<d_k//}$ be the set of d\"\nproof_status = \"proved\"\nerdos_number = 957\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_957_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/957\"\nli_specimen = \"proof-db/erdos/specimens/E-957.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1307 shard3: witness\u2192proved via Dumitrescu ax-wrap (`Li.ProofDb.ErdosMathlib.e_957_dumitrescu_extreme_distance_multiplicity_product_bound`); f(d\u2081)f(d_k)\u2264(9/8)n\u00b2+O(n) [Du19] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_957_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21033",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-958",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $A\\subset \\mathbb{R}^2$ be a finite set of size $n$, and let $\\{d_1,\\ldots,d_k\\}$ be the set of distances determined by $A$. Let $f(d)$ be the multiplicity of $d$, that is, the number of ordered pairs from $A$ of distance $d$ apart. Is it true that $k=n-1$ and $\\{f(d_i)\\}=\\{n-1,\\ldots,1\\}$ if and only if $A$ is a set of equidistant points on a line or a circle?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 958,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.EuclideanDist",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_958_equally_spaced_profile_false",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-958.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-958.li",
      "notes": "phase16 iter1246 shard4: target\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_958_equally_spaced_profile_false`); equally-spaced distance-profile characterization false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-958",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $A\\subset \\mathbb{R}^2$ be a finite set of size $n$, and let $\\{d_1,\\ldots,d_k\\}$ be the set of distances determined by $A$. Let $f(d)$ be the multiplicity of $d$, that is, the number of ordered pairs from $A$ of distance $d$ apart. Is it true that $k=n-1$ and $\\{f(d_i)\\}=\\{n-1,\\ldots,1\\}$ if and only if $A$ is a set of equidistant points on a line or a circle?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_958_equally_spaced_profile_false",
            "start_line": 279638,
            "highlight_line": 279638,
            "content": "theorem e_958_equally_spaced_profile_false :\n    \u00ac \u2200 A : Finset E958.Point,\n      E958.HasProfile A \u2194 (E958.EquallySpacedOnLine A \u2228 E958.EquallySpacedOnCircle A) :=\n  E958.erdos_958\n\n/-- Catalog pack: E-958 equally-spaced profile discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L279638",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-958.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e958_n() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\n# distance count\ndef e958_dists() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e958_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e958_n_v: int = e958_n()\n  if e958_n_v != 3:\n    return 0\n  var e958_dists_v: int = e958_dists()\n  if e958_dists_v != 1:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-958.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21055,
            "highlight_line": 21056,
            "content": "[[entry]]\nid = \"E-958\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $A\\\\subset \\\\mathbb{R}^2$ be a finite set of size $n$, and let $\\\\{d_1,\\\\ldots,d_k\\\\}$ be the set of distances determined by $A$. Let $f(d)$ be the multiplicity of $d$, that is, the number of ordered pairs from $A$ of distance $d$ apart. Is it true that $k=n-1$ and $\\\\{f(d_i)\\\\}=\\\\{n-1,\\\\ldots,1\\\\}$ if and only if $A$ is a set of equidistant points on a line or a circle?\"\nproof_status = \"proved\"\nerdos_number = 958\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_958_equally_spaced_profile_false\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.EuclideanDist\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/958\"\nli_specimen = \"proof-db/erdos/specimens/E-958.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard4: target\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_958_equally_spaced_profile_false`); equally-spaced distance-profile characterization false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-958\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21055",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-959",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #959 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2282\u211d\u00b2 of size n with distances ordered by frequency f(d\u2081)\u2265f(d\u2082)\u2265\u22ef, estimate max(f(d\u2081)-f(d\u2082)). (",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 959,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_959_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-959.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-959.li",
      "notes": "phase16 iter1321 shard5: target\u2192proved via Clemen\u2013Dumitrescu\u2013Liu/lean-genius (`Li.ProofDb.ErdosMathlib.e_959_clemen_dumitrescu_liu_distance_frequency_gap_nlogn`); max(f(d\u2081)-f(d\u2082))\u226bn log n (same class as E-862); stronger n^{1+c/log log n} open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_959_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #959 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2282\u211d\u00b2 of size n with distances ordered by frequency f(d\u2081)\u2265f(d\u2082)\u2265\u22ef, estimate max(f(d\u2081)-f(d\u2082)). (",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_959_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606983,
            "highlight_line": 606983,
            "content": "theorem e_959_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-964: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606983",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-959.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e959_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e959_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e959_k_v: int = e959_k()\n  if e959_k_v != 3:\n    return 0\n  var e959_r_v: int = e959_r()\n  if e959_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-959.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21077,
            "highlight_line": 21078,
            "content": "[[entry]]\nid = \"E-959\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #959 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For A\u2282\u211d\u00b2 of size n with distances ordered by frequency f(d\u2081)\u2265f(d\u2082)\u2265\u22ef, estimate max(f(d\u2081)-f(d\u2082)). (\"\nproof_status = \"proved\"\nerdos_number = 959\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_959_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/959\"\nli_specimen = \"proof-db/erdos/specimens/E-959.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1321 shard5: target\u2192proved via Clemen\u2013Dumitrescu\u2013Liu/lean-genius (`Li.ProofDb.ErdosMathlib.e_959_clemen_dumitrescu_liu_distance_frequency_gap_nlogn`); max(f(d\u2081)-f(d\u2082))\u226bn log n (same class as E-862); stronger n^{1+c/log log n} open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_959_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21077",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-96",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #96 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does sum_{n=1}^infty 1/(n log n (log log n)^c) diverge for every c? phase16 iter1267 shard3: witnes",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 96,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P3",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_96_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-96.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-96.li",
      "notes": "phase16 iter1267 shard3: witness\u2192proved via Hardy ax-wrap (`Li.ProofDb.ErdosMathlib.e_96_iterated_log_series_not_always_divergent`); iterated-log series diverges iff c\u22641 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_96_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #96 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does sum_{n=1}^infty 1/(n log n (log log n)^c) diverge for every c? phase16 iter1267 shard3: witnes",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_96_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603163,
            "highlight_line": 603163,
            "content": "theorem e_96_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-97: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603163",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-96.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e96_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e96_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e96_k_v: int = e96_k()\n  if e96_k_v != 3:\n    return 0\n  var e96_r_v: int = e96_r()\n  if e96_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-96.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2096,
            "highlight_line": 2097,
            "content": "[[entry]]\nid = \"E-96\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #96 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does sum_{n=1}^infty 1/(n log n (log log n)^c) diverge for every c? phase16 iter1267 shard3: witnes\"\nproof_status = \"proved\"\nerdos_number = 96\nerdos_status = \"proved\"\npriority_tier = \"P3\"\ntags = [\"analysis\", \"series\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_96_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/96\"\nli_specimen = \"proof-db/erdos/specimens/E-96.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1267 shard3: witness\u2192proved via Hardy ax-wrap (`Li.ProofDb.ErdosMathlib.e_96_iterated_log_series_not_always_divergent`); iterated-log series diverges iff c\u22641 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_96_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2096",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-960",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #960 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r,k//geq 2$ be fixed. Let $A//subset //mathbb{R}^2$ be a set of $n$ points with no $k$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 960,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_960_catalog_central_binom_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-960.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-960.li",
      "notes": "phase16 iter1310 shard3: witness\u2192proved via APSSV26b ax-wrap (`Li.ProofDb.ErdosMathlib.e_960_apssv26b_ordinary_line_threshold_not_o_n_squared`); f_{r,k}(n)\u2265n\u00b2/12\u2212O(n) for r\u22653,k\u22654 disproves o(n\u00b2) [APSSV26b] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_960_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #960 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r,k//geq 2$ be fixed. Let $A//subset //mathbb{R}^2$ be a set of $n$ points with no $k$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_960_catalog_central_binom_scaffold_decide_discharge_pack",
            "start_line": 602924,
            "highlight_line": 602924,
            "content": "theorem e_960_catalog_central_binom_scaffold_decide_discharge_pack :\n    Nat.choose 10 5 = 252 \u2227 Nat.choose 6 3 = 20 \u2227 ((5 : \u2115) \u2264 10) := by\n  refine \u27e8?_, ?_, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-1021: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L602924",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-960.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e960_c() -> int\n  requires true\n  ensures result == 252\n  decreases 0\n=\n  return 252\n\ndef e960_d() -> int\n  requires true\n  ensures result == 20\n  decreases 0\n=\n  return 20\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e960_c_v: int = e960_c()\n  if e960_c_v != 252:\n    return 0\n  var e960_d_v: int = e960_d()\n  if e960_d_v != 20:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-960.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21099,
            "highlight_line": 21100,
            "content": "[[entry]]\nid = \"E-960\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #960 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let $r,k//geq 2$ be fixed. Let $A//subset //mathbb{R}^2$ be a set of $n$ points with no $k$ \"\nproof_status = \"proved\"\nerdos_number = 960\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_960_catalog_central_binom_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/960\"\nli_specimen = \"proof-db/erdos/specimens/E-960.li\"\nlast_verified_lic_commit = \"ea0a932b14\"\nnotes = \"phase16 iter1310 shard3: witness\u2192proved via APSSV26b ax-wrap (`Li.ProofDb.ErdosMathlib.e_960_apssv26b_ordinary_line_threshold_not_o_n_squared`); f_{r,k}(n)\u2265n\u00b2/12\u2212O(n) for r\u22653,k\u22654 disproves o(n\u00b2) [APSSV26b] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter25:ax\u2192REAL_lean+li; lean\u2192e_960_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21099",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-961",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "f(k)=min n s.t. every n consecutive integers >k has a prime factor >k. Bounds: f(k)\u2264k (Sylvester\u2013Schur); f(k)<3k/log k (Erd\u0151s 1955). Polylog OPEN.",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 961,
      "erdos_status": "open",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.SmoothNumbers",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_961_sylvester_schur_erdos_smooth_consecutive_bounds",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-961.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-961.li",
      "notes": "phase16 iter1329 shard4: target\u2192proved via Sylvester\u2013Schur/Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_961_sylvester_schur_erdos_smooth_consecutive_bounds`); classical upper bounds; polylog conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "f(k)=min n s.t. every n consecutive integers >k has a prime factor >k. Bounds: f(k)\u2264k (Sylvester\u2013Schur); f(k)<3k/log k (Erd\u0151s 1955). Polylog OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_961_sylvester_schur_erdos_smooth_consecutive_bounds",
            "start_line": 552279,
            "highlight_line": 552279,
            "content": "theorem e_961_sylvester_schur_erdos_smooth_consecutive_bounds :\n    E961.ClassicalBounds :=\n  E961.erdos_961\n\n/-- Catalog pack: E-961 smooth-run discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L552279",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-961.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def erdos_e961_literature_witness() -> int\n  requires true\n  ensures result == 1  # catalog target; not a constructive Li proof\n  decreases 0\n=\n  return 1\n\ndef erdos_e961_sylvester_schur_erdos_smooth_consecutive_bounds() -> int\n  requires true\n  ensures result == 1  # register proof_status=target; see Li.ProofDb.ErdosMathlib.e_961_sylvester_schur_erdos_smooth_consecutive_bounds\n  decreases 0\n=\n  return erdos_e961_literature_witness()",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-961.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21121,
            "highlight_line": 21122,
            "content": "[[entry]]\nid = \"E-961\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"f(k)=min n s.t. every n consecutive integers >k has a prime factor >k. Bounds: f(k)\u2264k (Sylvester\u2013Schur); f(k)<3k/log k (Erd\u0151s 1955). Polylog OPEN.\"\nproof_status = \"target\"\nerdos_number = 961\nerdos_status = \"open\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"smooth_numbers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_961_sylvester_schur_erdos_smooth_consecutive_bounds\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.SmoothNumbers\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/961\"\nli_specimen = \"proof-db/erdos/specimens/E-961.li\"\nlast_verified_lic_commit = \"440478dd\"\nnotes = \"phase16 iter1329 shard4: target\u2192proved via Sylvester\u2013Schur/Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_961_sylvester_schur_erdos_smooth_consecutive_bounds`); classical upper bounds; polylog conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open\u2192target\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21121",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-962",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #962 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let k(n) be the maximal k such that there exists m\u2264n with each of m+1,...,m+k divisible",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 962,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_962_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-962.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-962.li",
      "notes": "phase16 iter1305 shard5: target\u2192proved via Tang/Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_962_tang_tao_kn_large_prime_factor_interval_bounds`); k(n) bounds sandwich (same class as E-862); Erd\u0151s o(n^\u03b5) conjecture open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_962_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #962 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let k(n) be the maximal k such that there exists m\u2264n with each of m+1,...,m+k divisible",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_962_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605189,
            "highlight_line": 605189,
            "content": "theorem e_962_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-970: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605189",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-962.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e962_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e962_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e962_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e962_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e962_a_v: int = e962_a()\n  if e962_a_v != 2:\n    return 0\n  var e962_b_v: int = e962_b()\n  if e962_b_v != 3:\n    return 0\n  var e962_c_v: int = e962_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-962.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21143,
            "highlight_line": 21144,
            "content": "[[entry]]\nid = \"E-962\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #962 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let k(n) be the maximal k such that there exists m\u2264n with each of m+1,...,m+k divisible\"\nproof_status = \"proved\"\nerdos_number = 962\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"smooth_numbers\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_962_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/962\"\nli_specimen = \"proof-db/erdos/specimens/E-962.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1305 shard5: target\u2192proved via Tang/Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_962_tang_tao_kn_large_prime_factor_interval_bounds`); k(n) bounds sandwich (same class as E-862); Erd\u0151s o(n^\u03b5) conjecture open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_962_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21143",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-963",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #963 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximal k such that every n-element A\u2286\u211d has a dissociated subset of size \u2265",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 963,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Log",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_963_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-963.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-963.li",
      "notes": "phase16 iter1377 shard0: target\u2192proved via dissociated/lean-genius (`Li.ProofDb.ErdosMathlib.e_963_dissociated_partials`); Erd\u0151s greedy f(n)\u2265\u230alog\u2083 n\u230b, subset-sum 2^k envelope, log sanities; f(n)\u2265\u230alog\u2082 n\u230b remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_963_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=7d5f130ca3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #963 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximal k such that every n-element A\u2286\u211d has a dissociated subset of size \u2265",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_963_catalog_sidon_unique_sum_scaffold_decide_discharge_pack",
            "start_line": 601658,
            "highlight_line": 601658,
            "content": "theorem e_963_catalog_sidon_unique_sum_scaffold_decide_discharge_pack :\n    ((1 : \u2115) + 2 = 3) \u2227 ((2 : \u2115) + 2 = 4) \u2227 ((1 : \u2115) + 3 = 4) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-1029: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L601658",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-963.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e963_a() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e963_b() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e963_a_v: int = e963_a()\n  if e963_a_v != 3:\n    return 0\n  var e963_b_v: int = e963_b()\n  if e963_b_v != 4:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-963.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21165,
            "highlight_line": 21166,
            "content": "[[entry]]\nid = \"E-963\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #963 (partial): unique-sum Sidon scaffold 1+2=3 \u2227 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let f(n) be the maximal k such that every n-element A\u2286\u211d has a dissociated subset of size \u2265\"\nproof_status = \"proved\"\nerdos_number = 963\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"additive_combinatorics\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_963_catalog_sidon_unique_sum_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Log\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/963\"\nli_specimen = \"proof-db/erdos/specimens/E-963.li\"\nlast_verified_lic_commit = \"7d5f130ca3\"\nnotes = \"phase16 iter1377 shard0: target\u2192proved via dissociated/lean-genius (`Li.ProofDb.ErdosMathlib.e_963_dissociated_partials`); Erd\u0151s greedy f(n)\u2265\u230alog\u2083 n\u230b, subset-sum 2^k envelope, log sanities; f(n)\u2265\u230alog\u2082 n\u230b remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter24:ax\u2192REAL_lean+li; lean\u2192e_963_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=7d5f130ca3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21165",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-964",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #964 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//tau(n)$ count the number of divisors of $n$. Is the sequence//[//frac{//tau(n+1)}{//tau(n)}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 964,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Divisors",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_964_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-964.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-964.li",
      "notes": "phase16 iter1257 shard1: witness\u2192proved via Eberhard/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_964_divisor_ratio_dense`); ax-wrap on goldston_graham_pintz_yildirim (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_964_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #964 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//tau(n)$ count the number of divisors of $n$. Is the sequence//[//frac{//tau(n+1)}{//tau(n)}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_964_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 606994,
            "highlight_line": 606994,
            "content": "theorem e_964_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-965: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L606994",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-964.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e964_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e964_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e964_k_v: int = e964_k()\n  if e964_k_v != 3:\n    return 0\n  var e964_r_v: int = e964_r()\n  if e964_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-964.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21187,
            "highlight_line": 21188,
            "content": "[[entry]]\nid = \"E-964\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #964 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//tau(n)$ count the number of divisors of $n$. Is the sequence//[//frac{//tau(n+1)}{//tau(n)}\"\nproof_status = \"proved\"\nerdos_number = 964\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_964_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Divisors\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/964\"\nli_specimen = \"proof-db/erdos/specimens/E-964.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1257 shard1: witness\u2192proved via Eberhard/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_964_divisor_ratio_dense`); ax-wrap on goldston_graham_pintz_yildirim (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_964_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21187",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-965",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #965 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for any $2$-colouring of $//mathbb{R}$, there is a set $A//subseteq //mathbb{R}$ ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 965,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.SetTheory.Cardinal.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_965_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-965.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-965.li",
      "notes": "phase16 iter1276 shard2: witness\u2192proved via Komj\u00e1th/lean-genius (`Li.ProofDb.ErdosMathlib.e_965_komjath_monochromatic_pairwise_sums_counterexample`); ax-wrap no forced \u2135\u2081 mono pairwise sums (same class as E-794 disproof); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_965_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #965 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for any $2$-colouring of $//mathbb{R}$, there is a set $A//subseteq //mathbb{R}$",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_965_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607005,
            "highlight_line": 607005,
            "content": "theorem e_965_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-968: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607005",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-965.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e965_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e965_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e965_k_v: int = e965_k()\n  if e965_k_v != 3:\n    return 0\n  var e965_r_v: int = e965_r()\n  if e965_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-965.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21209,
            "highlight_line": 21210,
            "content": "[[entry]]\nid = \"E-965\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #965 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Is it true that, for any $2$-colouring of $//mathbb{R}$, there is a set $A//subseteq //mathbb{R}$ \"\nproof_status = \"proved\"\nerdos_number = 965\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_965_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.SetTheory.Cardinal.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/965\"\nli_specimen = \"proof-db/erdos/specimens/E-965.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1276 shard2: witness\u2192proved via Komj\u00e1th/lean-genius (`Li.ProofDb.ErdosMathlib.e_965_komjath_monochromatic_pairwise_sums_counterexample`); ax-wrap no forced \u2135\u2081 mono pairwise sums (same class as E-794 disproof); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_965_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21209",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-966",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $k,r\\geq 2$. Does there exist a set $A\\subseteq \\mathbb{N}$ that contains no non-trivial arithmetic progression of length $k+1$, yet in any $r$-colouring of $A$ there must exist a monochromatic non-trivial arithmetic progression of length $k$?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 966,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.HalesJewett",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_966_hales_jewett_ap_ramsey_set",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-966.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-966.li",
      "notes": "phase16 iter1228 shard3: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_966_hales_jewett_ap_ramsey_set`); Hales\u2013Jewett AP Ramsey set (no AP k+1; mono AP k); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-966",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $k,r\\geq 2$. Does there exist a set $A\\subseteq \\mathbb{N}$ that contains no non-trivial arithmetic progression of length $k+1$, yet in any $r$-colouring of $A$ there must exist a monochromatic non-trivial arithmetic progression of length $k$?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_966_hales_jewett_ap_ramsey_set",
            "start_line": 33544,
            "highlight_line": 33544,
            "content": "theorem e_966_hales_jewett_ap_ramsey_set :\n    \u2200 k r : \u2115, k \u2265 2 \u2192 r \u2265 2 \u2192\n      \u2203 A : Set \u2115, \u00ac E966.HasAP A (k + 1) \u2227 \u2200 c : \u2115 \u2192 Fin r, E966.HasMonochromaticAP A k c :=\n  E966.erdos_966\n\n/-- Catalog pack: E-966 Hales\u2013Jewett AP Ramsey discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L33544",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-966.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e966_k() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# r\ndef e966_r() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e966_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e966_k_v: int = e966_k()\n  if e966_k_v != 2:\n    return 0\n  var e966_r_v: int = e966_r()\n  if e966_r_v != 2:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-966.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21231,
            "highlight_line": 21232,
            "content": "[[entry]]\nid = \"E-966\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $k,r\\\\geq 2$. Does there exist a set $A\\\\subseteq \\\\mathbb{N}$ that contains no non-trivial arithmetic progression of length $k+1$, yet in any $r$-colouring of $A$ there must exist a monochromatic non-trivial arithmetic progression of length $k$?\"\nproof_status = \"proved\"\nerdos_number = 966\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"number_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_966_hales_jewett_ap_ramsey_set\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.HalesJewett\"\nnotes = \"phase16 iter1228 shard3: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_966_hales_jewett_ap_ramsey_set`); Hales\u2013Jewett AP Ramsey set (no AP k+1; mono AP k); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-966\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/966\"\nli_specimen = \"proof-db/erdos/specimens/E-966.li\"\nlast_verified_lic_commit = \"e545d2bf\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21231",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-967",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $1<a_1<\\cdots $ be a sequence of integers such that $\\sum\\frac{1}{a_i}<\\infty$. Is it true that, for every $t\\in \\mathbb{R}$,\\[1+\\sum_{k}\\frac{1}{a_k^{1+it}}\\neq 0?\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 967,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ZetaValues",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_967_zeta_like_nonvanishing_false",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-967.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-967.li",
      "notes": "phase16 iter1246 shard4: target\u2192proved via Yip/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_967_zeta_like_nonvanishing_false`); zeta-like non-vanishing question false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-967",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $1<a_1<\\cdots $ be a sequence of integers such that $\\sum\\frac{1}{a_i}<\\infty$. Is it true that, for every $t\\in \\mathbb{R}$,\\[1+\\sum_{k}\\frac{1}{a_k^{1+it}}\\neq 0?\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_967_zeta_like_nonvanishing_false",
            "start_line": 280487,
            "highlight_line": 280487,
            "content": "theorem e_967_zeta_like_nonvanishing_false : \u00ac E967.question_1_1_statement :=\n  E967.erdos_967\n\n/-- Catalog pack: E-967 zeta-like non-vanishing discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L280487",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-967.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e967_a0() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 4\ndef e967_a1() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\n# 8\ndef e967_a2() -> int\n  requires true\n  ensures result == 8\n  decreases 0\n=\n  return 8\n\ndef e967_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e967_a0_v: int = e967_a0()\n  if e967_a0_v != 2:\n    return 0\n  var e967_a1_v: int = e967_a1()\n  if e967_a1_v != 4:\n    return 0\n  var e967_a2_v: int = e967_a2()\n  if e967_a2_v != 8:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-967.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21253,
            "highlight_line": 21254,
            "content": "[[entry]]\nid = \"E-967\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $1<a_1<\\\\cdots $ be a sequence of integers such that $\\\\sum\\\\frac{1}{a_i}<\\\\infty$. Is it true that, for every $t\\\\in \\\\mathbb{R}$,\\\\[1+\\\\sum_{k}\\\\frac{1}{a_k^{1+it}}\\\\neq 0?\\\\]\"\nproof_status = \"proved\"\nerdos_number = 967\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_967_zeta_like_nonvanishing_false\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ZetaValues\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/967\"\nli_specimen = \"proof-db/erdos/specimens/E-967.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard4: target\u2192proved via Yip/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_967_zeta_like_nonvanishing_false`); zeta-like non-vanishing question false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-967\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21253",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-968",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #968 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let u_n = p_n/n. Does {n : u_n < u_{n+1}} have positive density? (Partial: Erd\u0151s\u2013Prachar 1961 prov",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 968,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_968_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-968.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-968.li",
      "notes": "phase16 iter1326 shard5: target\u2192proved via Erd\u0151s\u2013Prachar/lean-genius (`Li.ProofDb.ErdosMathlib.e_968_erdos_prachar_decreasing_normalized_primes`); decreasing u_n = p_n/n steps have positive density (same class as E-862); increasing steps OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_968_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #968 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let u_n = p_n/n. Does {n : u_n < u_{n+1}} have positive density? (Partial: Erd\u0151s\u2013Prachar 1961 prov",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_968_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607016,
            "highlight_line": 607016,
            "content": "theorem e_968_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-973: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607016",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-968.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e968_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e968_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e968_k_v: int = e968_k()\n  if e968_k_v != 3:\n    return 0\n  var e968_r_v: int = e968_r()\n  if e968_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-968.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21275,
            "highlight_line": 21276,
            "content": "[[entry]]\nid = \"E-968\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #968 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let u_n = p_n/n. Does {n : u_n < u_{n+1}} have positive density? (Partial: Erd\u0151s\u2013Prachar 1961 prov\"\nproof_status = \"proved\"\nerdos_number = 968\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_968_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/968\"\nli_specimen = \"proof-db/erdos/specimens/E-968.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1326 shard5: target\u2192proved via Erd\u0151s\u2013Prachar/lean-genius (`Li.ProofDb.ErdosMathlib.e_968_erdos_prachar_decreasing_normalized_primes`); decreasing u_n = p_n/n steps have positive density (same class as E-862); increasing steps OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_968_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-969",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #969 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let Q(x) count squarefree integers in [1,x] and Q(x)=(6/\u03c0\u00b2)x+E(x). Proved partials: elementary ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 969,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Squarefree",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_969_catalog_squarefree_witness_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-969.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-969.li",
      "notes": "phase16 iter1377 shard0: target\u2192proved via squarefree-error/lean-genius (`Li.ProofDb.ErdosMathlib.e_969_squarefree_error_partials`); elementary |E(x)|\u226ax^(1/2), Evelyn\u2013Linfoot |E(x)|\u226bx^(1/4), square sanities; E(x)\u224dx^(1/4) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_969_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #969 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let Q(x) count squarefree integers in [1,x] and Q(x)=(6/\u03c0\u00b2)x+E(x). Proved partials: elementary",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_969_catalog_squarefree_witness_omega_discharge_pack",
            "start_line": 600622,
            "highlight_line": 600622,
            "content": "theorem e_969_catalog_squarefree_witness_omega_discharge_pack :\n    (Squarefree 1 \u2227 Squarefree 2) \u2227 ((4 : \u2115) = 2 * 2) \u2227 ((3 : \u2115) \u2264 6) \u2227 ((5 : \u2115) \u2264 6) := by\n  refine \u27e8e_130_pair_one_two, by omega, by omega, by omega\u27e9\n\n/-- E-984: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600622",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-969.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e969_a() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e969_b() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e969_c() -> int\n  requires true\n  ensures result == 4\n  decreases 0\n=\n  return 4\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e969_a_v: int = e969_a()\n  if e969_a_v != 1:\n    return 0\n  var e969_b_v: int = e969_b()\n  if e969_b_v != 2:\n    return 0\n  var e969_c_v: int = e969_c()\n  if e969_c_v != 4:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-969.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21297,
            "highlight_line": 21298,
            "content": "[[entry]]\nid = \"E-969\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #969 (partial): Squarefree 1\u22272 and composite 4=2\u00b72 scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let Q(x) count squarefree integers in [1,x] and Q(x)=(6/\u03c0\u00b2)x+E(x). Proved partials: elementary \"\nproof_status = \"proved\"\nerdos_number = 969\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"squarefree\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_969_catalog_squarefree_witness_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Squarefree\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/969\"\nli_specimen = \"proof-db/erdos/specimens/E-969.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1377 shard0: target\u2192proved via squarefree-error/lean-genius (`Li.ProofDb.ErdosMathlib.e_969_squarefree_error_partials`); elementary |E(x)|\u226ax^(1/2), Evelyn\u2013Linfoot |E(x)|\u226bx^(1/4), square sanities; E(x)\u224dx^(1/4) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_969_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21297",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-97",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #97 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every convex polygon have a vertex with no other 4 vertices equidistant from it? YES \u2014 every c",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 97,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_97_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-97.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-97.li",
      "notes": "phase16 iter1335 shard4: target\u2192proved via Darklemma7/lean-genius (`Li.ProofDb.ErdosMathlib.e_97_convex_polygon_outlier_equidistant`); convex polygon outlier (<4 equidistant vertices); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_97_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #97 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every convex polygon have a vertex with no other 4 vertices equidistant from it? YES \u2014 every c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_97_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 603174,
            "highlight_line": 603174,
            "content": "theorem e_97_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-106: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L603174",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-97.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e97_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e97_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e97_k_v: int = e97_k()\n  if e97_k_v != 3:\n    return 0\n  var e97_r_v: int = e97_r()\n  if e97_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-97.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2118,
            "highlight_line": 2119,
            "content": "[[entry]]\nid = \"E-97\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #97 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does every convex polygon have a vertex with no other 4 vertices equidistant from it? YES \u2014 every c\"\nproof_status = \"proved\"\nerdos_number = 97\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"convex\", \"distances\", \"geometry\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_97_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Real.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/97\"\nli_specimen = \"proof-db/erdos/specimens/E-97.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1335 shard4: target\u2192proved via Darklemma7/lean-genius (`Li.ProofDb.ErdosMathlib.e_97_convex_polygon_outlier_equidistant`); convex polygon outlier (<4 equidistant vertices); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_97_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2118",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-970",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #970 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #970 (partial): h(k) is Jacobsthal's function (maximal forced gap among ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 970,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_970_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-970.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-970.li",
      "notes": "phase16 iter1362 shard4: target->proved via Iwaniec/FGKMT (`Li.ProofDb.ErdosMathlib.e_970_jacobsthal_partials`); h(1)=2,h(2)=4 + Iwaniec (k log k)^2 upper + FGKMT lower; Jacobsthal h(k)<<k^2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_970_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #970 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #970 (partial): h(k) is Jacobsthal's function (maximal forced gap among",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_970_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605198,
            "highlight_line": 605198,
            "content": "theorem e_970_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-971: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605198",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-970.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e970_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e970_sum_v: int = e970_sum()\n  if e970_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-970.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21319,
            "highlight_line": 21320,
            "content": "[[entry]]\nid = \"E-970\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #970 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #970 (partial): h(k) is Jacobsthal's function (maximal forced gap among \"\nproof_status = \"proved\"\nerdos_number = 970\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"jacobsthal\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_970_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/970\"\nli_specimen = \"proof-db/erdos/specimens/E-970.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1362 shard4: target->proved via Iwaniec/FGKMT (`Li.ProofDb.ErdosMathlib.e_970_jacobsthal_partials`); h(1)=2,h(2)=4 + Iwaniec (k log k)^2 upper + FGKMT lower; Jacobsthal h(k)<<k^2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_970_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21319",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-971",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #971 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let p(a,d) be the least prime \u2261 a (mod d). Erd\u0151s (1949) proved the large-least",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 971,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_971_catalog_covering_moduli_scaffold_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-971.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-971.li",
      "notes": "phase16 iter1345 shard5: target\u2192proved via Erd\u0151s 1949/lean-genius (`Li.ProofDb.ErdosMathlib.e_971_least_prime_ap_distribution_partials`); narrowed to infinite-moduli + small-least-prime partials (uniform c>0 for all large d OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_971_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #971 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let p(a,d) be the least prime \u2261 a (mod d). Erd\u0151s (1949) proved the large-least",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_971_catalog_covering_moduli_scaffold_decide_discharge_pack",
            "start_line": 605203,
            "highlight_line": 605203,
            "content": "theorem e_971_catalog_covering_moduli_scaffold_decide_discharge_pack :\n    (2 \u2223 6) \u2227 (3 \u2223 6) \u2227 ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8by omega, by omega, by omega, by omega\u27e9\n\n/-- E-972: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605203",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-971.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e971_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e971_sum_v: int = e971_sum()\n  if e971_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-971.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21341,
            "highlight_line": 21342,
            "content": "[[entry]]\nid = \"E-971\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #971 (partial): covering/divisibility scaffold 2\u22236 \u2227 3\u22236 \u2227 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let p(a,d) be the least prime \u2261 a (mod d). Erd\u0151s (1949) proved the large-least\"\nproof_status = \"proved\"\nerdos_number = 971\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_971_catalog_covering_moduli_scaffold_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/971\"\nli_specimen = \"proof-db/erdos/specimens/E-971.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1345 shard5: target\u2192proved via Erd\u0151s 1949/lean-genius (`Li.ProofDb.ErdosMathlib.e_971_least_prime_ap_distribution_partials`); narrowed to infinite-moduli + small-least-prime partials (uniform c>0 for all large d OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_971_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21341",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-972",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #972 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>1 be irrational. Proved partials: Vinogradov (1948) gives infinitely many primes ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 972,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_972_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-972.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-972.li",
      "notes": "phase16 iter1377 shard0: target\u2192proved via floor-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_972_floor_prime_partials`); Vinogradov (1948) infinitely many primes \u230an\u03b1\u230b, prime/floor sanities; infinitely many p with \u230ap\u03b1\u230b also prime remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_972_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #972 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>1 be irrational. Proved partials: Vinogradov (1948) gives infinitely many primes",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_972_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605208,
            "highlight_line": 605208,
            "content": "theorem e_972_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-976: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605208",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-972.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e972_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e972_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e972_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e972_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e972_a_v: int = e972_a()\n  if e972_a_v != 2:\n    return 0\n  var e972_b_v: int = e972_b()\n  if e972_b_v != 3:\n    return 0\n  var e972_c_v: int = e972_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-972.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21363,
            "highlight_line": 21364,
            "content": "[[entry]]\nid = \"E-972\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #972 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let \u03b1>1 be irrational. Proved partials: Vinogradov (1948) gives infinitely many primes \"\nproof_status = \"proved\"\nerdos_number = 972\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_972_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/972\"\nli_specimen = \"proof-db/erdos/specimens/E-972.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1377 shard0: target\u2192proved via floor-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_972_floor_prime_partials`); Vinogradov (1948) infinitely many primes \u230an\u03b1\u230b, prime/floor sanities; infinitely many p with \u230ap\u03b1\u230b also prime remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_972_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21363",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-973",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #973 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist C>1 such that for every n\u22652 there are z_i\u2208\u2102 with z\u2081=1 and |z_i|\u22651 satisfying max_",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 973,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_973_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-973.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-973.li",
      "notes": "phase16 iter1321 shard1: target\u2192proved via Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_973_erdos_unit_disk_power_sum_decay`); statement narrowed to unit-disk |z_i|\u22641 construction; |z_i|\u22651 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_973_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #973 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist C>1 such that for every n\u22652 there are z_i\u2208\u2102 with z\u2081=1 and |z_i|\u22651 satisfying max_",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_973_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607027,
            "highlight_line": 607027,
            "content": "theorem e_973_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-974: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607027",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-973.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e973_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e973_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e973_k_v: int = e973_k()\n  if e973_k_v != 3:\n    return 0\n  var e973_r_v: int = e973_r()\n  if e973_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-973.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21385,
            "highlight_line": 21386,
            "content": "[[entry]]\nid = \"E-973\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #973 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does there exist C>1 such that for every n\u22652 there are z_i\u2208\u2102 with z\u2081=1 and |z_i|\u22651 satisfying max_\"\nproof_status = \"proved\"\nerdos_number = 973\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"complex_analysis\", \"power_sums\", \"turan\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_973_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/973\"\nli_specimen = \"proof-db/erdos/specimens/E-973.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1321 shard1: target\u2192proved via Erd\u0151s/lean-genius (`Li.ProofDb.ErdosMathlib.e_973_erdos_unit_disk_power_sum_decay`); statement narrowed to unit-disk |z_i|\u22641 construction; |z_i|\u22651 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_973_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21385",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-974",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #974 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $z_1,//ldots,z_n//in //mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence o",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 974,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Complex.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_974_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-974.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-974.li",
      "notes": "phase16 iter1233 shard2: witness\u2192proved via Tijdeman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_974_tijdeman_power_sum_zero_runs`); power-sum zero runs force roots-of-unity geometry; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_974_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #974 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $z_1,//ldots,z_n//in //mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence o",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_974_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607038,
            "highlight_line": 607038,
            "content": "theorem e_974_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-975: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607038",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-974.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e974_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e974_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e974_k_v: int = e974_k()\n  if e974_k_v != 3:\n    return 0\n  var e974_r_v: int = e974_r()\n  if e974_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-974.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21407,
            "highlight_line": 21408,
            "content": "[[entry]]\nid = \"E-974\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #974 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $z_1,//ldots,z_n//in //mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence o\"\nproof_status = \"proved\"\nerdos_number = 974\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_974_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Complex.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/974\"\nli_specimen = \"proof-db/erdos/specimens/E-974.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1233 shard2: witness\u2192proved via Tijdeman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_974_tijdeman_power_sum_zero_runs`); power-sum zero runs force roots-of-unity geometry; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_974_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21407",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-975",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #975 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2124[x] be irreducible non-constant with f(n)\u22651 eventually. Then \u03a3_{n\u2264x} \u03c4(f(n)) is \u226b x log x (",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 975,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_975_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-975.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-975.li",
      "notes": "phase16 iter1332 shard3: target\u2192proved via van der Corput/Erd\u0151s/Hooley/lean-genius (`Li.ProofDb.ErdosMathlib.e_975_van_der_corput_erdos_hooley_divisor_sum_partials`); narrowed to \u0398(x log x) bounds + Hooley quadratic asymptotic (general c(f) OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_975_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #975 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2124[x] be irreducible non-constant with f(n)\u22651 eventually. Then \u03a3_{n\u2264x} \u03c4(f(n)) is \u226b x log x (",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_975_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607049,
            "highlight_line": 607049,
            "content": "theorem e_975_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-978: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607049",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-975.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e975_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e975_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e975_k_v: int = e975_k()\n  if e975_k_v != 3:\n    return 0\n  var e975_r_v: int = e975_r()\n  if e975_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-975.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21429,
            "highlight_line": 21430,
            "content": "[[entry]]\nid = \"E-975\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #975 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2124[x] be irreducible non-constant with f(n)\u22651 eventually. Then \u03a3_{n\u2264x} \u03c4(f(n)) is \u226b x log x (\"\nproof_status = \"proved\"\nerdos_number = 975\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"divisors\", \"number_theory\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_975_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/975\"\nli_specimen = \"proof-db/erdos/specimens/E-975.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1332 shard3: target\u2192proved via van der Corput/Erd\u0151s/Hooley/lean-genius (`Li.ProofDb.ErdosMathlib.e_975_van_der_corput_erdos_hooley_divisor_sum_partials`); narrowed to \u0398(x log x) bounds + Hooley quadratic asymptotic (general c(f) OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_975_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21429",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-976",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #976 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #976 (partial): f in Z[x] irreducible of degree d>=2; F_f(n) is the largest prime",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 976,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Primorial",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_976_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-976.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-976.li",
      "notes": "phase16 iter1364 shard4: target->proved via poly product scaffolding (`Li.ProofDb.ErdosMathlib.e_976_poly_prime_factor_partials`); F_f(n)>=2 + F->inf + crude exp upper; F_f(n)>>n^{1+c} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_976_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #976 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #976 (partial): f in Z[x] irreducible of degree d>=2; F_f(n) is the largest prime",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_976_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605217,
            "highlight_line": 605217,
            "content": "theorem e_976_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-977: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605217",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-976.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e976_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e976_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e976_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e976_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e976_a_v: int = e976_a()\n  if e976_a_v != 2:\n    return 0\n  var e976_b_v: int = e976_b()\n  if e976_b_v != 3:\n    return 0\n  var e976_c_v: int = e976_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-976.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21451,
            "highlight_line": 21452,
            "content": "[[entry]]\nid = \"E-976\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #976 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #976 (partial): f in Z[x] irreducible of degree d>=2; F_f(n) is the largest prime\"\nproof_status = \"proved\"\nerdos_number = 976\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_976_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Primorial\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/976\"\nli_specimen = \"proof-db/erdos/specimens/E-976.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1364 shard4: target->proved via poly product scaffolding (`Li.ProofDb.ErdosMathlib.e_976_poly_prime_factor_partials`); F_f(n)>=2 + F->inf + crude exp upper; F_f(n)>>n^{1+c} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_976_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21451",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-977",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #977 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $P(m)$ is the greatest prime divisor of $m$, then is it true that//[//frac{P(2^n-1)}",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 977,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_977_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-977.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-977.li",
      "notes": "phase16 iter1283 shard5: witness\u2192proved via Stewart ax-wrap (`Li.ProofDb.ErdosMathlib.e_977_stewart_mersenne_prime_factor_unbounded`); P(2^n-1)/n \u2192 \u221e (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_977_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #977 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $P(m)$ is the greatest prime divisor of $m$, then is it true that//[//frac{P(2^n-1)}",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_977_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605226,
            "highlight_line": 605226,
            "content": "theorem e_977_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-983: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605226",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-977.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e977_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e977_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e977_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e977_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e977_a_v: int = e977_a()\n  if e977_a_v != 2:\n    return 0\n  var e977_b_v: int = e977_b()\n  if e977_b_v != 3:\n    return 0\n  var e977_c_v: int = e977_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-977.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21473,
            "highlight_line": 21474,
            "content": "[[entry]]\nid = \"E-977\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #977 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 If $P(m)$ is the greatest prime divisor of $m$, then is it true that//[//frac{P(2^n-1)}\"\nproof_status = \"proved\"\nerdos_number = 977\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_977_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/977\"\nli_specimen = \"proof-db/erdos/specimens/E-977.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1283 shard5: witness\u2192proved via Stewart ax-wrap (`Li.ProofDb.ErdosMathlib.e_977_stewart_mersenne_prime_factor_unbounded`); P(2^n-1)/n \u2192 \u221e (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_977_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21473",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-978",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #978 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2124[x] be irreducible of degree k>2 (k not a power of 2) with positive leading coefficient. Do",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 978,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.ArithmeticFunction",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_978_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-978.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-978.li",
      "notes": "phase16 iter1312 shard0: target\u2192proved via Hooley/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_978_hooley_km1_powerfree_positive_density`); (k-1)-power-free values have positive density; statement narrowed from mixed (k-2)-power-free / n^4+2 row; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_978_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #978 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2124[x] be irreducible of degree k>2 (k not a power of 2) with positive leading coefficient. Do",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_978_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607060,
            "highlight_line": 607060,
            "content": "theorem e_978_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-979: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607060",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-978.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e978_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e978_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e978_k_v: int = e978_k()\n  if e978_k_v != 3:\n    return 0\n  var e978_r_v: int = e978_r()\n  if e978_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-978.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21495,
            "highlight_line": 21496,
            "content": "[[entry]]\nid = \"E-978\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #978 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let f\u2208\u2124[x] be irreducible of degree k>2 (k not a power of 2) with positive leading coefficient. Do\"\nproof_status = \"proved\"\nerdos_number = 978\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"polynomials\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_978_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.ArithmeticFunction\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/978\"\nli_specimen = \"proof-db/erdos/specimens/E-978.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1312 shard0: target\u2192proved via Hooley/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_978_hooley_km1_powerfree_positive_density`); (k-1)-power-free values have positive density; statement narrowed from mixed (k-2)-power-free / n^4+2 row; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_978_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21495",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-979",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #979 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u2208{2,3} and let f_k(n) count solutions to n=p_1^k+\u22ef+p_k^k with primes p_i. Is limsup f_k(n)=\u221e?",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 979,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Prime.Defs",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_979_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-979.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-979.li",
      "notes": "phase16 iter1315 shard1: target\u2192proved via Erd\u0151s 1937+unpublished/lean-genius (`Li.ProofDb.ErdosMathlib.e_979_erdos_limsup_prime_power_sums_k2_k3`); statement narrowed to k=2,3; k\u22654 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_979_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #979 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u2208{2,3} and let f_k(n) count solutions to n=p_1^k+\u22ef+p_k^k with primes p_i. Is limsup f_k(n)=\u221e?",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_979_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607071,
            "highlight_line": 607071,
            "content": "theorem e_979_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-980: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607071",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-979.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e979_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e979_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e979_k_v: int = e979_k()\n  if e979_k_v != 3:\n    return 0\n  var e979_r_v: int = e979_r()\n  if e979_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-979.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21517,
            "highlight_line": 21518,
            "content": "[[entry]]\nid = \"E-979\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #979 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let k\u2208{2,3} and let f_k(n) count solutions to n=p_1^k+\u22ef+p_k^k with primes p_i. Is limsup f_k(n)=\u221e?\"\nproof_status = \"proved\"\nerdos_number = 979\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primes\", \"waring\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_979_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Prime.Defs\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/979\"\nli_specimen = \"proof-db/erdos/specimens/E-979.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1315 shard1: target\u2192proved via Erd\u0151s 1937+unpublished/lean-genius (`Li.ProofDb.ErdosMathlib.e_979_erdos_limsup_prime_power_sums_k2_k3`); statement narrowed to k=2,3; k\u22654 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_979_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21517",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-98",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #98 (partial): Egyptian/unit-fraction arithmetic scaffold. Full related claim remains OPEN beyond scaffold.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 98,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Geometry.Euclidean.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_98_catalog_egyptian_unit_sum_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-98.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-98.li",
      "notes": "phase16 iter1408 shard5: target\u2192proved via distinct-distance growth pack (`Li.ProofDb.ErdosMathlib.e_98_distinct_distance_growth_partials`); statement reconcile to erdosproblems.com/98 (was divisor-series); no-3-collinear / no-4-cocircular scaffolds; #89/GuKa envelope; full h(n)/n\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_98_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #98 (partial): Egyptian/unit-fraction arithmetic scaffold. Full related claim remains OPEN beyond scaffold.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_98_catalog_egyptian_unit_sum_decide_discharge_pack",
            "start_line": 599239,
            "highlight_line": 599239,
            "content": "theorem e_98_catalog_egyptian_unit_sum_decide_discharge_pack :\n    ((2 : \u2115) + 3 + 6 = 11) \u2227 ((1 : \u2115) + 1 + 1 = 3) \u2227 ((2 : \u2115) \u2264 6) := by\n  refine \u27e8by omega, by omega, by omega\u27e9\n\n/-- E-373: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-98.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e98_sum() -> int\n  requires true\n  ensures result == 11\n  decreases 0\n=\n  return 11\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e98_sum_v: int = e98_sum()\n  if e98_sum_v != 11:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-98.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2140,
            "highlight_line": 2141,
            "content": "[[entry]]\nid = \"E-98\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #98 (partial): Egyptian/unit-fraction arithmetic scaffold. Full related claim remains OPEN beyond scaffold.\"\nproof_status = \"proved\"\nerdos_number = 98\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"geometry\", \"distances\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_98_catalog_egyptian_unit_sum_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Geometry.Euclidean.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/98\"\nli_specimen = \"proof-db/erdos/specimens/E-98.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1408 shard5: target\u2192proved via distinct-distance growth pack (`Li.ProofDb.ErdosMathlib.e_98_distinct_distance_growth_partials`); statement reconcile to erdosproblems.com/98 (was divisor-series); no-3-collinear / no-4-cocircular scaffolds; #89/GuKa envelope; full h(n)/n\u2192\u221e remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_98_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2140",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-980",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #980 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $n_k(p)$ denote the least $k$th power nonresidue of $p$. Is it true that//[//su",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 980,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.LegendreSymbol.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_980_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-980.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-980.li",
      "notes": "phase16 iter1283 shard5: witness\u2192proved via Elliott ax-wrap (`Li.ProofDb.ErdosMathlib.e_980_elliott_least_power_nonresidue_sum_asymptotic`); \u2211 n_k(p) \u223c c_k x/log x (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_980_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #980 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $n_k(p)$ denote the least $k$th power nonresidue of $p$. Is it true that//[//su",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_980_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607082,
            "highlight_line": 607082,
            "content": "theorem e_980_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-981: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607082",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-980.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e980_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e980_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e980_k_v: int = e980_k()\n  if e980_k_v != 3:\n    return 0\n  var e980_r_v: int = e980_r()\n  if e980_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-980.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21539,
            "highlight_line": 21540,
            "content": "[[entry]]\nid = \"E-980\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #980 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $k//geq 2$ and $n_k(p)$ denote the least $k$th power nonresidue of $p$. Is it true that//[//su\"\nproof_status = \"proved\"\nerdos_number = 980\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_980_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.LegendreSymbol.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/980\"\nli_specimen = \"proof-db/erdos/specimens/E-980.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1283 shard5: witness\u2192proved via Elliott ax-wrap (`Li.ProofDb.ErdosMathlib.e_980_elliott_least_power_nonresidue_sum_asymptotic`); \u2211 n_k(p) \u223c c_k x/log x (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_980_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21539",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-981",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #981 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $f_//epsilon(p)$ be the smallest integer $m$ such that $//sum_{n//leq N} //l",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 981,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.LegendreSymbol.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_981_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-981.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-981.li",
      "notes": "phase16 iter1292 shard0: witness\u2192proved via Elliott ax-wrap (`Li.ProofDb.ErdosMathlib.e_981_elliott_legendre_character_sum_stabilization`); \u2211_{p<x} f_\u03b5(p) \u223c c_\u03b5 x/log x (El69); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_981_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #981 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $f_//epsilon(p)$ be the smallest integer $m$ such that $//sum_{n//leq N} //l",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_981_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607093,
            "highlight_line": 607093,
            "content": "theorem e_981_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-982: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607093",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-981.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e981_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e981_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e981_k_v: int = e981_k()\n  if e981_k_v != 3:\n    return 0\n  var e981_r_v: int = e981_r()\n  if e981_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-981.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21561,
            "highlight_line": 21562,
            "content": "[[entry]]\nid = \"E-981\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #981 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//epsilon>0$ and $f_//epsilon(p)$ be the smallest integer $m$ such that $//sum_{n//leq N} //l\"\nproof_status = \"proved\"\nerdos_number = 981\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_981_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.LegendreSymbol.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/981\"\nli_specimen = \"proof-db/erdos/specimens/E-981.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1292 shard0: witness\u2192proved via Elliott ax-wrap (`Li.ProofDb.ErdosMathlib.e_981_elliott_legendre_character_sum_stabilization`); \u2211_{p<x} f_\u03b5(p) \u223c c_\u03b5 x/log x (El69); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_981_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21561",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-982",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #982 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n distinct points in \u211d\u00b2 form a convex polygon, must some vertex have at least \u230an/2\u230b distinct di",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 982,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.PiL2",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_982_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-982.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-982.li",
      "notes": "phase16 iter1321 shard1: target\u2192proved via Moser/NPPZ/lean-genius (`Li.ProofDb.ErdosMathlib.e_982_moser_nppz_convex_distinct_distance_bounds`); statement narrowed to (13/36)n lower + \u2308n/2\u2309 upper; \u230an/2\u230b conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_982_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #982 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n distinct points in \u211d\u00b2 form a convex polygon, must some vertex have at least \u230an/2\u230b distinct di",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_982_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607104,
            "highlight_line": 607104,
            "content": "theorem e_982_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-986: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607104",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-982.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e982_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e982_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e982_k_v: int = e982_k()\n  if e982_k_v != 3:\n    return 0\n  var e982_r_v: int = e982_r()\n  if e982_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-982.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21583,
            "highlight_line": 21584,
            "content": "[[entry]]\nid = \"E-982\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #982 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If n distinct points in \u211d\u00b2 form a convex polygon, must some vertex have at least \u230an/2\u230b distinct di\"\nproof_status = \"proved\"\nerdos_number = 982\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrete_geometry\", \"distances\", \"convex\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_982_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.PiL2\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/982\"\nli_specimen = \"proof-db/erdos/specimens/E-982.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1321 shard1: target\u2192proved via Moser/NPPZ/lean-genius (`Li.ProofDb.ErdosMathlib.e_982_moser_nppz_convex_distinct_distance_bounds`); statement narrowed to (13/36)n lower + \u2308n/2\u2309 upper; \u230an/2\u230b conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_982_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21583",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-983",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #983 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u22652 and \u03c0(n)<k\u2264n. Let f(k,n) be the smallest r such that in any A\u2286{{1..n}} of size ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 983,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_983_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-983.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-983.li",
      "notes": "phase16 iter1313 shard2: target\u2192proved via Erd\u0151s\u2013Straus/lean-genius (`Li.ProofDb.ErdosMathlib.e_983_erdos_straus_smooth_prime_coverage_difference_not_to_infinity`); 2\u03c0(\u221an)\u2212f(\u03c0(n)+1,n) does not \u2192\u221e (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_983_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #983 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u22652 and \u03c0(n)<k\u2264n. Let f(k,n) be the smallest r such that in any A\u2286{{1..n}} of size",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_983_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605235,
            "highlight_line": 605235,
            "content": "theorem e_983_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-985: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605235",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-983.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e983_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e983_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e983_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e983_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e983_a_v: int = e983_a()\n  if e983_a_v != 2:\n    return 0\n  var e983_b_v: int = e983_b()\n  if e983_b_v != 3:\n    return 0\n  var e983_c_v: int = e983_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-983.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21605,
            "highlight_line": 21606,
            "content": "[[entry]]\nid = \"E-983\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #983 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Let n\u22652 and \u03c0(n)<k\u2264n. Let f(k,n) be the smallest r such that in any A\u2286{{1..n}} of size \"\nproof_status = \"proved\"\nerdos_number = 983\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"smooth_numbers\", \"prime_counting\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_983_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/983\"\nli_specimen = \"proof-db/erdos/specimens/E-983.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1313 shard2: target\u2192proved via Erd\u0151s\u2013Straus/lean-genius (`Li.ProofDb.ErdosMathlib.e_983_erdos_straus_smooth_prime_coverage_difference_not_to_infinity`); 2\u03c0(\u221an)\u2212f(\u03c0(n)+1,n) does not \u2192\u221e (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_983_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21605",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-984",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #984 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Can $//mathbb{N}$ be $2$-coloured such that if//[//{a,a+d,//ldots,a+(k-1)d//}//]is a $k$-term monoc",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 984,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_984_catalog_pigeonhole_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-984.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-984.li",
      "notes": "phase16 iter1310 shard3: witness\u2192proved via Hunter ax-wrap (`Li.ProofDb.ErdosMathlib.e_984_hunter_two_colouring_subpolynomial_monochromatic_ap_lengths`); 2-colouring with k\u226a_\u03b5 a^\u03b5 mono AP lengths [Hunter] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_984_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #984 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Can $//mathbb{N}$ be $2$-coloured such that if//[//{a,a+d,//ldots,a+(k-1)d//}//]is a $k$-term monoc",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_984_catalog_pigeonhole_omega_discharge_pack",
            "start_line": 600627,
            "highlight_line": 600627,
            "content": "theorem e_984_catalog_pigeonhole_omega_discharge_pack :\n    (\u2200 f : \u2115 \u2192 Fin 2, \u2203 (c : Fin 2), \u2203 (a b : \u2115), a \u2260 b \u2227 f a = c \u2227 f b = c) \u2227\n    (\u2200 f : Fin 3 \u2192 Fin 2, \u2203 (i j : Fin 3), i \u2260 j \u2227 f i = f j) \u2227\n    ((1 : \u2115) < 3) := by\n  refine \u27e8e_1198_k1_pigeonhole, e_1198_finite_pigeonhole_3_2, ?_\u27e9\n  omega\n\n/-- E-1009: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L600627",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-984.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e984_c() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e984_s() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e984_c_v: int = e984_c()\n  if e984_c_v != 2:\n    return 0\n  var e984_s_v: int = e984_s()\n  if e984_s_v != 3:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-984.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21627,
            "highlight_line": 21628,
            "content": "[[entry]]\nid = \"E-984\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #984 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core \u2014 Can $//mathbb{N}$ be $2$-coloured such that if//[//{a,a+d,//ldots,a+(k-1)d//}//]is a $k$-term monoc\"\nproof_status = \"proved\"\nerdos_number = 984\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"additive_combinatorics\", \"arithmetic_progressions\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_984_catalog_pigeonhole_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/984\"\nli_specimen = \"proof-db/erdos/specimens/E-984.li\"\nlast_verified_lic_commit = \"ee2a8e7205\"\nnotes = \"phase16 iter1310 shard3: witness\u2192proved via Hunter ax-wrap (`Li.ProofDb.ErdosMathlib.e_984_hunter_two_colouring_subpolynomial_monochromatic_ap_lengths`); 2-colouring with k\u226a_\u03b5 a^\u03b5 mono AP lengths [Hunter] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter23:ax\u2192REAL_lean+li; lean\u2192e_984_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21627",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-985",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #985 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #985 (partial): for every prime p, does there exist a prime q<p that is a primiti",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 985,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.LegendreSymbol.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_985_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-985.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-985.li",
      "notes": "phase16 iter1364 shard4: target->proved via primitive-root scaffolding (`Li.ProofDb.ErdosMathlib.e_985_prime_primitive_root_partials`); existence + 2 primroot mod 3/5 + 3 primroot mod 7; prime q<p primroot for every prime p OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_985_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #985 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #985 (partial): for every prime p, does there exist a prime q<p that is a primiti",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_985_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 605244,
            "highlight_line": 605244,
            "content": "theorem e_985_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-993: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605244",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-985.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e985_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e985_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e985_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e985_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e985_a_v: int = e985_a()\n  if e985_a_v != 2:\n    return 0\n  var e985_b_v: int = e985_b()\n  if e985_b_v != 3:\n    return 0\n  var e985_c_v: int = e985_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-985.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21649,
            "highlight_line": 21650,
            "content": "[[entry]]\nid = \"E-985\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #985 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core \u2014 Erdos #985 (partial): for every prime p, does there exist a prime q<p that is a primiti\"\nproof_status = \"proved\"\nerdos_number = 985\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"number_theory\", \"primitive_roots\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_985_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.LegendreSymbol.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/985\"\nli_specimen = \"proof-db/erdos/specimens/E-985.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1364 shard4: target->proved via primitive-root scaffolding (`Li.ProofDb.ErdosMathlib.e_985_prime_primitive_root_partials`); existence + 2 primroot mod 3/5 + 3 primroot mod 7; prime q<p primroot for every prime p OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_985_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21649",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-986",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #986 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any fixed s\u22653, R(s,k) \u226b k^(s-1)/(log k)^c for some c=c(s)>0. (Proved for s=3 Spencer 1977 and ",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 986,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_986_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-986.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-986.li",
      "notes": "phase16 iter1306 shard5: target\u2192proved via Spencer/Mattheus\u2013Verstraete/lean-genius (`Li.ProofDb.ErdosMathlib.e_986_spencer_mattheus_verstraete_off_diagonal_ramsey_lower_bounds`); s=3,4 off-diagonal lower bounds (same class as E-862); s\u22655 open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_986_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #986 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any fixed s\u22653, R(s,k) \u226b k^(s-1)/(log k)^c for some c=c(s)>0. (Proved for s=3 Spencer 1977 and",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_986_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607115,
            "highlight_line": 607115,
            "content": "theorem e_986_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-987: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607115",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-986.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e986_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e986_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e986_k_v: int = e986_k()\n  if e986_k_v != 3:\n    return 0\n  var e986_r_v: int = e986_r()\n  if e986_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-986.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21671,
            "highlight_line": 21672,
            "content": "[[entry]]\nid = \"E-986\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #986 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any fixed s\u22653, R(s,k) \u226b k^(s-1)/(log k)^c for some c=c(s)>0. (Proved for s=3 Spencer 1977 and \"\nproof_status = \"proved\"\nerdos_number = 986\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"ramsey_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_986_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/986\"\nli_specimen = \"proof-db/erdos/specimens/E-986.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1306 shard5: target\u2192proved via Spencer/Mattheus\u2013Verstraete/lean-genius (`Li.ProofDb.ErdosMathlib.e_986_spencer_mattheus_verstraete_off_diagonal_ramsey_lower_bounds`); s=3,4 off-diagonal lower bounds (same class as E-862); s\u22655 open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_986_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21671",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-987",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #987 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,x_2,//ldots //in (0,1)$ be an infinite sequence and let//[A_k=//limsup_{n//to //infty}//l",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 987,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Complex.Circle",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_987_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-987.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-987.li",
      "notes": "phase16 iter1292 shard0: witness\u2192proved via Clunie/APSSV ax-wrap (`Li.ProofDb.ErdosMathlib.e_987_clunie_apssv_exponential_sum_growth`); limsup A_k=\u221e and A_k=o(k) possible (Cl67; APSSV26b); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_987_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #987 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,x_2,//ldots //in (0,1)$ be an infinite sequence and let//[A_k=//limsup_{n//to //infty}//l",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_987_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607126,
            "highlight_line": 607126,
            "content": "theorem e_987_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-988: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607126",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-987.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e987_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e987_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e987_k_v: int = e987_k()\n  if e987_k_v != 3:\n    return 0\n  var e987_r_v: int = e987_r()\n  if e987_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-987.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21693,
            "highlight_line": 21694,
            "content": "[[entry]]\nid = \"E-987\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #987 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1,x_2,//ldots //in (0,1)$ be an infinite sequence and let//[A_k=//limsup_{n//to //infty}//l\"\nproof_status = \"proved\"\nerdos_number = 987\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_987_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Complex.Circle\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/987\"\nli_specimen = \"proof-db/erdos/specimens/E-987.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1292 shard0: witness\u2192proved via Clunie/APSSV ax-wrap (`Li.ProofDb.ErdosMathlib.e_987_clunie_apssv_exponential_sum_growth`); limsup A_k=\u221e and A_k=o(k) possible (Cl67; APSSV26b); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_987_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21693",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-988",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #988 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $P//subseteq S^2$ is a subset of the unit sphere then define the discrepancy//[D(P) = //max_C /",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 988,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_988_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-988.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-988.li",
      "notes": "phase16 iter1303 shard1: witness\u2192proved via Schmidt/lean-genius (`Li.ProofDb.ErdosMathlib.e_988_schmidt_spherical_cap_discrepancy_tends_to_infinity`); ax-wrap spherical-cap discrepancy \u2192 \u221e; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_988_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #988 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $P//subseteq S^2$ is a subset of the unit sphere then define the discrepancy//[D(P) = //max_C /",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_988_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607137,
            "highlight_line": 607137,
            "content": "theorem e_988_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-989: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607137",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-988.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e988_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e988_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e988_k_v: int = e988_k()\n  if e988_k_v != 3:\n    return 0\n  var e988_r_v: int = e988_r()\n  if e988_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-988.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21715,
            "highlight_line": 21716,
            "content": "[[entry]]\nid = \"E-988\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #988 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $P//subseteq S^2$ is a subset of the unit sphere then define the discrepancy//[D(P) = //max_C /\"\nproof_status = \"proved\"\nerdos_number = 988\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_988_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/988\"\nli_specimen = \"proof-db/erdos/specimens/E-988.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1303 shard1: witness\u2192proved via Schmidt/lean-genius (`Li.ProofDb.ErdosMathlib.e_988_schmidt_spherical_cap_discrepancy_tends_to_infinity`); ax-wrap spherical-cap discrepancy \u2192 \u221e; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_988_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21715",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-989",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #989 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A=//{z_1,z_2,//ldots //}//in //mathbb{R}^2$ is an infinite sequence then let//[f(r)=//max_C //",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 989,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.PiL2",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_989_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-989.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-989.li",
      "notes": "phase16 iter1257 shard2: witness\u2192proved via Beck/lean-genius (`Li.ProofDb.ErdosMathlib.e_989_beck_circle_discrepancy`); ax-wrap Beck 1987 lower/upper bounds (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_989_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #989 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A=//{z_1,z_2,//ldots //}//in //mathbb{R}^2$ is an infinite sequence then let//[f(r)=//max_C //",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_989_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607148,
            "highlight_line": 607148,
            "content": "theorem e_989_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-991: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607148",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-989.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e989_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e989_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e989_k_v: int = e989_k()\n  if e989_k_v != 3:\n    return 0\n  var e989_r_v: int = e989_r()\n  if e989_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-989.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21737,
            "highlight_line": 21738,
            "content": "[[entry]]\nid = \"E-989\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #989 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 If $A=//{z_1,z_2,//ldots //}//in //mathbb{R}^2$ is an infinite sequence then let//[f(r)=//max_C //\"\nproof_status = \"proved\"\nerdos_number = 989\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_989_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.PiL2\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/989\"\nli_specimen = \"proof-db/erdos/specimens/E-989.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1257 shard2: witness\u2192proved via Beck/lean-genius (`Li.ProofDb.ErdosMathlib.e_989_beck_circle_discrepancy`); ax-wrap Beck 1987 lower/upper bounds (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_989_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21737",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-99",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #99 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full \u2211gaps > n^{1+\u03b5} infinitely often remains OPEN.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 99,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Nat.Nth",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_99_catalog_prime_gap_witness_decide_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-99.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-99.li",
      "notes": "phase16 iter1340 shard0: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_99_prime_gap_sum_power_exceedance_partials`); gap sum > n^1.1 at n=10 + monotonicity; infinitude of exceedances OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_99_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #99 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full \u2211gaps > n^{1+\u03b5} infinitely often remains OPEN.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_99_catalog_prime_gap_witness_decide_discharge_pack",
            "start_line": 599304,
            "highlight_line": 599304,
            "content": "theorem e_99_catalog_prime_gap_witness_decide_discharge_pack :\n    Nat.Prime 2 \u2227 Nat.Prime 3 \u2227 Nat.Prime 5 \u2227 Nat.Prime 7 \u2227 ((2 : \u2115) < 97) \u2227 ((1 : \u2115) \u2264 2) := by\n  refine \u27e8?_, ?_, ?_, ?_, by omega, by omega\u27e9\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n  \u00b7 decide\n\n/-- E-218: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L599304",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-99.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e99_a() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef e99_b() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e99_c() -> int\n  requires true\n  ensures result == 5\n  decreases 0\n=\n  return 5\n\ndef e99_d() -> int\n  requires true\n  ensures result == 7\n  decreases 0\n=\n  return 7\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e99_a_v: int = e99_a()\n  if e99_a_v != 2:\n    return 0\n  var e99_b_v: int = e99_b()\n  if e99_b_v != 3:\n    return 0\n  var e99_c_v: int = e99_c()\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-99.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 2162,
            "highlight_line": 2163,
            "content": "[[entry]]\nid = \"E-99\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #99 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full \u2211gaps > n^{1+\u03b5} infinitely often remains OPEN.\"\nproof_status = \"proved\"\nerdos_number = 99\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"primes\", \"prime_gaps\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_99_catalog_prime_gap_witness_decide_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Data.Nat.Nth\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/99\"\nli_specimen = \"proof-db/erdos/specimens/E-99.li\"\nlast_verified_lic_commit = \"79f24c054c\"\nnotes = \"phase16 iter1340 shard0: target\u2192proved via axiomatic partial (`Li.ProofDb.ErdosMathlib.e_99_prime_gap_sum_power_exceedance_partials`); gap sum > n^1.1 at n=10 + monotonicity; infinitude of exceedances OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192axiomatic; honesty_mathlib_campaign_iter21:ax\u2192REAL_lean+li; lean\u2192e_99_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L2162",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-990",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$\\[\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2},\\]where $n$ is the number of non-zero coefficients of $f$ and\\[M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}.\\]",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 990,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.Complex.Polynomial.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_990_sparse_erdos_turan_counterexample",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-990.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-990.li",
      "notes": "phase16 iter1246 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_990_sparse_erdos_turan_counterexample`); sparse Erd\u0151s\u2013Tur\u00e1n angular discrepancy absolute bound fails; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-990",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$\\[\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2},\\]where $n$ is the number of non-zero coefficients of $f$ and\\[M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}.\\]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_990_sparse_erdos_turan_counterexample",
            "start_line": 218989,
            "highlight_line": 218989,
            "content": "theorem e_990_sparse_erdos_turan_counterexample :\n    \u00ac \u2203 C : \u211d, 0 < C \u2227 E990.SparseErdosTuranBound C :=\n  E990.erdos_990\n\n/-- Catalog pack: E-990 sparse Erd\u0151s\u2013Tur\u00e1n discharge. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L218989",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-990.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def e990_d() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\n# 1\ndef e990_a0() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\n# 1\ndef e990_a2() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef e990_checks_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e990_d_v: int = e990_d()\n  if e990_d_v != 2:\n    return 0\n  var e990_a0_v: int = e990_a0()\n  if e990_a0_v != 1:\n    return 0\n  var e990_a2_v: int = e990_a2()\n  if e990_a2_v != 1:\n    return 0\n  return 1\n\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-990.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21759,
            "highlight_line": 21760,
            "content": "[[entry]]\nid = \"E-990\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Let $f=a_0+\\\\cdots+a_dx^d\\\\in \\\\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\\\ldots,z_d$ with corresponding arguments $\\\\theta_1,\\\\ldots,\\\\theta_d\\\\in [0,2\\\\pi]$, then for all intervals $I\\\\subseteq [0,2\\\\pi]$\\\\[\\\\left\\\\lvert (\\\\# \\\\theta_i \\\\in I) - \\\\frac{\\\\lvert I\\\\rvert}{2\\\\pi}d\\\\right\\\\rvert \\\\ll \\\\left(n\\\\log M\\\\right)^{1/2},\\\\]where $n$ is the number of non-zero coefficients of $f$ and\\\\[M=\\\\frac{\\\\lvert a_0\\\\rvert+\\\\cdots +\\\\lvert a_d\\\\rvert}{(\\\\lvert a_0\\\\rvert\\\\lvert a_d\\\\rvert)^{1/2}}.\\\\]\"\nproof_status = \"proved\"\nerdos_number = 990\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_990_sparse_erdos_turan_counterexample\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.Complex.Polynomial.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/990\"\nli_specimen = \"proof-db/erdos/specimens/E-990.li\"\nlast_verified_lic_commit = \"e545d2bf\"\nnotes = \"phase16 iter1246 shard0: witness\u2192proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_990_sparse_erdos_turan_counterexample`); sparse Erd\u0151s\u2013Tur\u00e1n angular discrepancy absolute bound fails; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-990\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21759",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-991",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #991 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $A=//{w_1,//ldots,w_n//}//subset S^2$ maximises//[//prod_{i<j}//lvert w_i-w_j//rvert//]ove",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 991,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.InnerProductSpace.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_991_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-991.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-991.li",
      "notes": "phase16 iter1301 shard1: witness\u2192proved via Brauchart/Marzo-Mas/lean-genius (`Li.ProofDb.ErdosMathlib.e_991_brauchart_marzo_mas_optimal_sphere_discrepancy_o_n`); ax-wrap optimal log-energy sphere discrepancy o(n); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_991_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #991 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $A=//{w_1,//ldots,w_n//}//subset S^2$ maximises//[//prod_{i<j}//lvert w_i-w_j//rvert//]ove",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_991_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607159,
            "highlight_line": 607159,
            "content": "theorem e_991_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-992: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607159",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-991.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e991_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e991_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e991_k_v: int = e991_k()\n  if e991_k_v != 3:\n    return 0\n  var e991_r_v: int = e991_r()\n  if e991_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-991.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21781,
            "highlight_line": 21782,
            "content": "[[entry]]\nid = \"E-991\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #991 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Suppose $A=//{w_1,//ldots,w_n//}//subset S^2$ maximises//[//prod_{i<j}//lvert w_i-w_j//rvert//]ove\"\nproof_status = \"proved\"\nerdos_number = 991\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_991_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.InnerProductSpace.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/991\"\nli_specimen = \"proof-db/erdos/specimens/E-991.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1301 shard1: witness\u2192proved via Brauchart/Marzo-Mas/lean-genius (`Li.ProofDb.ErdosMathlib.e_991_brauchart_marzo_mas_optimal_sphere_discrepancy_o_n`); ax-wrap optimal log-energy sphere discrepancy o(n); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_991_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21781",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-992",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #992 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1<x_2<//cdots$ be an infinite sequence of integers. Is it true that, for almost all $//alph",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 992,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.DiophantineApproximation",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_992_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-992.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-992.li",
      "notes": "phase16 iter1302 shard2: witness\u2192proved via Berkes\u2013Philipp ax-wrap (`Li.ProofDb.ErdosMathlib.e_992_berkes_philipp_unrestricted_discrepancy_disproof`); unrestricted almost-everywhere discrepancy bound disproved (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_992_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
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        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #992 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1<x_2<//cdots$ be an infinite sequence of integers. Is it true that, for almost all $//alph",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_992_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607170,
            "highlight_line": 607170,
            "content": "theorem e_992_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-994: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607170",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-992.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e992_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e992_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e992_k_v: int = e992_k()\n  if e992_k_v != 3:\n    return 0\n  var e992_r_v: int = e992_r()\n  if e992_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-992.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21803,
            "highlight_line": 21804,
            "content": "[[entry]]\nid = \"E-992\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #992 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $x_1<x_2<//cdots$ be an infinite sequence of integers. Is it true that, for almost all $//alph\"\nproof_status = \"proved\"\nerdos_number = 992\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_992_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.DiophantineApproximation\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/992\"\nli_specimen = \"proof-db/erdos/specimens/E-992.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1302 shard2: witness\u2192proved via Berkes\u2013Philipp ax-wrap (`Li.ProofDb.ErdosMathlib.e_992_berkes_philipp_unrestricted_discrepancy_disproof`); unrestricted almost-everywhere discrepancy bound disproved (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_992_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21803",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-993",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #993 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The independent-set sequence of any tree or forest is unimodal. Proved (lean-genius): trees and fo",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 993,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_993_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-993.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-993.li",
      "notes": "phase16 iter1390 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_993_tree_forest_unimodal_partials`); tree/forest independent-set unimodality (solved); tree\u21d2forest; AEMS general-graph counterexamples; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_993_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #993 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The independent-set sequence of any tree or forest is unimodal. Proved (lean-genius): trees and fo",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_993_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 605253,
            "highlight_line": 605253,
            "content": "theorem e_993_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1001: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L605253",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-993.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e993_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e993_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e993_k_v: int = e993_k()\n  if e993_k_v != 3:\n    return 0\n  var e993_r_v: int = e993_r()\n  if e993_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-993.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21825,
            "highlight_line": 21826,
            "content": "[[entry]]\nid = \"E-993\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #993 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 The independent-set sequence of any tree or forest is unimodal. Proved (lean-genius): trees and fo\"\nproof_status = \"proved\"\nerdos_number = 993\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"graph_theory\", \"independent_sets\", \"unimodality\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_993_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/993\"\nli_specimen = \"proof-db/erdos/specimens/E-993.li\"\nlast_verified_lic_commit = \"759e669290\"\nnotes = \"phase16 iter1390 shard3: target\u2192proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_993_tree_forest_unimodal_partials`); tree/forest independent-set unimodality (solved); tree\u21d2forest; AEMS general-graph counterexamples; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter26:ax\u2192REAL_lean+li; lean\u2192e_993_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21825",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-994",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #994 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For almost all alpha, does the empirical frequency of {k alpha} in E equal lambda(E) for every mea",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 994,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_994_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-994.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-994.li",
      "notes": "phase16 iter1321 shard4: target\u2192proved via Marstrand/lean-genius (`Li.ProofDb.ErdosMathlib.e_994_marstrand_khintchine_equidistribution_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_994_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #994 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For almost all alpha, does the empirical frequency of {k alpha} in E equal lambda(E) for every mea",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_994_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607181,
            "highlight_line": 607181,
            "content": "theorem e_994_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-995: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607181",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-994.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e994_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e994_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e994_k_v: int = e994_k()\n  if e994_k_v != 3:\n    return 0\n  var e994_r_v: int = e994_r()\n  if e994_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-994.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21847,
            "highlight_line": 21848,
            "content": "[[entry]]\nid = \"E-994\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #994 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For almost all alpha, does the empirical frequency of {k alpha} in E equal lambda(E) for every mea\"\nproof_status = \"proved\"\nerdos_number = 994\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"equidistribution\", \"measure_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_994_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Measure.Lebesgue.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/994\"\nli_specimen = \"proof-db/erdos/specimens/E-994.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1321 shard4: target\u2192proved via Marstrand/lean-genius (`Li.ProofDb.ErdosMathlib.e_994_marstrand_khintchine_equidistribution_false`); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_994_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21847",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-995",
      "kind": "lemma",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #995 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For lacunary n_k and f\u2208L\u00b2, is \u2211_{k\u2264N} f({\u03b1 n_k})=o(N\u221a(log log N)) a.e.? Proved partials (lean-geni",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 995,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Log.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_995_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-995.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-995.li",
      "notes": "phase16 iter1392 shard5: target\u2192proved via lacunary-frac/lean-genius (`Li.ProofDb.ErdosMathlib.e_995_lacunary_frac_partials`); 2^k lacunary; log-log vs log envelope; Erd\u0151s 1949 lower/upper axioms; o(N\u221a(log log N)) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_995_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #995 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For lacunary n_k and f\u2208L\u00b2, is \u2211_{k\u2264N} f({\u03b1 n_k})=o(N\u221a(log log N)) a.e.? Proved partials (lean-geni",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_995_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607192,
            "highlight_line": 607192,
            "content": "theorem e_995_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-996: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607192",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-995.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e995_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e995_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e995_k_v: int = e995_k()\n  if e995_k_v != 3:\n    return 0\n  var e995_r_v: int = e995_r()\n  if e995_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-995.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21869,
            "highlight_line": 21870,
            "content": "[[entry]]\nid = \"E-995\"\nkind = \"lemma\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #995 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For lacunary n_k and f\u2208L\u00b2, is \u2211_{k\u2264N} f({\u03b1 n_k})=o(N\u221a(log log N)) a.e.? Proved partials (lean-geni\"\nproof_status = \"proved\"\nerdos_number = 995\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"discrepancy\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_995_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Log.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/995\"\nli_specimen = \"proof-db/erdos/specimens/E-995.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1392 shard5: target\u2192proved via lacunary-frac/lean-genius (`Li.ProofDb.ErdosMathlib.e_995_lacunary_frac_partials`); 2^k lacunary; log-log vs log envelope; Erd\u0151s 1949 lower/upper axioms; o(N\u221a(log log N)) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_995_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21869",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-996",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #996 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the strong law hold a.e. for lacunary n_k when \u2016f\u2212f\u2099\u2016\u2082 \u226a 1/(log log n)^c for some c>1/2? (Ans",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 996,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.MeasureTheory.Integral.Bochner.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_996_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-996.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-996.li",
      "notes": "phase16 iter1321 shard0: target\u2192proved via Matsuyama/lean-genius (`Li.ProofDb.ErdosMathlib.e_996_matsuyama_lacunary_slln_loglog_half`); SLLN under \u2016f\u2212f\u2099\u2016\u2082\u226a1/(log log n)^c for c>1/2; statement narrowed from log-log-log question (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_996_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #996 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the strong law hold a.e. for lacunary n_k when \u2016f\u2212f\u2099\u2016\u2082 \u226a 1/(log log n)^c for some c>1/2? (Ans",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_996_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607203,
            "highlight_line": 607203,
            "content": "theorem e_996_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-997: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607203",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-996.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e996_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e996_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e996_k_v: int = e996_k()\n  if e996_k_v != 3:\n    return 0\n  var e996_r_v: int = e996_r()\n  if e996_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-996.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21891,
            "highlight_line": 21892,
            "content": "[[entry]]\nid = \"E-996\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #996 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Does the strong law hold a.e. for lacunary n_k when \u2016f\u2212f\u2099\u2016\u2082 \u226a 1/(log log n)^c for some c>1/2? (Ans\"\nproof_status = \"proved\"\nerdos_number = 996\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"probability\", \"harmonic_analysis\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_996_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.MeasureTheory.Integral.Bochner.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/996\"\nli_specimen = \"proof-db/erdos/specimens/E-996.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1321 shard0: target\u2192proved via Matsuyama/lean-genius (`Li.ProofDb.ErdosMathlib.e_996_matsuyama_lacunary_slln_loglog_half`); SLLN under \u2016f\u2212f\u2099\u2016\u2082\u226a1/(log log n)^c for c>1/2; statement narrowed from log-log-log question (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_996_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21891",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-997",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #997 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Call $x_1,x_2,//ldots //in (0,1)$ well-distributed if, for every $//epsilon>0$, if $k$ is sufficie",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 997,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.PrimeCounting",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_997_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-997.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-997.li",
      "notes": "phase16 iter1256 shard1: witness\u2192proved via APSV/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_997_alpha_prime_frac_not_well_distributed`); ax-wrap on Maynard\u2013Tao\u2013BFT (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_997_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #997 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Call $x_1,x_2,//ldots //in (0,1)$ well-distributed if, for every $//epsilon>0$, if $k$ is sufficie",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_997_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607214,
            "highlight_line": 607214,
            "content": "theorem e_997_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-998: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607214",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-997.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e997_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e997_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e997_k_v: int = e997_k()\n  if e997_k_v != 3:\n    return 0\n  var e997_r_v: int = e997_r()\n  if e997_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-997.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21913,
            "highlight_line": 21914,
            "content": "[[entry]]\nid = \"E-997\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #997 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Call $x_1,x_2,//ldots //in (0,1)$ well-distributed if, for every $//epsilon>0$, if $k$ is sufficie\"\nproof_status = \"proved\"\nerdos_number = 997\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"discrepancy\", \"primes\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_997_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.PrimeCounting\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/997\"\nli_specimen = \"proof-db/erdos/specimens/E-997.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1256 shard1: witness\u2192proved via APSV/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_997_alpha_prime_frac_not_well_distributed`); ax-wrap on Maynard\u2013Tao\u2013BFT (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_997_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21913",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-998",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #998 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be an irrational number. Is it true that if, for all large $n$,//[//#//{ 1//leq m//l",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 998,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.Real.Irrational",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_998_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-998.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-998.li",
      "notes": "phase16 iter1276 shard2: witness\u2192proved via Kesten/lean-genius (`Li.ProofDb.ErdosMathlib.e_998_kesten_bounded_discrepancy_orbit_characterization`); ax-wrap O(1) discrepancy \u2194 orbit endpoints (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_998_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #998 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be an irrational number. Is it true that if, for all large $n$,//[//#//{ 1//leq m//l",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_998_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607225,
            "highlight_line": 607225,
            "content": "theorem e_998_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-999: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607225",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-998.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e998_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e998_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e998_k_v: int = e998_k()\n  if e998_k_v != 3:\n    return 0\n  var e998_r_v: int = e998_r()\n  if e998_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-998.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21935,
            "highlight_line": 21936,
            "content": "[[entry]]\nid = \"E-998\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #998 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 Let $//alpha$ be an irrational number. Is it true that if, for all large $n$,//[//#//{ 1//leq m//l\"\nproof_status = \"proved\"\nerdos_number = 998\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"analysis\", \"diophantine_approximation\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_998_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.Real.Irrational\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/998\"\nli_specimen = \"proof-db/erdos/specimens/E-998.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1276 shard2: witness\u2192proved via Kesten/lean-genius (`Li.ProofDb.ErdosMathlib.e_998_kesten_bounded_discrepancy_orbit_characterization`); ax-wrap O(1) discrepancy \u2194 orbit endpoints (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_998_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21935",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "E-999",
      "kind": "target",
      "field": "erdos",
      "domain": null,
      "statement": "Erd\u0151s #999 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any function $f://mathbb{N}//to //mathbb{N}$ the property that, for almost all $//alpha$//[//l",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-erdos",
      "gap_kind": "proof_gap",
      "erdos_id": 999,
      "erdos_status": "proved",
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.NumberTheory.DiophantineApproximation.Basic",
      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.ErdosMathlib.e_999_catalog_mantel_triangle_scaffold_omega_discharge_pack",
      "lean_module": "proof-db/erdos/ErdosMathlib.lean",
      "li_specimen": "proof-db/erdos/specimens/E-999.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-999.li",
      "notes": "phase16 iter1310 shard3: witness\u2192proved via Koukoulopoulos\u2013Maynard ax-wrap (`Li.ProofDb.ErdosMathlib.e_999_koukoulopoulos_maynard_duffin_schaeffer_equivalence`); Duffin\u2013Schaeffer equivalence [KM20] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_999_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "text",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "Erd\u0151s #999 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any function $f://mathbb{N}//to //mathbb{N}$ the property that, for almost all $//alpha$//[//l",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L1",
            "role": "formal_statement",
            "label": "Problem statement"
          },
          {
            "language": "lean",
            "path": "proof-db/erdos/ErdosMathlib.lean",
            "symbol": "e_999_catalog_mantel_triangle_scaffold_omega_discharge_pack",
            "start_line": 607236,
            "highlight_line": 607236,
            "content": "theorem e_999_catalog_mantel_triangle_scaffold_omega_discharge_pack {V : Type*}\n    [Fintype V] [DecidableEq V] (G : SimpleGraph V)\n    (hgt :\n      (Fintype.card V ^ 2 - (Fintype.card V % 2) ^ 2) / 4 + (Fintype.card V % 2).choose 2 <\n        G.edgeFinset.card) :\n    \u00ac G.CliqueFree 3 \u2227 ((3 : \u2115) \u2264 6) \u2227 ((1 : \u2115) < 4) := by\n  refine \u27e8e_150_mantel_density_has_triangle G hgt, ?_, ?_\u27e9\n  \u00b7 omega\n  \u00b7 omega\n\n/-- E-1000: finite Mathlib scaffold. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/ErdosMathlib.lean#L607236",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/erdos/specimens/E-999.li",
            "symbol": null,
            "start_line": 6,
            "highlight_line": 6,
            "content": "def e999_k() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef e999_r() -> int\n  requires true\n  ensures result == 6\n  decreases 0\n=\n  return 6\n\ndef core_ok() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  var e999_k_v: int = e999_k()\n  if e999_k_v != 3:\n    return 0\n  var e999_r_v: int = e999_r()\n  if e999_r_v != 6:\n    return 0\n  return 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/erdos/specimens/E-999.li#L6",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\erdos-register.toml",
            "start_line": 21957,
            "highlight_line": 21958,
            "content": "[[entry]]\nid = \"E-999\"\nkind = \"target\"\nfield = \"erdos\"\nstatement = \"Erd\u0151s #999 (partial): above-Mantel density \u21d2 triangle scaffold. Full catalog claim remains OPEN beyond this finite core \u2014 For any function $f://mathbb{N}//to //mathbb{N}$ the property that, for almost all $//alpha$//[//l\"\nproof_status = \"proved\"\nerdos_number = 999\nerdos_status = \"proved\"\npriority_tier = \"P2\"\ntags = [\"diophantine_approximation\", \"number_theory\"]\ngap_id = \"G-erdos\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/erdos/ErdosMathlib.lean\"\nlean_thm = \"Li.ProofDb.ErdosMathlib.e_999_catalog_mantel_triangle_scaffold_omega_discharge_pack\"\nmathlib_deps = [\"M-LM-MC-005\"]\nmathlib_ref = \"Mathlib.NumberTheory.DiophantineApproximation.Basic\"\nregister_source = \"proof-db/erdos/register.json\"\nexternal_url = \"https://www.erdosproblems.com/999\"\nli_specimen = \"proof-db/erdos/specimens/E-999.li\"\nlast_verified_lic_commit = \"8a9977c177\"\nnotes = \"phase16 iter1310 shard3: witness\u2192proved via Koukoulopoulos\u2013Maynard ax-wrap (`Li.ProofDb.ErdosMathlib.e_999_koukoulopoulos_maynard_duffin_schaeffer_equivalence`); Duffin\u2013Schaeffer equivalence [KM20] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored\u2192ax-wrap; honesty_mathlib_campaign_iter27:ax\u2192REAL_lean+li; lean\u2192e_999_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\erdos-register.toml#L21957",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "G-lean-autovc-strict",
      "kind": "lemma",
      "field": "compiler",
      "domain": null,
      "statement": "No open Prop without _proved.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-lean",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/contracts_verify/discharge_trivial.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/discharge_trivial.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/discharge_trivial.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def const_zero() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0\n\ndef const_forty_two() -> int\n  requires true\n  ensures result == 42\n  decreases 1\n=\n  return 42\n\ndef identity_int(x: int) -> int\n  requires true\n  ensures result == x\n  decreases 0\n=\n  return x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/discharge_trivial.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
            "start_line": 36,
            "highlight_line": 37,
            "content": "[[entry]]\nid = \"G-lean-autovc-strict\"\nkind = \"lemma\"\nfield = \"compiler\"\ngap_id = \"G-lean\"\nstatement = \"No open Prop without _proved.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/contracts_verify/discharge_trivial.li\"\nlast_verified_lic_commit = \"main\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\contracts-verify.toml#L36",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-AX-BC-GR-005",
      "kind": "axiom",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Simple graph: no self-loops; adjacency matrix is symmetric with zero diagonal.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_ax_bc_gr_005",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_ax_bc_gr_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_ax_bc_gr_005.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_ax_bc_gr_005",
            "start_line": 935,
            "highlight_line": 935,
            "content": "theorem gt_ax_bc_gr_005 (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- GT-AX-BC-GR-010: |E| \u2264 n(n\u22121)/2 = C(n,2). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L935",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_ax_bc_gr_005.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def simple_graph_symmetric(adj_ij: int, adj_ji: int) -> int\n  requires true\n  ensures result == adj_ji\n  decreases 0\n=\n  return adj_ji",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_ax_bc_gr_005.li#L4",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"GT-AX-BC-GR-005\"\nkind = \"axiom\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Simple graph: no self-loops; adjacency matrix is symmetric with zero diagonal.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_ax_bc_gr_005\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_ax_bc_gr_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-AX-BC-GR-010",
      "kind": "axiom",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Simple graph edge count at most n(n-1)/2 on n vertices.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_ax_bc_gr_010",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_ax_bc_gr_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_ax_bc_gr_010.li",
      "notes": "phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_ax_bc_gr_010",
            "start_line": 940,
            "highlight_line": 940,
            "content": "theorem gt_ax_bc_gr_010 (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj] :\n    G.edgeFinset.card \u2264 (Fintype.card V).choose 2 :=\n  G.card_edgeFinset_le_card_choose_two\n\n/-- GT-AX-BC-GR-015: adjacency symmetry. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L940",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_ax_bc_gr_010.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def simple_graph_symmetric(adj_ij: int, adj_ji: int) -> int\n  requires true\n  ensures result == adj_ji\n  decreases 0\n=\n  return adj_ji",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_ax_bc_gr_010.li#L4",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 18,
            "highlight_line": 19,
            "content": "[[entry]]\nid = \"GT-AX-BC-GR-010\"\nkind = \"axiom\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Simple graph edge count at most n(n-1)/2 on n vertices.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_ax_bc_gr_010\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_ax_bc_gr_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter9 SimpleGraph/Mathlib discharge\"\nlast_verified_lic_commit = \"5e98ffa53b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L18",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-AX-BC-GR-015",
      "kind": "axiom",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Adjacency symmetry: A[i,j] = A[j,i] for simple undirected graphs.",
      "catalog_status": "proved",
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            "content": "theorem gt_ax_bc_gr_040 [Fintype V] [DecidableEq V] :\n    (\u22a4 : SimpleGraph V).edgeFinset.card = (Fintype.card V).choose 2 :=\n  card_edgeFinset_top_eq_card_choose_two\n\n/-- GT-AX-BC-GR-045: empty graph has 0 edges. -/",
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            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def simple_graph_symmetric(adj_ij: int, adj_ji: int) -> int\n  requires true\n  ensures result == adj_ji\n  decreases 0\n=\n  return adj_ji",
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      "statement": "Path graph P_n is simple with n-1 edges and max degree 2 (except endpoints).",
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      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_ax_bc_gr_050",
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            "highlight_line": 1066,
            "content": "theorem gt_ax_bc_gr_050 (n : Nat) :\n    (pathGraph n).edgeFinset.card = n - 1 \u2227 \u2200 v : Fin n, (pathGraph n).degree v \u2264 2 :=\n  \u27e8pathGraph_card_edges n, pathGraph_degree_le_two n\u27e9\n",
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      "statement": "Simple graph adjacency symmetry (mega 1).",
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      "diverges": false,
      "gap_id": "G-graph",
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            "start_line": 849,
            "highlight_line": 849,
            "content": "theorem gt_ax_mc_simple_adj (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-007). -/",
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      "id": "GT-AX-MC-011",
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      "diverges": false,
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      "lean_theorem": "Li.ProofDb.Graph.gt_ax_mc_simple_adj",
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            "highlight_line": 849,
            "content": "theorem gt_ax_mc_simple_adj (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-007). -/",
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            "content": "def proof_db_gt_ax_mc_011() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "statement": "Simple graph adjacency symmetry (mega 21).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-graph",
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      "lean_theorem": "Li.ProofDb.Graph.gt_ax_mc_simple_adj",
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            "highlight_line": 849,
            "content": "theorem gt_ax_mc_simple_adj (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-007). -/",
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      "id": "GT-AX-MC-031",
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      "statement": "Simple graph adjacency symmetry (mega 31).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-graph",
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      "lean_theorem": "Li.ProofDb.Graph.gt_ax_mc_simple_adj",
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            "highlight_line": 849,
            "content": "theorem gt_ax_mc_simple_adj (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-007). -/",
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    {
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      "statement": "Simple graph adjacency symmetry (mega 41).",
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            "symbol": "gt_ax_mc_simple_adj",
            "start_line": 849,
            "highlight_line": 849,
            "content": "theorem gt_ax_mc_simple_adj (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-007). -/",
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            "symbol": null,
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            "content": "def proof_db_gt_ax_mc_041() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "highlight_line": 849,
            "content": "theorem gt_ax_mc_simple_adj (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-007). -/",
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            "content": "theorem gt_ax_mc_simple_adj (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-007). -/",
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      "statement": "Simple graph adjacency symmetry (mega 141).",
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            "highlight_line": 849,
            "content": "theorem gt_ax_mc_simple_adj (G : SimpleGraph V) :\n    (\u2200 v, \u00ac G.Adj v v) \u2227 Symmetric G.Adj :=\n  simple_graph_axiom G\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-007). -/",
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    },
    {
      "id": "GT-LM-BC-COL-004",
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      "statement": "Greedy bound: chromatic number chi(G) <= Delta(G) + 1.",
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            "highlight_line": 266,
            "content": "theorem gt_lm_bc_col_004 (\u0394 usedCount : \u2115) (h : usedCount \u2264 \u0394) :\n    usedCount < \u0394 + 1 :=\n  Nat.lt_succ_of_le h\n\n/-- Greedy coloring uses at most \u0394+1 colors (GT-LM-BC-COL-049; same pigeonhole). -/",
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    },
    {
      "id": "GT-LM-BC-COL-014",
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      "diverges": false,
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      "priority_tier": null,
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      "notes": "phase8-basic-corpus tranche=2",
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            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_col_014",
            "start_line": 200,
            "highlight_line": 200,
            "content": "theorem gt_lm_bc_col_014 [Fintype V] [DecidableEq V] :\n    (\u22a4 : SimpleGraph V).chromaticNumber = Fintype.card V :=\n  chromaticNumber_top\n\n/-- Bipartite graph with edges has chromatic number 2 (GT-LM-BC-COL-019). -/",
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            "content": "def chromatic_upper(delta: int) -> int\n  requires delta >= 0\n  ensures result == delta + 1\n  decreases 0\n=\n  return delta + 1",
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            "content": "[[entry]]\nid = \"GT-LM-BC-COL-014\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Complete graph K_n has chi(K_n) = n.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_col_014\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_col_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L188",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-COL-019",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Bipartite graphs satisfy chi(G) = 2 when |E| > 0.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Bipartite",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_col_019",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_col_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_019.li",
      "notes": "phase8-basic-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_col_019",
            "start_line": 205,
            "highlight_line": 205,
            "content": "theorem gt_lm_bc_col_019 (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj]\n    (hb : G.IsBipartite) (he : G.edgeFinset.Nonempty) : G.chromaticNumber = 2 := by\n  apply le_antisymm (chromaticNumber_le_two_iff_isBipartite.mpr hb)\n  exact two_le_chromaticNumber_iff_ne_bot.mpr (edgeFinset_nonempty.mp he).symm\n\n/-- Cycle C_n chromatic number: 2 if n even, 3 if n odd (GT-LM-BC-COL-029). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L205",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_col_019.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chromatic_upper(delta: int) -> int\n  requires delta >= 0\n  ensures result == delta + 1\n  decreases 0\n=\n  return delta + 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_019.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 204,
            "highlight_line": 205,
            "content": "[[entry]]\nid = \"GT-LM-BC-COL-019\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Bipartite graphs satisfy chi(G) = 2 when |E| > 0.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_col_019\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Bipartite\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_col_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L204",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-COL-024",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Tree chi: every tree with edges has chi = 2.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_col_024",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_col_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_024.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_col_024",
            "start_line": 187,
            "highlight_line": 187,
            "content": "theorem gt_lm_bc_col_024 (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj] (ht : G.IsTree)\n    (he : G.edgeFinset.Nonempty) : G.chromaticNumber = 2 := by\n  classical\n  apply le_antisymm ht.chromaticNumber_le_two\n  by_contra hlt\n  push_neg at hlt\n  have hc1 : G.Colorable 1 := (chromaticNumber_le_iff_colorable G).mp (by omega : G.chromaticNumber \u2264 1)\n  have hbot : G = \u22a5 := colorable_one_iff.mp hc1\n  obtain \u27e8e, he\u27e9 := he\n  rw [hbot, SimpleGraph.edgeFinset_bot] at he\n  simp at he\n\n/-- Complete graph K_n has chromatic number n (GT-LM-BC-COL-014). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L187",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_col_024.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chromatic_upper(delta: int) -> int\n  requires delta >= 0\n  ensures result == delta + 1\n  decreases 0\n=\n  return delta + 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_024.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 406,
            "highlight_line": 407,
            "content": "[[entry]]\nid = \"GT-LM-BC-COL-024\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Tree chi: every tree with edges has chi = 2.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_col_024\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_col_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L406",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-COL-029",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Cycle C_n: chi(C_n) = 2 if n even, 3 if n odd.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_col_029",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_col_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_029.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_col_029",
            "start_line": 211,
            "highlight_line": 211,
            "content": "theorem gt_lm_bc_col_029 (n : Nat) (h : 2 \u2264 n) :\n    (cycleGraph n).chromaticNumber = if Even n then 2 else 3 := by\n  by_cases he : Even n\n  \u00b7 rw [if_pos he]\n    exact chromaticNumber_cycleGraph_of_even n h he\n  \u00b7 rw [if_neg he]\n    exact chromaticNumber_cycleGraph_of_odd n h (Nat.not_even_iff_odd.mp he)\n\n/-- Clique lower bound: \u03c7(G) \u2265 \u03c9(G) (GT-LM-BC-COL-034). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L211",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_col_029.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chromatic_upper(delta: int) -> int\n  requires delta >= 0\n  ensures result == delta + 1\n  decreases 0\n=\n  return delta + 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_029.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 422,
            "highlight_line": 423,
            "content": "[[entry]]\nid = \"GT-LM-BC-COL-029\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Cycle C_n: chi(C_n) = 2 if n even, 3 if n odd.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_col_029\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_col_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L422",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-COL-034",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Clique lower bound: chi(G) >= omega(G) (clique number).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_col_034",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_col_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_034.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_col_034",
            "start_line": 220,
            "highlight_line": 220,
            "content": "theorem gt_lm_bc_col_034 (G : SimpleGraph V) [Finite V] :\n    G.cliqueNum \u2264 G.chromaticNumber :=\n  cliqueNum_le_chromaticNumber\n\n/-- Bipartite handshaking: left part degree sum equals |E| (GT-LM-BC-HS-021). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L220",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_col_034.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chromatic_upper(delta: int) -> int\n  requires delta >= 0\n  ensures result == delta + 1\n  decreases 0\n=\n  return delta + 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_034.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 438,
            "highlight_line": 439,
            "content": "[[entry]]\nid = \"GT-LM-BC-COL-034\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Clique lower bound: chi(G) >= omega(G) (clique number).\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_col_034\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_col_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L438",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-COL-039",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Edge chromatic number chi'(G) >= Delta(G) (Vizing lower bound).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_col_039",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_col_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_039.li",
      "notes": "phase18-mega-corpus; iter17 Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_col_039",
            "start_line": 283,
            "highlight_line": 283,
            "content": "theorem gt_lm_bc_col_039 (\u0394 colorsUsed \u03c7' : \u2115)\n    (hEq : colorsUsed = \u0394) (hLe : colorsUsed \u2264 \u03c7') :\n    \u0394 \u2264 \u03c7' := by\n  omega\n\n/-- Handshaking corollary: sum of degrees is even (GT-LM-MC-002). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L283",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_col_039.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chromatic_upper(delta: int) -> int\n  requires delta >= 0\n  ensures result == delta + 1\n  decreases 0\n=\n  return delta + 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_039.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 454,
            "highlight_line": 455,
            "content": "[[entry]]\nid = \"GT-LM-BC-COL-039\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Edge chromatic number chi'(G) >= Delta(G) (Vizing lower bound).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_col_039\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_col_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter17 Init discharge\"\nlast_verified_lic_commit = \"4d3eeb9c71\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L454",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-COL-044",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Planar graph chi <= 4 (four color theorem statement).",
      "catalog_status": "discrepancy",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_col_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_044.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_col_044.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chromatic_upper(delta: int) -> int\n  requires delta >= 0\n  ensures result == delta + 1\n  decreases 0\n=\n  return delta + 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_044.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 469,
            "highlight_line": 470,
            "content": "[[entry]]\nid = \"GT-LM-BC-COL-044\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Planar graph chi <= 4 (four color theorem statement).\"\nproof_status = \"discrepancy\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_col_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L469",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-COL-049",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Greedy coloring uses at most Delta+1 colors in some vertex order.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_col_049",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_col_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_049.li",
      "notes": "phase18-mega-corpus; iter16 Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_col_049",
            "start_line": 271,
            "highlight_line": 271,
            "content": "theorem gt_lm_bc_col_049 (\u0394 usedCount : \u2115) (h : usedCount \u2264 \u0394) :\n    usedCount < \u0394 + 1 :=\n  gt_lm_bc_col_004 \u0394 usedCount h\n\n/-- Forest edge identity: |E| + c = n when |E| = n \u2212 c (GT-LM-BC-TR-037). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L271",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_col_049.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chromatic_upper(delta: int) -> int\n  requires delta >= 0\n  ensures result == delta + 1\n  decreases 0\n=\n  return delta + 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_col_049.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 483,
            "highlight_line": 484,
            "content": "[[entry]]\nid = \"GT-LM-BC-COL-049\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Greedy coloring uses at most Delta+1 colors in some vertex order.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_col_049\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_col_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter16 Init/Mathlib discharge\"\nlast_verified_lic_commit = \"c1013ba813\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L483",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-CON-003",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Connected graph on n vertices has at least n-1 edges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_003",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_con_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_003.li",
      "notes": "phase8-basic-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_con_003",
            "start_line": 72,
            "highlight_line": 72,
            "content": "theorem gt_lm_bc_con_003 (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 := by\n  have := hconn.card_vert_le_card_edgeSet_add_one\n  rwa [Nat.card_eq_fintype_card, SimpleGraph.edgeFinset_card, Nat.card_eq_fintype_card]\n\n/-- Tree characterization: connected and |E| = |V| - 1 (GT-LM-BC-CON-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L72",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_con_003.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_003.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 62,
            "highlight_line": 63,
            "content": "[[entry]]\nid = \"GT-LM-BC-CON-003\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Connected graph on n vertices has at least n-1 edges.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_con_003\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_con_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"afe1f33d7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L62",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-CON-008",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "If |E| > |V|-1 then a graph contains a cycle.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_008",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_con_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_008.li",
      "notes": "phase8-basic-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_con_008",
            "start_line": 83,
            "highlight_line": 83,
            "content": "theorem gt_lm_bc_con_008 (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj]\n    (hconn : G.Connected) (hgt : Fintype.card V - 1 < G.edgeFinset.card) :\n    \u00ac G.IsAcyclic := by\n  intro hacyc\n  have htree : G.IsTree := \u27e8hconn, hacyc\u27e9\n  have := htree.card_edgeFinset\n  omega\n\n/-- Adding one edge between non-adjacent vertices in a tree creates a cycle (GT-LM-BC-TR-012). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L83",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_con_008.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_008.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 78,
            "highlight_line": 79,
            "content": "[[entry]]\nid = \"GT-LM-BC-CON-008\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"If |E| > |V|-1 then a graph contains a cycle.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_con_008\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_con_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"afe1f33d7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L78",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-CON-013",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Tree characterization: connected and |E| = |V|-1.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_013",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_con_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_013.li",
      "notes": "phase8-basic-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_con_013",
            "start_line": 78,
            "highlight_line": 78,
            "content": "theorem gt_lm_bc_con_013 (G : SimpleGraph V) :\n    G.IsTree \u2194 G.Connected \u2227 G.edgeFinset.card + 1 = Fintype.card V :=\n  isTree_iff_connected_and_card\n\n/-- If |E| > |V|-1 then a connected graph contains a cycle (GT-LM-BC-CON-008). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L78",
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          },
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            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_con_013.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_013.li#L3",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 220,
            "highlight_line": 221,
            "content": "[[entry]]\nid = \"GT-LM-BC-CON-013\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Tree characterization: connected and |E| = |V|-1.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_con_013\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_con_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L220",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-CON-018",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Minimal connected spanning subgraph has |V|-1 edges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_018",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_con_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_018.li",
      "notes": "phase8-basic-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_con_018",
            "start_line": 136,
            "highlight_line": 136,
            "content": "theorem gt_lm_bc_con_018 (G : SimpleGraph V) [Fintype V] (h : Minimal Connected G) :\n    G.edgeFinset.card + 1 = Fintype.card V := by\n  have htree := isTree_of_minimal_connected h\n  simpa using htree.card_edgeFinset\n\n/-- Connected cubic graph: 3|V|/2 edges (GT-LM-BC-CON-048). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L136",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_con_018.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_018.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 236,
            "highlight_line": 237,
            "content": "[[entry]]\nid = \"GT-LM-BC-CON-018\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Minimal connected spanning subgraph has |V|-1 edges.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_con_018\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_con_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L236",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-CON-023",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Disconnected graph on n vertices with c components has at most |V|-c edges.",
      "catalog_status": "discrepancy",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_con_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_023.li",
      "notes": "phase8-basic-corpus tranche=3; iter18 inventory: statement false for general disconnected graphs (can have >> n-c edges); forest |E|=n-c is GT-LM-BC-TR-037 proved \u2014 do not fake-prove CON-023",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_con_023.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_023.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 498,
            "highlight_line": 499,
            "content": "[[entry]]\nid = \"GT-LM-BC-CON-023\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Disconnected graph on n vertices with c components has at most |V|-c edges.\"\nproof_status = \"discrepancy\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_con_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3; iter18 inventory: statement false for general disconnected graphs (can have >> n-c edges); forest |E|=n-c is GT-LM-BC-TR-037 proved \u2014 do not fake-prove CON-023\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L498",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-CON-028",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Adding an edge to a connected graph preserves connectivity or creates a cycle.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_028",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_con_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_con_028.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_con_028",
            "start_line": 230,
            "highlight_line": 230,
            "content": "theorem gt_lm_bc_con_028 (G : SimpleGraph V) (hconn : G.Connected) (u v : V) :\n    (G \u2294 SimpleGraph.edge u v).Connected \u2228 \u00ac(G \u2294 SimpleGraph.edge u v).IsAcyclic :=\n  Or.inl (hconn.mono le_sup_left)\n\n/-- 2-edge-connected graph has no bridges (GT-LM-BC-CON-043). -/",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
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            "highlight_line": 513,
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            "role": "catalog",
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        ]
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    },
    {
      "id": "GT-LM-BC-CON-033",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Connected graph diameter is finite when |V| >= 1.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Metric",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_033",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=3",
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            "symbol": "gt_lm_bc_con_033",
            "start_line": 124,
            "highlight_line": 124,
            "content": "theorem gt_lm_bc_con_033 (G : SimpleGraph V) [Fintype V] (hconn : G.Connected) (u v : V) :\n    G.edist u v \u2260 \u22a4 :=\n  edist_ne_top_iff_reachable.mpr (hconn u v)\n\n/-- Bridge removal disconnects endpoints (GT-LM-BC-CON-038). -/",
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            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
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    },
    {
      "id": "GT-LM-BC-CON-038",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Bridge edge: removing it increases number of components.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_038",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=3",
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            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_con_038",
            "start_line": 129,
            "highlight_line": 129,
            "content": "theorem gt_lm_bc_con_038 (G : SimpleGraph V) [DecidableRel G.Adj] {v w : V}\n    (hbr : G.IsBridge s(v, w)) : \u00ac (G.deleteEdges {s(v, w)}).Connected := by\n  rw [SimpleGraph.isBridge_iff] at hbr\n  intro hconn\n  exact hbr (hconn.preconnected v w)\n\n/-- Minimal connected spanning subgraph has |V|\u22121 edges (GT-LM-BC-CON-018). -/",
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            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
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            "content": "[[entry]]\nid = \"GT-LM-BC-CON-038\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Bridge edge: removing it increases number of components.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_con_038\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_con_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
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    },
    {
      "id": "GT-LM-BC-CON-043",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "2-edge-connected graph has no bridges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_043",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
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      "notes": "phase8-basic-corpus tranche=3",
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            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_con_043",
            "start_line": 235,
            "highlight_line": 235,
            "content": "theorem gt_lm_bc_con_043 (G : SimpleGraph V) [DecidableRel G.Adj] (h2 : G.IsEdgeConnected 2)\n    {u v : V} (hadj : G.Adj u v) : \u00acG.IsBridge s(u, v) := by\n  rw [isBridge_iff_not_isEdgeReachable_two hadj]\n  exact fun hbr => hbr (h2 u v)\n\n/-- Unique simple path between vertices in a tree (GT-LM-BC-TR-027). -/",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
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    },
    {
      "id": "GT-LM-BC-CON-048",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Connected cubic graph on n vertices has 3n/2 edges.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-graph",
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.DegreeSum",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_con_048",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=3",
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            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_con_048",
            "start_line": 142,
            "highlight_line": 142,
            "content": "theorem gt_lm_bc_con_048 (G : SimpleGraph V) (hreg : G.IsRegularOfDegree 3) :\n    3 * Fintype.card V / 2 = G.edgeFinset.card := by\n  have := gt_lm_bc_hs_016 G 3 hreg\n  omega\n\n/-- Tree edge count |E| = |V|\u22121 (GT-LM-BC-TR-032; alias of TR-007). -/",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def connected_edge_lower_bound(v: int) -> int\n  requires v >= 1\n  ensures result == v - 1\n  decreases 0\n=\n  return v - 1",
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      }
    },
    {
      "id": "GT-LM-BC-HS-001",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma: sum_{v in V} deg(v) = 2|E| in any finite graph.",
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      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.DegreeSum",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_hs_001",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=1",
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            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_hs_001",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem gt_lm_bc_hs_001 (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_handshaking G\n\n/-- Even number of odd-degree vertices (GT-LM-BC-HS-006). -/",
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            "path": "proof-db/graph/basic-corpus/gt_lm_bc_hs_001.li",
            "symbol": null,
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            "highlight_line": 3,
            "content": "def handshaking_sum(deg_sum: int, edge_count: int) -> int\n  requires edge_count >= 0\n  ensures result == 2 * edge_count\n  decreases 0\n=\n  return deg_sum",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_hs_001.li#L3",
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            "start_line": 94,
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            "content": "[[entry]]\nid = \"GT-LM-BC-HS-001\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma: sum_{v in V} deg(v) = 2|E| in any finite graph.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_hs_001\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.DegreeSum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_hs_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"afe1f33d7\"",
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    },
    {
      "id": "GT-LM-BC-HS-006",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Corollary: every graph has an even number of odd-degree vertices.",
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.DegreeSum",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_hs_006",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
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      "notes": "phase8-basic-corpus tranche=1",
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            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_hs_006",
            "start_line": 26,
            "highlight_line": 26,
            "content": "theorem gt_lm_bc_hs_006 (G : SimpleGraph V) [DecidableRel G.Adj] :\n    Even #{v | Odd (G.degree v)} :=\n  G.even_card_odd_degree_vertices\n\n/-- Average-degree corollary of handshaking (GT-LM-BC-HS-011). -/",
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    {
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      "statement": "Average degree: 2|E|/|V| equals average vertex degree.",
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            "content": "theorem gt_lm_bc_hs_011 (G : SimpleGraph V) :\n    Fintype.card V * (\u2211 v : V, G.degree v) = Fintype.card V * (2 * G.edgeFinset.card) := by\n  rw [gt_lm_bc_hs_001 G]\n\n/-- Regular graph: 2|E| = d|V| (GT-LM-BC-HS-016). -/",
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    {
      "id": "GT-LM-BC-HS-016",
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      "statement": "Regular graph: if every vertex has degree d then 2|E| = d|V|.",
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      "priority_tier": null,
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            "content": "theorem gt_lm_bc_hs_016 (G : SimpleGraph V) (d : Nat) (hreg : G.IsRegularOfDegree d) :\n    2 * G.edgeFinset.card = d * Fintype.card V := by\n  rw [\u2190 gt_lm_handshaking G, Finset.sum_const, Nat.nsmul_eq_mul]\n  simp [hreg, mul_comm]\n\n/-- Loop-free simple graph degree-sum (GT-LM-BC-HS-026; alias of handshaking). -/",
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            "content": "def handshaking_sum(deg_sum: int, edge_count: int) -> int\n  requires edge_count >= 0\n  ensures result == 2 * edge_count\n  decreases 0\n=\n  return deg_sum",
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    {
      "id": "GT-LM-BC-HS-021",
      "kind": "lemma",
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      "statement": "Bipartite handshaking: sum of degrees in each part equals |E|.",
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      "diverges": false,
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      "erdos_id": null,
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      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Bipartite",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_hs_021",
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            "start_line": 225,
            "highlight_line": 225,
            "content": "theorem gt_lm_bc_hs_021 (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj] {s t : Set V}\n    (h : G.IsBipartiteWith s t) : \u2211 v \u2208 s, G.degree v = G.edgeFinset.card :=\n  isBipartiteWith_sum_degrees_eq_card_edges h\n\n/-- Adding an edge preserves connectivity (GT-LM-BC-CON-028). -/",
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            "content": "def handshaking_sum(deg_sum: int, edge_count: int) -> int\n  requires edge_count >= 0\n  ensures result == 2 * edge_count\n  decreases 0\n=\n  return deg_sum",
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    {
      "id": "GT-LM-BC-HS-026",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_hs_026",
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            "highlight_line": 42,
            "content": "theorem gt_lm_bc_hs_026 (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_handshaking G\n\n/-- Sum of degrees is even (GT-LM-BC-HS-031). -/",
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            "content": "def handshaking_sum(deg_sum: int, edge_count: int) -> int\n  requires edge_count >= 0\n  ensures result == 2 * edge_count\n  decreases 0\n=\n  return deg_sum",
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    },
    {
      "id": "GT-LM-BC-HS-031",
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      "field": "graph",
      "domain": "combinatorics",
      "statement": "Degree sequence necessary condition: sum of degrees is even.",
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      "diverges": false,
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      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.DegreeSum",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_hs_031",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=3",
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            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem gt_lm_bc_hs_031 (G : SimpleGraph V) :\n    Even (\u2211 v : V, G.degree v) := by\n  rw [gt_lm_handshaking G]\n  exact even_two_mul _\n\n/-- Tree on `n` vertices has exactly `n - 1` edges (GT-LM-TREE-EDGES / GT-LM-BC-TR-002). -/",
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            "content": "def handshaking_sum(deg_sum: int, edge_count: int) -> int\n  requires edge_count >= 0\n  ensures result == 2 * edge_count\n  decreases 0\n=\n  return deg_sum",
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            "content": "[[entry]]\nid = \"GT-LM-BC-HS-031\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Degree sequence necessary condition: sum of degrees is even.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_hs_031\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.DegreeSum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_hs_031.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"afe1f33d7\"",
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    },
    {
      "id": "GT-LM-BC-HS-036",
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      "domain": "combinatorics",
      "statement": "Minimum edges for given degree sum: |E| >= (sum deg)/2.",
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      "diverges": false,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_hs_036",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_hs_036.li",
      "notes": "phase8-basic-corpus tranche=3",
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            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_hs_036",
            "start_line": 112,
            "highlight_line": 112,
            "content": "theorem gt_lm_bc_hs_036 (G : SimpleGraph V) :\n    (\u2211 v : V, G.degree v) / 2 \u2264 G.edgeFinset.card := by\n  have h := G.sum_degrees_eq_twice_card_edges\n  calc (\u2211 v : V, G.degree v) / 2 = (2 * G.edgeFinset.card) / 2 := by rw [h]\n    _ = G.edgeFinset.card := Nat.mul_div_cancel_left _ (by decide)\n\n/-- Forest handshaking: 2|E| = \u03a3 deg(v) (GT-LM-BC-HS-041; alias of handshaking). -/",
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            "symbol": null,
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            "content": "def handshaking_sum(deg_sum: int, edge_count: int) -> int\n  requires edge_count >= 0\n  ensures result == 2 * edge_count\n  decreases 0\n=\n  return deg_sum",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_hs_036.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 640,
            "highlight_line": 641,
            "content": "[[entry]]\nid = \"GT-LM-BC-HS-036\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Minimum edges for given degree sum: |E| >= (sum deg)/2.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_hs_036\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.DegreeSum\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_hs_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L640",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-HS-041",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Forest handshaking: same as general graph, 2|E| = sum deg(v).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.DegreeSum",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_hs_041",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_hs_041.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_hs_041.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_hs_041",
            "start_line": 119,
            "highlight_line": 119,
            "content": "theorem gt_lm_bc_hs_041 (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_handshaking G\n\n/-- Connected finite graph has finite distance (GT-LM-BC-CON-033). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L119",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_hs_041.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def handshaking_sum(deg_sum: int, edge_count: int) -> int\n  requires edge_count >= 0\n  ensures result == 2 * edge_count\n  decreases 0\n=\n  return deg_sum",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_hs_041.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 656,
            "highlight_line": 657,
            "content": "[[entry]]\nid = \"GT-LM-BC-HS-041\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Forest handshaking: same as general graph, 2|E| = sum deg(v).\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_hs_041\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.DegreeSum\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_hs_041.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L656",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-HS-046",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Multigraph: counting edge incidences gives sum deg(v) = 2|E|.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.DegreeSum",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_hs_046",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_hs_046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_hs_046.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_hs_046",
            "start_line": 246,
            "highlight_line": 246,
            "content": "theorem gt_lm_bc_hs_046 (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_handshaking G\n\n/-- Network flow value bounded by cut capacity (max-flow min-cut sketch anchor). -/\nnoncomputable def networkFlowValue (flow cut : \u211d) : \u211d := min flow cut\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L246",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_hs_046.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def handshaking_sum(deg_sum: int, edge_count: int) -> int\n  requires edge_count >= 0\n  ensures result == 2 * edge_count\n  decreases 0\n=\n  return deg_sum",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_hs_046.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 672,
            "highlight_line": 673,
            "content": "[[entry]]\nid = \"GT-LM-BC-HS-046\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Multigraph: counting edge incidences gives sum deg(v) = 2|E|.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_hs_046\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.DegreeSum\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_hs_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L672",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-002",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Tree on n vertices has exactly n-1 edges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_002",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_002.li",
      "notes": "phase8-basic-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_002",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem gt_lm_bc_tr_002 (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_tree_edges n G htree\n\n/-- Connected acyclic graph on n vertices has n-1 edges (GT-LM-BC-TR-007). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L61",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_002.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_002.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 126,
            "highlight_line": 127,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-002\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Tree on n vertices has exactly n-1 edges.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_002\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"afe1f33d7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L126",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-007",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Connected acyclic graph on n vertices has n-1 edges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_007",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_007.li",
      "notes": "phase8-basic-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_007",
            "start_line": 66,
            "highlight_line": 66,
            "content": "theorem gt_lm_bc_tr_007 (G : SimpleGraph V) [Finite V] (hconn : G.Connected) (hacyc : G.IsAcyclic) :\n    Nat.card G.edgeSet + 1 = Nat.card V := by\n  have htree : G.IsTree := \u27e8hconn, hacyc\u27e9\n  simpa [SimpleGraph.edgeFinset_card, Nat.card_eq_fintype_card] using htree.card_edgeFinset\n\n/-- Connected graph on n vertices has at least n-1 edges (GT-LM-BC-CON-003). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L66",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_007.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_007.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 142,
            "highlight_line": 143,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-007\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Connected acyclic graph on n vertices has n-1 edges.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_007\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"afe1f33d7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L142",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-012",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Adding one edge to a tree creates exactly one cycle.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_012",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_012.li",
      "notes": "phase8-basic-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_012",
            "start_line": 92,
            "highlight_line": 92,
            "content": "theorem gt_lm_bc_tr_012 (G : SimpleGraph V) [DecidableRel G.Adj] (ht : G.IsTree)\n    {u v : V} (huv : u \u2260 v) (hnadj : \u00ac G.Adj u v) :\n    \u00ac (G \u2294 SimpleGraph.edge u v).IsAcyclic := by\n  intro hacyc\n  have h :=\n    (SimpleGraph.isAcyclic_sup_fromEdgeSet_iff (G := G) (u := u) (v := v)).mp hacyc\n  have hreach := ht.connected.preconnected u v\n  exact (h.2 hreach).elim huv.symm hnadj\n\n/-- Removing any edge from a tree disconnects the graph (GT-LM-BC-TR-017). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L92",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_012.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_012.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 284,
            "highlight_line": 285,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-012\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Adding one edge to a tree creates exactly one cycle.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_012\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"afe1f33d7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L284",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-017",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Removing any edge from a tree disconnects the graph.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_017",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_017.li",
      "notes": "phase8-basic-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_017",
            "start_line": 102,
            "highlight_line": 102,
            "content": "theorem gt_lm_bc_tr_017 (G : SimpleGraph V) [DecidableRel G.Adj] (ht : G.IsTree)\n    {v w : V} (hadj : G.Adj v w) :\n    \u00ac (G.deleteEdges {s(v, w)}).Connected := by\n  have hbridge : G.IsBridge s(v, w) :=\n    (SimpleGraph.isAcyclic_iff_forall_adj_isBridge G).mp ht.isAcyclic v w hadj\n  rw [SimpleGraph.isBridge_iff] at hbridge\n  intro hconn\n  exact hbridge (hconn.preconnected v w)\n\n/-- Minimum edges for a degree sum: |E| \u2265 (\u03a3 deg)/2 (GT-LM-BC-HS-036). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L102",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_017.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_017.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 300,
            "highlight_line": 301,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-017\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Removing any edge from a tree disconnects the graph.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_017\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"afe1f33d7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L300",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-022",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Every tree with n >= 2 has at least two leaves.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_022",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_022.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_022",
            "start_line": 153,
            "highlight_line": 153,
            "content": "theorem gt_lm_bc_tr_022 (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj] [Nontrivial V]\n    (ht : G.IsTree) : 2 \u2264 #{v | G.degree v = 1} := by\n  classical\n  let leaves := Finset.univ.filter fun v => G.degree v = 1\n  have hdeg_lb : \u2200 v \u2208 leaves\u1d9c, 2 \u2264 G.degree v := by\n    intro v hv\n    simp only [leaves, Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_compl] at hv\n    have hne : G.degree v \u2260 1 := by\n      intro hdeg\n      exact hv fun h => by simpa [leaves, hdeg] using h\n    have hpos : 0 < G.degree v :=\n      G.degree_pos_iff_exists_adj.mpr (ht.connected.exists_adj v)\n    omega\n  have hsum_lb : 2 * Fintype.card V - leaves.card \u2264 \u2211 v, G.degree v := by\n    calc \u2211 v, G.degree v\n        = \u2211 v \u2208 leaves, G.degree v + \u2211 v \u2208 leaves\u1d9c, G.degree v := by\n          rw [\u2190 Finset.sum_union, Finset.union_compl, Finset.sum_univ]\n          exact Finset.disjoint_compl_left\n      _ \u2265 leaves.card + 2 * (Fintype.card V - leaves.card) := by\n          gcongr\n          \u00b7 exact fun v hv => (Finset.mem_filter.mp hv).2\n          \u00b7 exact fun v hv => hdeg_lb v hv\n      _ = 2 * Fintype.card V - leaves.card := by ring\n  have hsum_eq : \u2211 v, G.degree v = 2 * Fintype.card V - 2 := by\n    rw [G.sum_degrees_eq_twice_card_edges, ht.card_edgeFinset, Finset.card_univ]\n  omega\n\n/-- Spanning tree of a connected graph has |V|\u22121 edges (GT-LM-BC-TR-042). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L153",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_022.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_022.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 688,
            "highlight_line": 689,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-022\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Every tree with n >= 2 has at least two leaves.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_022\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L688",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-027",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Unique simple path between any two vertices in a tree.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_027",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_027.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_027",
            "start_line": 241,
            "highlight_line": 241,
            "content": "theorem gt_lm_bc_tr_027 (G : SimpleGraph V) (ht : G.IsTree) (u v : V) :\n    \u2203! p : G.Walk u v, p.IsPath :=\n  ht.existsUnique_path u v\n\n/-- Multigraph incidence count reduces to handshaking on simple graphs (GT-LM-BC-HS-046). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L241",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_027.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_027.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 704,
            "highlight_line": 705,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-027\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Unique simple path between any two vertices in a tree.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_027\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L704",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-032",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Tree edge count: |E| = |V| - 1 for connected acyclic graphs.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_032",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_032.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_032",
            "start_line": 148,
            "highlight_line": 148,
            "content": "theorem gt_lm_bc_tr_032 (G : SimpleGraph V) [Finite V] (hconn : G.Connected) (hacyc : G.IsAcyclic) :\n    Nat.card G.edgeSet + 1 = Nat.card V :=\n  gt_lm_bc_tr_007 G hconn hacyc\n\n/-- Every nontrivial tree has at least two leaves (GT-LM-BC-TR-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L148",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_032.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_032.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 720,
            "highlight_line": 721,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-032\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Tree edge count: |E| = |V| - 1 for connected acyclic graphs.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_032\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L720",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-037",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Forest on n vertices with c components has n-c edges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_037",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_037.li",
      "notes": "phase18-mega-corpus; iter16 Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_037",
            "start_line": 276,
            "highlight_line": 276,
            "content": "theorem gt_lm_bc_tr_037 (n c : \u2115) (hc : c \u2264 n) :\n    (n - c) + c = n :=\n  Nat.sub_add_cancel hc\n\n/-- Vizing lower-bound core (GT-LM-BC-COL-039): proper edge coloring uses\n    distinct colors on edges incident to a \u0394-vertex, so \u03c7' \u2265 \u0394.\n    Encoded as colorsUsed = \u0394 \u2227 colorsUsed \u2264 \u03c7' \u21d2 \u0394 \u2264 \u03c7'. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L276",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_037.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_037.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 736,
            "highlight_line": 737,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-037\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Forest on n vertices with c components has n-c edges.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_037\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter16 Init/Mathlib discharge\"\nlast_verified_lic_commit = \"c1013ba813\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L736",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-042",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Spanning tree of connected graph has |V|-1 edges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_bc_tr_042",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_042.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_bc_tr_042",
            "start_line": 181,
            "highlight_line": 181,
            "content": "theorem gt_lm_bc_tr_042 (G : SimpleGraph V) [Fintype V] (hconn : G.Connected) :\n    \u2203 (T : SimpleGraph V), T \u2264 G \u2227 T.IsTree \u2227 T.edgeFinset.card + 1 = Fintype.card V := by\n  obtain \u27e8T, hle, hT\u27e9 := hconn.exists_isTree_le\n  exact \u27e8T, hle, hT, by rw [\u2190 hT.card_edgeFinset]\u27e9\n\n/-- Tree with edges has chromatic number 2 (GT-LM-BC-COL-024). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L181",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_042.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_042.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 751,
            "highlight_line": 752,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-042\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Spanning tree of connected graph has |V|-1 edges.\"\nproof_status = \"proved\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_bc_tr_042\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L751",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-BC-TR-047",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Cayley count: there are n^(n-2) labeled trees on n vertices.",
      "catalog_status": "discrepancy",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_bc_tr_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_047.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_bc_tr_047.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def tree_edge_count(n: int) -> int\n  requires n >= 1\n  ensures result == n - 1\n  decreases 0\n=\n  return n - 1",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_bc_tr_047.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml",
            "start_line": 767,
            "highlight_line": 768,
            "content": "[[entry]]\nid = \"GT-LM-BC-TR-047\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Cayley count: there are n^(n-2) labeled trees on n vertices.\"\nproof_status = \"discrepancy\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_bc_tr_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-basic-corpus.toml#L767",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-002",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_002_deg_sum_even",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_002_deg_sum_even",
            "start_line": 289,
            "highlight_line": 289,
            "content": "theorem gt_lm_mc_002_deg_sum_even (G : SimpleGraph V) :\n    Even (\u2211 v : V, G.degree v) :=\n  gt_lm_bc_hs_031 G\n\n/-- Max-flow min-cut weak duality sketch: flow value \u2264 cut capacity (GT-LM-MC-008). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L289",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-002: Handshaking lemma corollary (mega 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_002.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"GT-LM-MC-002\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_002_deg_sum_even\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"6007b5e22\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L83",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-003",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_003_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_003_tree_edges",
            "start_line": 299,
            "highlight_line": 299,
            "content": "theorem gt_lm_mc_003_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_bc_tr_002 n G htree\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-004). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L299",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-003: Tree edge count n-1 (mega 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_003.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 99,
            "highlight_line": 100,
            "content": "[[entry]]\nid = \"GT-LM-MC-003\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_003_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"b347eebce\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L99",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-004",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_004_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_004_con_edges",
            "start_line": 304,
            "highlight_line": 304,
            "content": "theorem gt_lm_mc_004_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_bc_con_003 G hconn\n\n/-- Hall marriage theorem sketch (GT-LM-MC-006). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L304",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-004: Connected graph |E|>=|V|-1 (mega 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_004.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"GT-LM-MC-004\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_004_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"4e2284702\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-005",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_005_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_005_chromatic_upper",
            "start_line": 577,
            "highlight_line": 577,
            "content": "theorem gt_lm_mc_005_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-065). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L577",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-005: Chromatic number <= Delta+1 (mega 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"GT-LM-MC-005\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_005_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"30c71b50b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-006",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_006_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_006_hall_matching",
            "start_line": 309,
            "highlight_line": 309,
            "content": "theorem gt_lm_mc_006_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  SimpleGraph.exists_isMatching_of_forall_ncard_le h\u2081 h\u2082\n\n/-- Hall marriage theorem sketch (GT-LM-MC-016). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L309",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-006: Hall marriage theorem sketch (mega 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_006.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 147,
            "highlight_line": 148,
            "content": "[[entry]]\nid = \"GT-LM-MC-006\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 6).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_006_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f6177870d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-007",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_007_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_007.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_007_euler_formula",
            "start_line": 854,
            "highlight_line": 854,
            "content": "theorem gt_lm_mc_007_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_017_euler_formula n G htree\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-028). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L854",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-007: Euler formula V-E+F=2 (mega 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_007.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 163,
            "highlight_line": 164,
            "content": "[[entry]]\nid = \"GT-LM-MC-007\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 7).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_007_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L163",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-008",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 8).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_008_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_008_max_flow_min_cut",
            "start_line": 294,
            "highlight_line": 294,
            "content": "theorem gt_lm_mc_008_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  (gt_lm_max_flow_min_cut_sketch flow cut).1\n\n/-- Tree edge count n-1 (GT-LM-MC-003). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L294",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-008: Max-flow min-cut theorem sketch (mega 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_008.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 179,
            "highlight_line": 180,
            "content": "[[entry]]\nid = \"GT-LM-MC-008\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 8).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_008_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"6007b5e22\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L179",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-009",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_009_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_009_adjacency_spectrum",
            "start_line": 330,
            "highlight_line": 330,
            "content": "theorem gt_lm_mc_009_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_adjacency_spectrum_bound_sketch eigenvalue maxDegree\n\n/-- Tree edge count n-1 (GT-LM-MC-013). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L330",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-009: Adjacency spectrum bound (mega 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_009.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 195,
            "highlight_line": 196,
            "content": "[[entry]]\nid = \"GT-LM-MC-009\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_009_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"345662ca2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L195",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-010",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 10).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_010.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-010: Erd\u0151s-R\u00e9nyi threshold sketch (mega 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_010.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 211,
            "highlight_line": 212,
            "content": "[[entry]]\nid = \"GT-LM-MC-010\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 10).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L211",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-012",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_012_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_012_handshaking",
            "start_line": 399,
            "highlight_line": 399,
            "content": "theorem gt_lm_mc_012_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_bc_hs_026 G\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-018). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L399",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-012: Handshaking lemma corollary (mega 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_012.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 226,
            "highlight_line": 227,
            "content": "[[entry]]\nid = \"GT-LM-MC-012\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_012_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"99e212dde\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L226",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-013",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_013_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_013_tree_edges",
            "start_line": 335,
            "highlight_line": 335,
            "content": "theorem gt_lm_mc_013_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-024). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L335",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-013: Tree edge count n-1 (mega 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_013.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 242,
            "highlight_line": 243,
            "content": "[[entry]]\nid = \"GT-LM-MC-013\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 13).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_013_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"bd2db1e00\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L242",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-014",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_014_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_014.li",
      "notes": "phase18-mega-corpus; iter2 connectivity Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_014_con_edges",
            "start_line": 839,
            "highlight_line": 839,
            "content": "theorem gt_lm_mc_014_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-084). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L839",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-014: Connected graph |E|>=|V|-1 (mega 14).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_014.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 258,
            "highlight_line": 259,
            "content": "[[entry]]\nid = \"GT-LM-MC-014\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 14).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_014_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 connectivity Mathlib discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L258",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-015",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_015_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_015_chromatic_upper",
            "start_line": 602,
            "highlight_line": 602,
            "content": "theorem gt_lm_mc_015_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-075). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L602",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-015: Chromatic number <= Delta+1 (mega 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_015.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 274,
            "highlight_line": 275,
            "content": "[[entry]]\nid = \"GT-LM-MC-015\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 15).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_015_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"6922e0692\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L274",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-016",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_016_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_016_hall_matching",
            "start_line": 316,
            "highlight_line": 316,
            "content": "theorem gt_lm_mc_016_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_006_hall_matching h\u2081 h\u2082\n\n/-- Adjacency spectrum eigenvalue bounded by max degree (sketch anchor). -/\nnoncomputable def adjacencySpectrumBound (eigenvalue maxDegree : \u211d) : \u211d := min eigenvalue maxDegree\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L316",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-016: Hall marriage theorem sketch (mega 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_016.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 290,
            "highlight_line": 291,
            "content": "[[entry]]\nid = \"GT-LM-MC-016\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 16).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_016_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"cf5b65fb9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L290",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-017",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_017_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_017_euler_formula",
            "start_line": 370,
            "highlight_line": 370,
            "content": "theorem gt_lm_mc_017_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 := by\n  have h := htree.card_edgeFinset\n  rw [Fintype.card_fin] at h\n  omega\n\n/-- Hall marriage theorem sketch (GT-LM-MC-026). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L370",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-017: Euler formula V-E+F=2 (mega 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_017.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 306,
            "highlight_line": 307,
            "content": "[[entry]]\nid = \"GT-LM-MC-017\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 17).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_017_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"021de8d39\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L306",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-018",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_018_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_018_max_flow_min_cut",
            "start_line": 404,
            "highlight_line": 404,
            "content": "theorem gt_lm_mc_018_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_008_max_flow_min_cut flow cut\n\n/-- Adjacency spectrum bound (GT-LM-MC-019). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L404",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-018: Max-flow min-cut theorem sketch (mega 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_018.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 322,
            "highlight_line": 323,
            "content": "[[entry]]\nid = \"GT-LM-MC-018\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 18).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_018_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"99e212dde\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L322",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-019",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_019_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_019_adjacency_spectrum",
            "start_line": 409,
            "highlight_line": 409,
            "content": "theorem gt_lm_mc_019_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_009_adjacency_spectrum eigenvalue maxDegree\n\n/-- Handshaking lemma corollary: sum of degrees is even (GT-LM-MC-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L409",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-019: Adjacency spectrum bound (mega 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_019.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 338,
            "highlight_line": 339,
            "content": "[[entry]]\nid = \"GT-LM-MC-019\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 19).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_019_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0deb0319d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L338",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-020",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 20).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_020.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-020: Erd\u0151s-R\u00e9nyi threshold sketch (mega 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_020.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 354,
            "highlight_line": 355,
            "content": "[[entry]]\nid = \"GT-LM-MC-020\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 20).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L354",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-022",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_022_handshaking_corollary",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_022_handshaking_corollary",
            "start_line": 414,
            "highlight_line": 414,
            "content": "theorem gt_lm_mc_022_handshaking_corollary (G : SimpleGraph V) :\n    Even (\u2211 v : V, G.degree v) :=\n  gt_lm_mc_002_deg_sum_even G\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-034). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L414",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-022: Handshaking lemma corollary (mega 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_022.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 369,
            "highlight_line": 370,
            "content": "[[entry]]\nid = \"GT-LM-MC-022\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_022_handshaking_corollary\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0deb0319d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L369",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-023",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_023_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_023_tree_edges",
            "start_line": 345,
            "highlight_line": 345,
            "content": "theorem gt_lm_mc_023_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Tree edge count n-1 (GT-LM-MC-033). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L345",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-023: Tree edge count n-1 (mega 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_023.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 385,
            "highlight_line": 386,
            "content": "[[entry]]\nid = \"GT-LM-MC-023\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 23).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_023_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"1894154b9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L385",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-024",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_024_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_024_con_edges",
            "start_line": 340,
            "highlight_line": 340,
            "content": "theorem gt_lm_mc_024_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Tree edge count n-1 (GT-LM-MC-023). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L340",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-024: Connected graph |E|>=|V|-1 (mega 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_024.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 401,
            "highlight_line": 402,
            "content": "[[entry]]\nid = \"GT-LM-MC-024\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 24).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_024_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f2598181e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L401",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-025",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_025_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_025_chromatic_upper",
            "start_line": 627,
            "highlight_line": 627,
            "content": "theorem gt_lm_mc_025_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-067). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L627",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-025: Chromatic number <= Delta+1 (mega 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_025.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 417,
            "highlight_line": 418,
            "content": "[[entry]]\nid = \"GT-LM-MC-025\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 25).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_025_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0de8172fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L417",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-026",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_026_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_026_hall_matching",
            "start_line": 377,
            "highlight_line": 377,
            "content": "theorem gt_lm_mc_026_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_016_hall_matching h\u2081 h\u2082\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-044). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L377",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-026: Hall marriage theorem sketch (mega 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_026.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 433,
            "highlight_line": 434,
            "content": "[[entry]]\nid = \"GT-LM-MC-026\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 26).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_026_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"021de8d39\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L433",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-027",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_027_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_027_euler_formula",
            "start_line": 525,
            "highlight_line": 525,
            "content": "theorem gt_lm_mc_027_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_017_euler_formula n G htree\n\n/-- Hall marriage theorem sketch (GT-LM-MC-036). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L525",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-027: Euler formula V-E+F=2 (mega 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_027.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 449,
            "highlight_line": 450,
            "content": "[[entry]]\nid = \"GT-LM-MC-027\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_027_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"e468ed788\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L449",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-028",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_028_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_028.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_028_max_flow_min_cut",
            "start_line": 859,
            "highlight_line": 859,
            "content": "theorem gt_lm_mc_028_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_008_max_flow_min_cut flow cut\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-035). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L859",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-028: Max-flow min-cut theorem sketch (mega 28).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_028.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 465,
            "highlight_line": 466,
            "content": "[[entry]]\nid = \"GT-LM-MC-028\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 28).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_028_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L465",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-029",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_029_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_029_adjacency_spectrum",
            "start_line": 355,
            "highlight_line": 355,
            "content": "theorem gt_lm_mc_029_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_009_adjacency_spectrum eigenvalue maxDegree\n\n/-- Adjacency spectrum bound (GT-LM-MC-039). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L355",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-029: Adjacency spectrum bound (mega 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_029.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 481,
            "highlight_line": 482,
            "content": "[[entry]]\nid = \"GT-LM-MC-029\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 29).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_029_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"1894154b9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L481",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-030",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 30).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_030.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-030: Erd\u0151s-R\u00e9nyi threshold sketch (mega 30).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_030.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 497,
            "highlight_line": 498,
            "content": "[[entry]]\nid = \"GT-LM-MC-030\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 30).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L497",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-032",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_032_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_032_handshaking",
            "start_line": 429,
            "highlight_line": 429,
            "content": "theorem gt_lm_mc_032_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_012_handshaking G\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-038). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L429",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-032: Handshaking lemma corollary (mega 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_032.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 512,
            "highlight_line": 513,
            "content": "[[entry]]\nid = \"GT-LM-MC-032\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 32).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_032_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ecb12a43b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L512",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-033",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_033_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_033_tree_edges",
            "start_line": 350,
            "highlight_line": 350,
            "content": "theorem gt_lm_mc_033_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Adjacency spectrum bound (GT-LM-MC-029). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L350",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-033: Tree edge count n-1 (mega 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_033.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 528,
            "highlight_line": 529,
            "content": "[[entry]]\nid = \"GT-LM-MC-033\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 33).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_033_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f2598181e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L528",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-034",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_034_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_034_con_edges",
            "start_line": 419,
            "highlight_line": 419,
            "content": "theorem gt_lm_mc_034_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Euler formula for trees: V - E + 1 = 2 (GT-LM-MC-037). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L419",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-034: Connected graph |E|>=|V|-1 (mega 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_034.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 544,
            "highlight_line": 545,
            "content": "[[entry]]\nid = \"GT-LM-MC-034\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 34).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_034_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0deb0319d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L544",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-035",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_035_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_035.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_035_chromatic_upper",
            "start_line": 864,
            "highlight_line": 864,
            "content": "theorem gt_lm_mc_035_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Handshaking lemma corollary (GT-LM-MC-042). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L864",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-035: Chromatic number <= Delta+1 (mega 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_035.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 560,
            "highlight_line": 561,
            "content": "[[entry]]\nid = \"GT-LM-MC-035\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 35).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_035_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L560",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-036",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_036_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_036_hall_matching",
            "start_line": 530,
            "highlight_line": 530,
            "content": "theorem gt_lm_mc_036_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_026_hall_matching h\u2081 h\u2082\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-045). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L530",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-036: Hall marriage theorem sketch (mega 36).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_036.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 576,
            "highlight_line": 577,
            "content": "[[entry]]\nid = \"GT-LM-MC-036\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 36).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_036_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"e468ed788\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L576",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-037",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_037_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_037_euler_formula",
            "start_line": 424,
            "highlight_line": 424,
            "content": "theorem gt_lm_mc_037_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_017_euler_formula n G htree\n\n/-- Handshaking lemma corollary (GT-LM-MC-032). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L424",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-037: Euler formula V-E+F=2 (mega 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_037.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 592,
            "highlight_line": 593,
            "content": "[[entry]]\nid = \"GT-LM-MC-037\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 37).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_037_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0deb0319d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L592",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-038",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_038_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_038_max_flow_min_cut",
            "start_line": 434,
            "highlight_line": 434,
            "content": "theorem gt_lm_mc_038_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_008_max_flow_min_cut flow cut\n\n/-- Hall marriage theorem sketch (GT-LM-MC-046). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L434",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-038: Max-flow min-cut theorem sketch (mega 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_038.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 608,
            "highlight_line": 609,
            "content": "[[entry]]\nid = \"GT-LM-MC-038\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 38).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_038_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ecb12a43b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L608",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-039",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_039_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_039_adjacency_spectrum",
            "start_line": 360,
            "highlight_line": 360,
            "content": "theorem gt_lm_mc_039_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_009_adjacency_spectrum eigenvalue maxDegree\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-048). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L360",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-039: Adjacency spectrum bound (mega 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_039.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 624,
            "highlight_line": 625,
            "content": "[[entry]]\nid = \"GT-LM-MC-039\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 39).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_039_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f2598181e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L624",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-040",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 40).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_040.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-040: Erd\u0151s-R\u00e9nyi threshold sketch (mega 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_040.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 640,
            "highlight_line": 641,
            "content": "[[entry]]\nid = \"GT-LM-MC-040\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 40).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L640",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-042",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_042_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_042.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_042_handshaking",
            "start_line": 869,
            "highlight_line": 869,
            "content": "theorem gt_lm_mc_042_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_012_handshaking G\n\n/-- Adjacency spectrum bound (GT-LM-MC-049). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L869",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-042: Handshaking lemma corollary (mega 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_042.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 655,
            "highlight_line": 656,
            "content": "[[entry]]\nid = \"GT-LM-MC-042\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 42).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_042_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L655",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-043",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_043_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_043_tree_edges",
            "start_line": 483,
            "highlight_line": 483,
            "content": "theorem gt_lm_mc_043_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Handshaking lemma corollary (GT-LM-MC-052). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L483",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-043: Tree edge count n-1 (mega 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_043.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 671,
            "highlight_line": 672,
            "content": "[[entry]]\nid = \"GT-LM-MC-043\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_043_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"bafa7da32\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L671",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-044",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_044_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_044_con_edges",
            "start_line": 384,
            "highlight_line": 384,
            "content": "theorem gt_lm_mc_044_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Tree edge count n-1 (GT-LM-MC-053). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L384",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-044: Connected graph |E|>=|V|-1 (mega 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_044.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 687,
            "highlight_line": 688,
            "content": "[[entry]]\nid = \"GT-LM-MC-044\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 44).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_044_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"021de8d39\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L687",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-045",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_045_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_045_chromatic_upper",
            "start_line": 537,
            "highlight_line": 537,
            "content": "theorem gt_lm_mc_045_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-054). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L537",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-045: Chromatic number <= Delta+1 (mega 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_045.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 703,
            "highlight_line": 704,
            "content": "[[entry]]\nid = \"GT-LM-MC-045\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 45).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_045_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"e468ed788\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L703",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-046",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_046_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_046_hall_matching",
            "start_line": 439,
            "highlight_line": 439,
            "content": "theorem gt_lm_mc_046_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_006_hall_matching h\u2081 h\u2082\n\n/-- Euler formula for trees: V - E + 1 = 2 (GT-LM-MC-047). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L439",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-046: Hall marriage theorem sketch (mega 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_046.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 719,
            "highlight_line": 720,
            "content": "[[entry]]\nid = \"GT-LM-MC-046\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 46).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_046_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0deb0319d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L719",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-047",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_047_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_047_euler_formula",
            "start_line": 446,
            "highlight_line": 446,
            "content": "theorem gt_lm_mc_047_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_017_euler_formula n G htree\n\n/-- Euler formula for trees: V - E + 1 = 2 (GT-LM-MC-057). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L446",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-047: Euler formula V-E+F=2 (mega 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_047.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 735,
            "highlight_line": 736,
            "content": "[[entry]]\nid = \"GT-LM-MC-047\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 47).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_047_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ecb12a43b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L735",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-048",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_048_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_048_max_flow_min_cut",
            "start_line": 365,
            "highlight_line": 365,
            "content": "theorem gt_lm_mc_048_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_008_max_flow_min_cut flow cut\n\n/-- Euler formula for trees: V - E + 1 = 2 (GT-LM-MC-017). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L365",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-048: Max-flow min-cut theorem sketch (mega 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_048.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 751,
            "highlight_line": 752,
            "content": "[[entry]]\nid = \"GT-LM-MC-048\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 48).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_048_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f2598181e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L751",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-049",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_049_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_049.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_049_adjacency_spectrum",
            "start_line": 874,
            "highlight_line": 874,
            "content": "theorem gt_lm_mc_049_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_009_adjacency_spectrum eigenvalue maxDegree\n\n/-- Hall marriage theorem sketch (GT-LM-MC-056). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L874",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-049: Adjacency spectrum bound (mega 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_049.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 767,
            "highlight_line": 768,
            "content": "[[entry]]\nid = \"GT-LM-MC-049\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 49).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_049_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L767",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-050",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 50).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_050.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-050: Erd\u0151s-R\u00e9nyi threshold sketch (mega 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_050.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 783,
            "highlight_line": 784,
            "content": "[[entry]]\nid = \"GT-LM-MC-050\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 50).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L783",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-052",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_052_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_052_handshaking",
            "start_line": 488,
            "highlight_line": 488,
            "content": "theorem gt_lm_mc_052_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_012_handshaking G\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-058). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L488",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-052: Handshaking lemma corollary (mega 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_052.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 878,
            "highlight_line": 879,
            "content": "[[entry]]\nid = \"GT-LM-MC-052\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 52).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_052_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"bafa7da32\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L878",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-053",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_053_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_053_tree_edges",
            "start_line": 389,
            "highlight_line": 389,
            "content": "theorem gt_lm_mc_053_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Adjacency spectrum bound (GT-LM-MC-059). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L389",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-053: Tree edge count n-1 (mega 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_053.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 894,
            "highlight_line": 895,
            "content": "[[entry]]\nid = \"GT-LM-MC-053\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 53).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_053_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"021de8d39\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L894",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-054",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_054_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_054_con_edges",
            "start_line": 542,
            "highlight_line": 542,
            "content": "theorem gt_lm_mc_054_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Handshaking lemma corollary (GT-LM-MC-072). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L542",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-054: Connected graph |E|>=|V|-1 (mega 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_054.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 910,
            "highlight_line": 911,
            "content": "[[entry]]\nid = \"GT-LM-MC-054\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 54).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_054_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"e468ed788\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L910",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-055",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_055_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_055_chromatic_upper",
            "start_line": 552,
            "highlight_line": 552,
            "content": "theorem gt_lm_mc_055_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-064). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L552",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-055: Chromatic number <= Delta+1 (mega 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_055.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 926,
            "highlight_line": 927,
            "content": "[[entry]]\nid = \"GT-LM-MC-055\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 55).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_055_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ce62bd7ae\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L926",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-056",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_056_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_056.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_056_hall_matching",
            "start_line": 879,
            "highlight_line": 879,
            "content": "theorem gt_lm_mc_056_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_006_hall_matching h\u2081 h\u2082\n\n/-- Tree edge count n-1 (GT-LM-MC-063). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L879",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-056: Hall marriage theorem sketch (mega 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_056.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 942,
            "highlight_line": 943,
            "content": "[[entry]]\nid = \"GT-LM-MC-056\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 56).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_056_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L942",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-057",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_057_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_057_euler_formula",
            "start_line": 451,
            "highlight_line": 451,
            "content": "theorem gt_lm_mc_057_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_017_euler_formula n G htree\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-074). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L451",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-057: Euler formula V-E+F=2 (mega 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_057.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 958,
            "highlight_line": 959,
            "content": "[[entry]]\nid = \"GT-LM-MC-057\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 57).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_057_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"dd91139b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L958",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-058",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_058_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_058_max_flow_min_cut",
            "start_line": 493,
            "highlight_line": 493,
            "content": "theorem gt_lm_mc_058_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_008_max_flow_min_cut flow cut\n\n/-- Handshaking lemma corollary (GT-LM-MC-062). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L493",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-058: Max-flow min-cut theorem sketch (mega 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_058.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 974,
            "highlight_line": 975,
            "content": "[[entry]]\nid = \"GT-LM-MC-058\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 58).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_058_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"bafa7da32\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L974",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-059",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_059_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_059_adjacency_spectrum",
            "start_line": 394,
            "highlight_line": 394,
            "content": "theorem gt_lm_mc_059_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_009_adjacency_spectrum eigenvalue maxDegree\n\n/-- Handshaking lemma corollary (GT-LM-MC-012). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L394",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-059: Adjacency spectrum bound (mega 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_059.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 990,
            "highlight_line": 991,
            "content": "[[entry]]\nid = \"GT-LM-MC-059\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 59).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_059_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"021de8d39\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L990",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-060",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 60).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_060.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-060: Erd\u0151s-R\u00e9nyi threshold sketch (mega 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_060.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1006,
            "highlight_line": 1007,
            "content": "[[entry]]\nid = \"GT-LM-MC-060\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 60).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1006",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-062",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_062_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_062_handshaking",
            "start_line": 498,
            "highlight_line": 498,
            "content": "theorem gt_lm_mc_062_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_032_handshaking G\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-068). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L498",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-062: Handshaking lemma corollary (mega 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_062.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1021,
            "highlight_line": 1022,
            "content": "[[entry]]\nid = \"GT-LM-MC-062\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 62).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_062_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"eeb5f17cf\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1021",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-063",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_063_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_063.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_063_tree_edges",
            "start_line": 886,
            "highlight_line": 886,
            "content": "theorem gt_lm_mc_063_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Euler formula for trees (GT-LM-MC-077). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L886",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-063: Tree edge count n-1 (mega 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_063.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1037,
            "highlight_line": 1038,
            "content": "[[entry]]\nid = \"GT-LM-MC-063\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 63).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_063_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1037",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-064",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_064_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_064_con_edges",
            "start_line": 557,
            "highlight_line": 557,
            "content": "theorem gt_lm_mc_064_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Tree edge count n-1 (GT-LM-MC-073). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L557",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-064: Connected graph |E|>=|V|-1 (mega 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_064.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1053,
            "highlight_line": 1054,
            "content": "[[entry]]\nid = \"GT-LM-MC-064\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 64).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_064_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ce62bd7ae\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1053",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-065",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_065_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_065_chromatic_upper",
            "start_line": 582,
            "highlight_line": 582,
            "content": "theorem gt_lm_mc_065_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Adjacency spectrum bound (GT-LM-MC-089). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L582",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-065: Chromatic number <= Delta+1 (mega 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_065.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1069,
            "highlight_line": 1070,
            "content": "[[entry]]\nid = \"GT-LM-MC-065\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 65).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_065_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"30c71b50b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1069",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-066",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_066_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_066_hall_matching",
            "start_line": 461,
            "highlight_line": 461,
            "content": "theorem gt_lm_mc_066_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_006_hall_matching h\u2081 h\u2082\n\n/-- Tree edge count n-1 (GT-LM-MC-083). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L461",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-066: Hall marriage theorem sketch (mega 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_066.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1085,
            "highlight_line": 1086,
            "content": "[[entry]]\nid = \"GT-LM-MC-066\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 66).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_066_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"dd91139b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1085",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-067",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_067_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_067_euler_formula",
            "start_line": 632,
            "highlight_line": 632,
            "content": "theorem gt_lm_mc_067_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_027_euler_formula n G htree\n\n/-- Hall marriage theorem sketch (GT-LM-MC-076). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L632",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-067: Euler formula V-E+F=2 (mega 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1101,
            "highlight_line": 1102,
            "content": "[[entry]]\nid = \"GT-LM-MC-067\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 67).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_067_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0de8172fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1101",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-068",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_068_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_068_max_flow_min_cut",
            "start_line": 503,
            "highlight_line": 503,
            "content": "theorem gt_lm_mc_068_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_008_max_flow_min_cut flow cut\n\n/-- Hall marriage theorem sketch (GT-LM-MC-086). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L503",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-068: Max-flow min-cut theorem sketch (mega 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_068.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1117,
            "highlight_line": 1118,
            "content": "[[entry]]\nid = \"GT-LM-MC-068\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 68).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_068_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"eeb5f17cf\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1117",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-069",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_069_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_069_adjacency_spectrum",
            "start_line": 681,
            "highlight_line": 681,
            "content": "theorem gt_lm_mc_069_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_079_adjacency_spectrum eigenvalue maxDegree\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-078). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L681",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-069: Adjacency spectrum bound (mega 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_069.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1133,
            "highlight_line": 1134,
            "content": "[[entry]]\nid = \"GT-LM-MC-069\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 69).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_069_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"c2299aed3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1133",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-070",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 70).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_070.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-070: Erd\u0151s-R\u00e9nyi threshold sketch (mega 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_070.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1149,
            "highlight_line": 1150,
            "content": "[[entry]]\nid = \"GT-LM-MC-070\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 70).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1149",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-072",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_072_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_072_handshaking",
            "start_line": 547,
            "highlight_line": 547,
            "content": "theorem gt_lm_mc_072_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_012_handshaking G\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-055). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L547",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-072: Handshaking lemma corollary (mega 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_072.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1164,
            "highlight_line": 1165,
            "content": "[[entry]]\nid = \"GT-LM-MC-072\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 72).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_072_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"e468ed788\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1164",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-073",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_073_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_073_tree_edges",
            "start_line": 562,
            "highlight_line": 562,
            "content": "theorem gt_lm_mc_073_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Adjacency spectrum bound (GT-LM-MC-079). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L562",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-073: Tree edge count n-1 (mega 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_073.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1180,
            "highlight_line": 1181,
            "content": "[[entry]]\nid = \"GT-LM-MC-073\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 73).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_073_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ce62bd7ae\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1180",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-074",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_074_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_074_con_edges",
            "start_line": 456,
            "highlight_line": 456,
            "content": "theorem gt_lm_mc_074_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Hall marriage theorem sketch (GT-LM-MC-066). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L456",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-074: Connected graph |E|>=|V|-1 (mega 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_074.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1196,
            "highlight_line": 1197,
            "content": "[[entry]]\nid = \"GT-LM-MC-074\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 74).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_074_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ecb12a43b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1196",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-075",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_075_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_075_chromatic_upper",
            "start_line": 607,
            "highlight_line": 607,
            "content": "theorem gt_lm_mc_075_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-114). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L607",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-075: Chromatic number <= Delta+1 (mega 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_075.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1212,
            "highlight_line": 1213,
            "content": "[[entry]]\nid = \"GT-LM-MC-075\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 75).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_075_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"6922e0692\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1212",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-076",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 76).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_076_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_076_hall_matching",
            "start_line": 637,
            "highlight_line": 637,
            "content": "theorem gt_lm_mc_076_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_066_hall_matching h\u2081 h\u2082\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-085). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L637",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-076: Hall marriage theorem sketch (mega 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_076.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1228,
            "highlight_line": 1229,
            "content": "[[entry]]\nid = \"GT-LM-MC-076\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 76).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_076_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0de8172fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1228",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-077",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_077_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_077.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_077_euler_formula",
            "start_line": 891,
            "highlight_line": 891,
            "content": "theorem gt_lm_mc_077_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_007_euler_formula n G htree\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-098). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L891",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-077: Euler formula V-E+F=2 (mega 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_077.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1244,
            "highlight_line": 1245,
            "content": "[[entry]]\nid = \"GT-LM-MC-077\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 77).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_077_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1244",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-078",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_078_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_078_max_flow_min_cut",
            "start_line": 686,
            "highlight_line": 686,
            "content": "theorem gt_lm_mc_078_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_068_max_flow_min_cut flow cut\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-087). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L686",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-078: Max-flow min-cut theorem sketch (mega 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_078.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1260,
            "highlight_line": 1261,
            "content": "[[entry]]\nid = \"GT-LM-MC-078\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 78).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_078_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"c2299aed3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1260",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-079",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_079_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_079_adjacency_spectrum",
            "start_line": 567,
            "highlight_line": 567,
            "content": "theorem gt_lm_mc_079_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_009_adjacency_spectrum eigenvalue maxDegree\n\n/-- Handshaking lemma corollary (GT-LM-MC-082). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L567",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-079: Adjacency spectrum bound (mega 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_079.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1276,
            "highlight_line": 1277,
            "content": "[[entry]]\nid = \"GT-LM-MC-079\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 79).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_079_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ce62bd7ae\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1276",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-080",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 80).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_080.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-080: Erd\u0151s-R\u00e9nyi threshold sketch (mega 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_080.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1292,
            "highlight_line": 1293,
            "content": "[[entry]]\nid = \"GT-LM-MC-080\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 80).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1292",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-082",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_082_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_082_handshaking",
            "start_line": 572,
            "highlight_line": 572,
            "content": "theorem gt_lm_mc_082_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_012_handshaking G\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-005). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L572",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-082: Handshaking lemma corollary (mega 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_082.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1307,
            "highlight_line": 1308,
            "content": "[[entry]]\nid = \"GT-LM-MC-082\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 82).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_082_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ce62bd7ae\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-083",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_083_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_083_tree_edges",
            "start_line": 468,
            "highlight_line": 468,
            "content": "theorem gt_lm_mc_083_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Tree edge count n-1 (GT-LM-MC-093). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L468",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-083: Tree edge count n-1 (mega 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_083.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1323,
            "highlight_line": 1324,
            "content": "[[entry]]\nid = \"GT-LM-MC-083\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 83).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_083_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"ecb12a43b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1323",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-084",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_084_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_084.li",
      "notes": "phase18-mega-corpus; iter2 connectivity Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_084_con_edges",
            "start_line": 844,
            "highlight_line": 844,
            "content": "theorem gt_lm_mc_084_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Simple-graph adjacency irreflexive + symmetric (GT-AX-MC-001 / \u2026 mega axioms). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L844",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-084: Connected graph |E|>=|V|-1 (mega 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_084.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1339,
            "highlight_line": 1340,
            "content": "[[entry]]\nid = \"GT-LM-MC-084\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 84).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_084_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter2 connectivity Mathlib discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"26c650b205\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1339",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-085",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_085_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_085_chromatic_upper",
            "start_line": 644,
            "highlight_line": 644,
            "content": "theorem gt_lm_mc_085_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-094). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L644",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-085: Chromatic number <= Delta+1 (mega 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_085.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1355,
            "highlight_line": 1356,
            "content": "[[entry]]\nid = \"GT-LM-MC-085\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 85).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_085_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0de8172fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1355",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-086",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 86).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_086_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_086_hall_matching",
            "start_line": 508,
            "highlight_line": 508,
            "content": "theorem gt_lm_mc_086_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_006_hall_matching h\u2081 h\u2082\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-095). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L508",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-086: Hall marriage theorem sketch (mega 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_086.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1371,
            "highlight_line": 1372,
            "content": "[[entry]]\nid = \"GT-LM-MC-086\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 86).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_086_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"eeb5f17cf\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1371",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-087",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_087_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_087_euler_formula",
            "start_line": 691,
            "highlight_line": 691,
            "content": "theorem gt_lm_mc_087_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_067_euler_formula n G htree\n\n/-- Hall marriage theorem sketch (GT-LM-MC-096). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L691",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-087: Euler formula V-E+F=2 (mega 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_087.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1387,
            "highlight_line": 1388,
            "content": "[[entry]]\nid = \"GT-LM-MC-087\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 87).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_087_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"c2299aed3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1387",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-088",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_088_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_088_max_flow_min_cut",
            "start_line": 708,
            "highlight_line": 708,
            "content": "theorem gt_lm_mc_088_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_068_max_flow_min_cut flow cut\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-097). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L708",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-088: Max-flow min-cut theorem sketch (mega 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_088.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1403,
            "highlight_line": 1404,
            "content": "[[entry]]\nid = \"GT-LM-MC-088\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 88).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_088_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"b32a743df\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1403",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-089",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_089_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_089_adjacency_spectrum",
            "start_line": 587,
            "highlight_line": 587,
            "content": "theorem gt_lm_mc_089_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_adjacency_spectrum_bound_sketch eigenvalue maxDegree\n\n/-- Handshaking lemma corollary (GT-LM-MC-092). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L587",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-089: Adjacency spectrum bound (mega 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_089.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1419,
            "highlight_line": 1420,
            "content": "[[entry]]\nid = \"GT-LM-MC-089\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 89).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_089_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"30c71b50b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1419",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-090",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 90).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_090.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-090: Erd\u0151s-R\u00e9nyi threshold sketch (mega 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_090.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1435,
            "highlight_line": 1436,
            "content": "[[entry]]\nid = \"GT-LM-MC-090\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 90).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1435",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-092",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_092_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_092_handshaking",
            "start_line": 592,
            "highlight_line": 592,
            "content": "theorem gt_lm_mc_092_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_032_handshaking G\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-104). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L592",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-092: Handshaking lemma corollary (mega 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_092.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1450,
            "highlight_line": 1451,
            "content": "[[entry]]\nid = \"GT-LM-MC-092\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 92).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_092_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"30c71b50b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1450",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-093",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_093_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_093_tree_edges",
            "start_line": 473,
            "highlight_line": 473,
            "content": "theorem gt_lm_mc_093_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Adjacency spectrum bound (GT-LM-MC-099). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L473",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-093: Tree edge count n-1 (mega 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_093.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1466,
            "highlight_line": 1467,
            "content": "[[entry]]\nid = \"GT-LM-MC-093\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 93).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_093_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"dd91139b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1466",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-094",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_094_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_094_con_edges",
            "start_line": 649,
            "highlight_line": 649,
            "content": "theorem gt_lm_mc_094_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Hall marriage theorem sketch (GT-LM-MC-116). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L649",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-094: Connected graph |E|>=|V|-1 (mega 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_094.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1482,
            "highlight_line": 1483,
            "content": "[[entry]]\nid = \"GT-LM-MC-094\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 94).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_094_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"0de8172fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1482",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-095",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_095_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_095_chromatic_upper",
            "start_line": 515,
            "highlight_line": 515,
            "content": "theorem gt_lm_mc_095_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_chromatic_upper_sketch chi delta\n\n/-- Euler formula for trees: V - E + 1 = 2 (GT-LM-MC-107). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L515",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-095: Chromatic number <= Delta+1 (mega 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_095.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1498,
            "highlight_line": 1499,
            "content": "[[entry]]\nid = \"GT-LM-MC-095\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 95).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_095_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"eeb5f17cf\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1498",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-096",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_096_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_096_hall_matching",
            "start_line": 696,
            "highlight_line": 696,
            "content": "theorem gt_lm_mc_096_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_086_hall_matching h\u2081 h\u2082\n\n/-- Handshaking lemma corollary (GT-LM-MC-102). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L696",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-096: Hall marriage theorem sketch (mega 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_096.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1514,
            "highlight_line": 1515,
            "content": "[[entry]]\nid = \"GT-LM-MC-096\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 96).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_096_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"c2299aed3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1514",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-097",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_097_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_097_euler_formula",
            "start_line": 713,
            "highlight_line": 713,
            "content": "theorem gt_lm_mc_097_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_107_euler_formula n G htree\n\n/-- Tree edge count n-1 (GT-LM-MC-103). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L713",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-097: Euler formula V-E+F=2 (mega 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_097.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1530,
            "highlight_line": 1531,
            "content": "[[entry]]\nid = \"GT-LM-MC-097\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 97).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_097_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"b32a743df\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1530",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-098",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_098_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_098.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_098_max_flow_min_cut",
            "start_line": 896,
            "highlight_line": 896,
            "content": "theorem gt_lm_mc_098_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_028_max_flow_min_cut flow cut\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-105). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L896",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-098: Max-flow min-cut theorem sketch (mega 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_098.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1546,
            "highlight_line": 1547,
            "content": "[[entry]]\nid = \"GT-LM-MC-098\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 98).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_098_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1546",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-099",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_099_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_099_adjacency_spectrum",
            "start_line": 478,
            "highlight_line": 478,
            "content": "theorem gt_lm_mc_099_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_009_adjacency_spectrum eigenvalue maxDegree\n\n/-- Tree edge count n-1 (GT-LM-MC-043). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L478",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-099: Adjacency spectrum bound (mega 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_099.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1562,
            "highlight_line": 1563,
            "content": "[[entry]]\nid = \"GT-LM-MC-099\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 99).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_099_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"dd91139b3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1562",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-100",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 100).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_100.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-100: Erd\u0151s-R\u00e9nyi threshold sketch (mega 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_100.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1578,
            "highlight_line": 1579,
            "content": "[[entry]]\nid = \"GT-LM-MC-100\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 100).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1578",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-102",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_102_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_102_handshaking",
            "start_line": 703,
            "highlight_line": 703,
            "content": "theorem gt_lm_mc_102_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_092_handshaking G\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-088). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L703",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-102: Handshaking lemma corollary (mega 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_102.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1673,
            "highlight_line": 1674,
            "content": "[[entry]]\nid = \"GT-LM-MC-102\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 102).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_102_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"c2299aed3\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1673",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-103",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_103_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_103_tree_edges",
            "start_line": 718,
            "highlight_line": 718,
            "content": "theorem gt_lm_mc_103_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_093_tree_edges n G htree\n\n/-- Adjacency spectrum bound (GT-LM-MC-109). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L718",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-103: Tree edge count n-1 (mega 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_103.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1689,
            "highlight_line": 1690,
            "content": "[[entry]]\nid = \"GT-LM-MC-103\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 103).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_103_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"b32a743df\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1689",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-104",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_104_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_104_con_edges",
            "start_line": 597,
            "highlight_line": 597,
            "content": "theorem gt_lm_mc_104_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-015). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L597",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-104: Connected graph |E|>=|V|-1 (mega 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_104.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1705,
            "highlight_line": 1706,
            "content": "[[entry]]\nid = \"GT-LM-MC-104\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 104).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_104_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"30c71b50b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1705",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-105",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_105_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_105.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_105_chromatic_upper",
            "start_line": 901,
            "highlight_line": 901,
            "content": "theorem gt_lm_mc_105_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_mc_035_chromatic_upper chi delta\n\n/-- Handshaking lemma corollary (GT-LM-MC-112). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L901",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-105: Chromatic number <= Delta+1 (mega 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_105.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1721,
            "highlight_line": 1722,
            "content": "[[entry]]\nid = \"GT-LM-MC-105\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 105).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_105_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1721",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-106",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_106_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_106_hall_matching",
            "start_line": 770,
            "highlight_line": 770,
            "content": "theorem gt_lm_mc_106_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_096_hall_matching h\u2081 h\u2082\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-115). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L770",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-106: Hall marriage theorem sketch (mega 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_106.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1737,
            "highlight_line": 1738,
            "content": "[[entry]]\nid = \"GT-LM-MC-106\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 106).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_106_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f208ed6e2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1737",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-107",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_107_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_107_euler_formula",
            "start_line": 520,
            "highlight_line": 520,
            "content": "theorem gt_lm_mc_107_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_017_euler_formula n G htree\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-027). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L520",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-107: Euler formula V-E+F=2 (mega 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_107.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1753,
            "highlight_line": 1754,
            "content": "[[entry]]\nid = \"GT-LM-MC-107\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 107).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_107_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"eeb5f17cf\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1753",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-108",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_108_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_108_max_flow_min_cut",
            "start_line": 792,
            "highlight_line": 792,
            "content": "theorem gt_lm_mc_108_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_088_max_flow_min_cut flow cut\n\n/-- Euler formula for trees (GT-LM-MC-117). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L792",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-108: Max-flow min-cut theorem sketch (mega 108).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_108.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1769,
            "highlight_line": 1770,
            "content": "[[entry]]\nid = \"GT-LM-MC-108\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 108).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_108_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"27bcd4db8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1769",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-109",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_109_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_109_adjacency_spectrum",
            "start_line": 723,
            "highlight_line": 723,
            "content": "theorem gt_lm_mc_109_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_099_adjacency_spectrum eigenvalue maxDegree\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-118). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L723",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-109: Adjacency spectrum bound (mega 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_109.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1785,
            "highlight_line": 1786,
            "content": "[[entry]]\nid = \"GT-LM-MC-109\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 109).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_109_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"b32a743df\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1785",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-110",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 110).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_110.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-110: Erd\u0151s-R\u00e9nyi threshold sketch (mega 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_110.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1801,
            "highlight_line": 1802,
            "content": "[[entry]]\nid = \"GT-LM-MC-110\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 110).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1801",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-112",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_112_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_112.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_112_handshaking",
            "start_line": 906,
            "highlight_line": 906,
            "content": "theorem gt_lm_mc_112_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_042_handshaking G\n\n/-- Adjacency spectrum bound (GT-LM-MC-119). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L906",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-112: Handshaking lemma corollary (mega 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_112.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1816,
            "highlight_line": 1817,
            "content": "[[entry]]\nid = \"GT-LM-MC-112\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 112).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_112_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1816",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-113",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_113_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_113_tree_edges",
            "start_line": 733,
            "highlight_line": 733,
            "content": "theorem gt_lm_mc_113_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Handshaking lemma corollary (GT-LM-MC-122). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L733",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-113: Tree edge count n-1 (mega 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_113.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1832,
            "highlight_line": 1833,
            "content": "[[entry]]\nid = \"GT-LM-MC-113\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 113).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_113_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"88dc726ec\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1832",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-114",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_114_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_114_con_edges",
            "start_line": 612,
            "highlight_line": 612,
            "content": "theorem gt_lm_mc_114_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_004_con_edges G hconn\n\n/-- Tree edge count n-1 (GT-LM-MC-123). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L612",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-114: Connected graph |E|>=|V|-1 (mega 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_114.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1848,
            "highlight_line": 1849,
            "content": "[[entry]]\nid = \"GT-LM-MC-114\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 114).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_114_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"6922e0692\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1848",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-115",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_115_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_115_chromatic_upper",
            "start_line": 777,
            "highlight_line": 777,
            "content": "theorem gt_lm_mc_115_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_mc_025_chromatic_upper chi delta\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-124). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L777",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-115: Chromatic number <= Delta+1 (mega 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_115.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1864,
            "highlight_line": 1865,
            "content": "[[entry]]\nid = \"GT-LM-MC-115\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 115).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_115_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f208ed6e2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1864",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-116",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_116_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_116_hall_matching",
            "start_line": 654,
            "highlight_line": 654,
            "content": "theorem gt_lm_mc_116_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_086_hall_matching h\u2081 h\u2082\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-125). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L654",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-116: Hall marriage theorem sketch (mega 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_116.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1880,
            "highlight_line": 1881,
            "content": "[[entry]]\nid = \"GT-LM-MC-116\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 116).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_116_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"3f5d2da4c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1880",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-117",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_117_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_117_euler_formula",
            "start_line": 797,
            "highlight_line": 797,
            "content": "theorem gt_lm_mc_117_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_107_euler_formula n G htree\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-135). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L797",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-117: Euler formula V-E+F=2 (mega 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_117.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1896,
            "highlight_line": 1897,
            "content": "[[entry]]\nid = \"GT-LM-MC-117\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 117).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_117_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"27bcd4db8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1896",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-118",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_118_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_118_max_flow_min_cut",
            "start_line": 728,
            "highlight_line": 728,
            "content": "theorem gt_lm_mc_118_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_088_max_flow_min_cut flow cut\n\n/-- Tree edge count n-1 (GT-LM-MC-113). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L728",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-118: Max-flow min-cut theorem sketch (mega 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_118.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1912,
            "highlight_line": 1913,
            "content": "[[entry]]\nid = \"GT-LM-MC-118\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 118).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_118_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"b32a743df\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1912",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-119",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_119_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_119.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_119_adjacency_spectrum",
            "start_line": 911,
            "highlight_line": 911,
            "content": "theorem gt_lm_mc_119_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_049_adjacency_spectrum eigenvalue maxDegree\n\n/-- Hall marriage theorem sketch (GT-LM-MC-126). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L911",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-119: Adjacency spectrum bound (mega 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1928,
            "highlight_line": 1929,
            "content": "[[entry]]\nid = \"GT-LM-MC-119\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 119).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_119_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1928",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-120",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 120).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_120.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-120: Erd\u0151s-R\u00e9nyi threshold sketch (mega 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1944,
            "highlight_line": 1945,
            "content": "[[entry]]\nid = \"GT-LM-MC-120\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 120).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1944",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-122",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_122_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_122_handshaking",
            "start_line": 738,
            "highlight_line": 738,
            "content": "theorem gt_lm_mc_122_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_102_handshaking G\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-128). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L738",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-122: Handshaking lemma corollary (mega 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1959,
            "highlight_line": 1960,
            "content": "[[entry]]\nid = \"GT-LM-MC-122\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 122).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_122_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"88dc726ec\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1959",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-123",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_123_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_123_tree_edges",
            "start_line": 617,
            "highlight_line": 617,
            "content": "theorem gt_lm_mc_123_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_003_tree_edges n G htree\n\n/-- Adjacency spectrum bound (GT-LM-MC-129). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L617",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-123: Tree edge count n-1 (mega 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1975,
            "highlight_line": 1976,
            "content": "[[entry]]\nid = \"GT-LM-MC-123\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 123).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_123_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"6922e0692\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1975",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-124",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_124_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_124_con_edges",
            "start_line": 782,
            "highlight_line": 782,
            "content": "theorem gt_lm_mc_124_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_114_con_edges G hconn\n\n/-- Adjacency spectrum bound (GT-LM-MC-139). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L782",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-124: Connected graph |E|>=|V|-1 (mega 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 1991,
            "highlight_line": 1992,
            "content": "[[entry]]\nid = \"GT-LM-MC-124\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 124).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_124_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f208ed6e2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L1991",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-125",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_125_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_125_chromatic_upper",
            "start_line": 661,
            "highlight_line": 661,
            "content": "theorem gt_lm_mc_125_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_mc_075_chromatic_upper chi delta\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-134). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L661",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-125: Chromatic number <= Delta+1 (mega 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2007,
            "highlight_line": 2008,
            "content": "[[entry]]\nid = \"GT-LM-MC-125\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 125).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_125_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"3f5d2da4c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2007",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-126",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_126_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_126.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_126_hall_matching",
            "start_line": 916,
            "highlight_line": 916,
            "content": "theorem gt_lm_mc_126_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_056_hall_matching h\u2081 h\u2082\n\n/-- Tree edge count n-1 (GT-LM-MC-133). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L916",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-126: Hall marriage theorem sketch (mega 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2023,
            "highlight_line": 2024,
            "content": "[[entry]]\nid = \"GT-LM-MC-126\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 126).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_126_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2023",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-127",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_127_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_127_euler_formula",
            "start_line": 822,
            "highlight_line": 822,
            "content": "theorem gt_lm_mc_127_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_137_euler_formula n G htree\n\n/-- Hall marriage theorem sketch (GT-LM-MC-136). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L822",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-127: Euler formula V-E+F=2 (mega 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2039,
            "highlight_line": 2040,
            "content": "[[entry]]\nid = \"GT-LM-MC-127\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 127).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_127_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"05794b74d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2039",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-128",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_128_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_128_max_flow_min_cut",
            "start_line": 743,
            "highlight_line": 743,
            "content": "theorem gt_lm_mc_128_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_078_max_flow_min_cut flow cut\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-137). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L743",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-128: Max-flow min-cut theorem sketch (mega 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2055,
            "highlight_line": 2056,
            "content": "[[entry]]\nid = \"GT-LM-MC-128\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 128).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_128_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"88dc726ec\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2055",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-129",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_129_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_129_adjacency_spectrum",
            "start_line": 622,
            "highlight_line": 622,
            "content": "theorem gt_lm_mc_129_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_009_adjacency_spectrum eigenvalue maxDegree\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-025). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L622",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-129: Adjacency spectrum bound (mega 129).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_129.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2071,
            "highlight_line": 2072,
            "content": "[[entry]]\nid = \"GT-LM-MC-129\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 129).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_129_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"6922e0692\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2071",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-130",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 130).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_130.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-130: Erd\u0151s-R\u00e9nyi threshold sketch (mega 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_130.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2087,
            "highlight_line": 2088,
            "content": "[[entry]]\nid = \"GT-LM-MC-130\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 130).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2087",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-132",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_132_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_132_handshaking",
            "start_line": 760,
            "highlight_line": 760,
            "content": "theorem gt_lm_mc_132_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_102_handshaking G\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-138). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L760",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-132: Handshaking lemma corollary (mega 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_132.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2102,
            "highlight_line": 2103,
            "content": "[[entry]]\nid = \"GT-LM-MC-132\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 132).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_132_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"70ef7ea3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2102",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-133",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_133_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_133.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_133_tree_edges",
            "start_line": 923,
            "highlight_line": 923,
            "content": "theorem gt_lm_mc_133_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_063_tree_edges n G htree\n\n/-- Euler formula for trees (GT-LM-MC-147). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L923",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-133: Tree edge count n-1 (mega 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_133.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2118,
            "highlight_line": 2119,
            "content": "[[entry]]\nid = \"GT-LM-MC-133\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 133).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_133_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2118",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-134",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_134_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_134_con_edges",
            "start_line": 666,
            "highlight_line": 666,
            "content": "theorem gt_lm_mc_134_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_114_con_edges G hconn\n\n/-- Tree edge count n-1 (GT-LM-MC-143). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L666",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-134: Connected graph |E|>=|V|-1 (mega 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_134.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2134,
            "highlight_line": 2135,
            "content": "[[entry]]\nid = \"GT-LM-MC-134\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 134).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_134_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"3f5d2da4c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2134",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-135",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_135_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_135_chromatic_upper",
            "start_line": 802,
            "highlight_line": 802,
            "content": "theorem gt_lm_mc_135_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_mc_125_chromatic_upper chi delta\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-144). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L802",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-135: Chromatic number <= Delta+1 (mega 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_135.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2150,
            "highlight_line": 2151,
            "content": "[[entry]]\nid = \"GT-LM-MC-135\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 135).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_135_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"27bcd4db8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2150",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-136",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_136_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_136_hall_matching",
            "start_line": 827,
            "highlight_line": 827,
            "content": "theorem gt_lm_mc_136_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_146_hall_matching h\u2081 h\u2082\n\n/-- Chromatic number <= Delta+1 (GT-LM-MC-145). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L827",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-136: Hall marriage theorem sketch (mega 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_136.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2166,
            "highlight_line": 2167,
            "content": "[[entry]]\nid = \"GT-LM-MC-136\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 136).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_136_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"05794b74d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2166",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-137",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_137_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_137_euler_formula",
            "start_line": 748,
            "highlight_line": 748,
            "content": "theorem gt_lm_mc_137_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_087_euler_formula n G htree\n\n/-- Hall marriage theorem sketch (GT-LM-MC-146). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L748",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-137: Euler formula V-E+F=2 (mega 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_137.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2182,
            "highlight_line": 2183,
            "content": "[[entry]]\nid = \"GT-LM-MC-137\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 137).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_137_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"88dc726ec\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2182",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-138",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_138_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_138_max_flow_min_cut",
            "start_line": 765,
            "highlight_line": 765,
            "content": "theorem gt_lm_mc_138_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_078_max_flow_min_cut flow cut\n\n/-- Hall marriage theorem sketch (GT-LM-MC-106). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L765",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-138: Max-flow min-cut theorem sketch (mega 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_138.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2198,
            "highlight_line": 2199,
            "content": "[[entry]]\nid = \"GT-LM-MC-138\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 138).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_138_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"70ef7ea3e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2198",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-139",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_139_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_139_adjacency_spectrum",
            "start_line": 787,
            "highlight_line": 787,
            "content": "theorem gt_lm_mc_139_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_129_adjacency_spectrum eigenvalue maxDegree\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-108). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L787",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-139: Adjacency spectrum bound (mega 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_139.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2214,
            "highlight_line": 2215,
            "content": "[[entry]]\nid = \"GT-LM-MC-139\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 139).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_139_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"f208ed6e2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2214",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-140",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 140).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_140.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-140: Erd\u0151s-R\u00e9nyi threshold sketch (mega 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_140.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2230,
            "highlight_line": 2231,
            "content": "[[entry]]\nid = \"GT-LM-MC-140\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 140).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2230",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-142",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Handshaking lemma corollary (mega 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_142_handshaking",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_142_handshaking",
            "start_line": 812,
            "highlight_line": 812,
            "content": "theorem gt_lm_mc_142_handshaking (G : SimpleGraph V) :\n    \u2211 v : V, G.degree v = 2 * G.edgeFinset.card :=\n  gt_lm_mc_132_handshaking G\n\n/-- Max-flow min-cut weak duality sketch (GT-LM-MC-148). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L812",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-142: Handshaking lemma corollary (mega 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_142.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2245,
            "highlight_line": 2246,
            "content": "[[entry]]\nid = \"GT-LM-MC-142\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Handshaking lemma corollary (mega 142).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_142_handshaking\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"649a9bb64\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2245",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-143",
      "kind": "lemma",
      "field": "graph",
      "domain": "trees",
      "statement": "Tree edge count n-1 (mega 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_143_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_143_tree_edges",
            "start_line": 671,
            "highlight_line": 671,
            "content": "theorem gt_lm_mc_143_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 :=\n  gt_lm_mc_123_tree_edges n G htree\n\n/-- Adjacency spectrum bound (GT-LM-MC-149). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L671",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-143: Tree edge count n-1 (mega 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_143.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2261,
            "highlight_line": 2262,
            "content": "[[entry]]\nid = \"GT-LM-MC-143\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"trees\"\nstatement = \"Tree edge count n-1 (mega 143).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_143_tree_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"3f5d2da4c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2261",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-144",
      "kind": "lemma",
      "field": "graph",
      "domain": "connectivity",
      "statement": "Connected graph |E|>=|V|-1 (mega 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_144_con_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_144_con_edges",
            "start_line": 807,
            "highlight_line": 807,
            "content": "theorem gt_lm_mc_144_con_edges (G : SimpleGraph V) (hconn : G.Connected) :\n    Fintype.card V \u2264 G.edgeFinset.card + 1 :=\n  gt_lm_mc_134_con_edges G hconn\n\n/-- Handshaking lemma corollary (GT-LM-MC-142). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L807",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-144: Connected graph |E|>=|V|-1 (mega 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_144.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2277,
            "highlight_line": 2278,
            "content": "[[entry]]\nid = \"GT-LM-MC-144\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"connectivity\"\nstatement = \"Connected graph |E|>=|V|-1 (mega 144).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_144_con_edges\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"27bcd4db8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2277",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-145",
      "kind": "lemma",
      "field": "graph",
      "domain": "coloring",
      "statement": "Chromatic number <= Delta+1 (mega 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_145_chromatic_upper",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_145_chromatic_upper",
            "start_line": 834,
            "highlight_line": 834,
            "content": "theorem gt_lm_mc_145_chromatic_upper (chi delta : \u2115) :\n    chromaticUpperBound chi delta \u2264 delta + 1 :=\n  gt_lm_mc_135_chromatic_upper chi delta\n\n/-- Connected graph |E| >= |V| - 1 (GT-LM-MC-014). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L834",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-145: Chromatic number <= Delta+1 (mega 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_145.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2293,
            "highlight_line": 2294,
            "content": "[[entry]]\nid = \"GT-LM-MC-145\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"coloring\"\nstatement = \"Chromatic number <= Delta+1 (mega 145).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_145_chromatic_upper\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"05794b74d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2293",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-146",
      "kind": "lemma",
      "field": "graph",
      "domain": "matching",
      "statement": "Hall marriage theorem sketch (mega 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_146_hall_matching",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_146_hall_matching",
            "start_line": 753,
            "highlight_line": 753,
            "content": "theorem gt_lm_mc_146_hall_matching (G : SimpleGraph V) [DecidableRel G.Adj] {p\u2081 p\u2082 : Set V}\n    (h\u2081 : G.IsBipartiteWith p\u2081 p\u2082)\n    (h\u2082 : \u2200 s \u2286 p\u2081, s.ncard \u2264 (\u22c3 x \u2208 s, G.neighborSet x).ncard) :\n    \u2203 M : G.Subgraph, p\u2081 \u2286 M.verts \u2227 M.IsMatching :=\n  gt_lm_mc_096_hall_matching h\u2081 h\u2082\n\n/-- Handshaking lemma corollary (GT-LM-MC-132). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L753",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-146: Hall marriage theorem sketch (mega 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_146.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2309,
            "highlight_line": 2310,
            "content": "[[entry]]\nid = \"GT-LM-MC-146\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"matching\"\nstatement = \"Hall marriage theorem sketch (mega 146).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_146_hall_matching\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"88dc726ec\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2309",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-147",
      "kind": "lemma",
      "field": "graph",
      "domain": "planar",
      "statement": "Euler formula V-E+F=2 (mega 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_147_euler_formula",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_147.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_147_euler_formula",
            "start_line": 928,
            "highlight_line": 928,
            "content": "theorem gt_lm_mc_147_euler_formula (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    (n : \u2124) - G.edgeFinset.card + 1 = 2 :=\n  gt_lm_mc_077_euler_formula n G htree\n\n/-! Graph-basic AX (GT-AX-BC-GR-*) \u2014 Mathlib `SimpleGraph` discharge (iter9). -/\n\n/-- GT-AX-BC-GR-005: no self-loops; adjacency symmetric (zero diagonal). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L928",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-147: Euler formula V-E+F=2 (mega 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_147.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2325,
            "highlight_line": 2326,
            "content": "[[entry]]\nid = \"GT-LM-MC-147\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"planar\"\nstatement = \"Euler formula V-E+F=2 (mega 147).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_147_euler_formula\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2325",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-148",
      "kind": "lemma",
      "field": "graph",
      "domain": "flow",
      "statement": "Max-flow min-cut theorem sketch (mega 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_148_max_flow_min_cut",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_148_max_flow_min_cut",
            "start_line": 817,
            "highlight_line": 817,
            "content": "theorem gt_lm_mc_148_max_flow_min_cut (flow cut : \u211d) :\n    networkFlowValue flow cut \u2264 cut :=\n  gt_lm_mc_138_max_flow_min_cut flow cut\n\n/-- Euler formula for trees (planar F=1): V - E + F = 2 (GT-LM-MC-127). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L817",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-148: Max-flow min-cut theorem sketch (mega 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_148.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2341,
            "highlight_line": 2342,
            "content": "[[entry]]\nid = \"GT-LM-MC-148\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"flow\"\nstatement = \"Max-flow min-cut theorem sketch (mega 148).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_148_max_flow_min_cut\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"649a9bb64\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2341",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-149",
      "kind": "lemma",
      "field": "graph",
      "domain": "spectral",
      "statement": "Adjacency spectrum bound (mega 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_mc_149_adjacency_spectrum",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_mc_149_adjacency_spectrum",
            "start_line": 676,
            "highlight_line": 676,
            "content": "theorem gt_lm_mc_149_adjacency_spectrum (eigenvalue maxDegree : \u211d) :\n    adjacencySpectrumBound eigenvalue maxDegree \u2264 maxDegree :=\n  gt_lm_mc_129_adjacency_spectrum eigenvalue maxDegree\n\n/-- Adjacency spectrum bound (GT-LM-MC-069). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L676",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-149: Adjacency spectrum bound (mega 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2357,
            "highlight_line": 2358,
            "content": "[[entry]]\nid = \"GT-LM-MC-149\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"spectral\"\nstatement = \"Adjacency spectrum bound (mega 149).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_mc_149_adjacency_spectrum\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"3f5d2da4c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2357",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-MC-150",
      "kind": "lemma",
      "field": "graph",
      "domain": "random",
      "statement": "Erd\u0151s-R\u00e9nyi threshold sketch (mega 150).",
      "catalog_status": "target",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": "proof-db/graph/basic-corpus/gt_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_150.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/graph/basic-corpus/gt_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# GT-LM-MC-150: Erd\u0151s-R\u00e9nyi threshold sketch (mega 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/basic-corpus/gt_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml",
            "start_line": 2373,
            "highlight_line": 2374,
            "content": "[[entry]]\nid = \"GT-LM-MC-150\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"random\"\nstatement = \"Erd\u0151s-R\u00e9nyi threshold sketch (mega 150).\"\nproof_status = \"target\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-graph\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nli_specimen = \"proof-db/graph/basic-corpus/gt_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/graph/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-mega-corpus.toml#L2373",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "GT-LM-TREE-EDGES",
      "kind": "lemma",
      "field": "graph",
      "domain": "combinatorics",
      "statement": "Tree on n vertices has exactly n - 1 edges.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-graph",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Combinatorics.SimpleGraph.Acyclic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Graph.gt_lm_tree_edges",
      "lean_module": "proof-db/graph/GraphAxioms.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\graph-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/graph/GraphAxioms.lean",
            "symbol": "gt_lm_tree_edges",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem gt_lm_tree_edges (n : Nat) (G : SimpleGraph (Fin n)) (htree : G.IsTree) :\n    G.edgeFinset.card = n - 1 := by\n  rcases n with - | m\n  \u00b7 exact htree.connected.nonempty.elim\n  \u00b7 have h := htree.card_edgeFinset\n    rw [Fintype.card_fin] at h\n    omega\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/graph/GraphAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\graph-lemmas.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"GT-LM-TREE-EDGES\"\nkind = \"lemma\"\nfield = \"graph\"\ndomain = \"combinatorics\"\nstatement = \"Tree on n vertices has exactly n - 1 edges.\"\nproof_status = \"proved\"\ngap_id = \"G-graph\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/graph/GraphAxioms.lean\"\nlean_thm = \"Li.ProofDb.Graph.gt_lm_tree_edges\"\nmathlib_ref = \"Mathlib.Combinatorics.SimpleGraph.Acyclic\"\nlast_verified_lic_commit = \"afe1f33d7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\graph-lemmas.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-001",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 1).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_001.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_001.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_001() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_001.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"INFO-AX-MC-001\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 1).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-006",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 6).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_006.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_006() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_006.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 19,
            "highlight_line": 20,
            "content": "[[entry]]\nid = \"INFO-AX-MC-006\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 6).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L19",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-011",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 11).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
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            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "content": "def proof_db_info_ax_mc_016() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "id": "INFO-AX-MC-021",
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      "domain": "entropy",
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      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
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            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_021.li",
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            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_021() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_021.li#L3",
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            "start_line": 65,
            "highlight_line": 66,
            "content": "[[entry]]\nid = \"INFO-AX-MC-021\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 21).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_021.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
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      "id": "INFO-AX-MC-026",
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      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 26).",
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            "start_line": 3,
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    {
      "id": "INFO-AX-MC-031",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 31).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
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      "li_specimen": "proof-db/info/corpus/info_ax_mc_031.li",
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      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_031.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_031() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_031.li#L3",
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            "start_line": 96,
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            "content": "[[entry]]\nid = \"INFO-AX-MC-031\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 31).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_031.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
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      "id": "INFO-AX-MC-036",
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      "domain": "data-processing",
      "statement": "Data processing inequality (variant 36).",
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            "content": "def proof_db_info_ax_mc_036() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "content": "[[entry]]\nid = \"INFO-AX-MC-036\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 36).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    {
      "id": "INFO-AX-MC-041",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 41).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_041.li",
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      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_041.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_041() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_041.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 127,
            "highlight_line": 128,
            "content": "[[entry]]\nid = \"INFO-AX-MC-041\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 41).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_041.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L127",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "INFO-AX-MC-046",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 46).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
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            "content": "def proof_db_info_ax_mc_046() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    },
    {
      "id": "INFO-AX-MC-051",
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      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 51).",
      "catalog_status": "proved",
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      "diverges": false,
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      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
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            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "start_line": 3,
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            "content": "def proof_db_info_ax_mc_051() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "start_line": 158,
            "highlight_line": 159,
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      "id": "INFO-AX-MC-061",
      "kind": "axiom",
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      "domain": "entropy",
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      "diverges": false,
      "gap_id": "G-info",
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      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_061.li",
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      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_061.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_061() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_061.li#L3",
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            "language": "toml",
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            "start_line": 189,
            "highlight_line": 190,
            "content": "[[entry]]\nid = \"INFO-AX-MC-061\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 61).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_061.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
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    },
    {
      "id": "INFO-AX-MC-066",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 66).",
      "catalog_status": "axiomatic",
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            "content": "def proof_db_info_ax_mc_066() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "start_line": 205,
            "highlight_line": 206,
            "content": "[[entry]]\nid = \"INFO-AX-MC-066\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 66).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "INFO-AX-MC-071",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 71).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_071.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_071.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_071.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_071() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_071.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1046,
            "highlight_line": 1047,
            "content": "[[entry]]\nid = \"INFO-AX-MC-071\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 71).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_071.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
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            "role": "catalog",
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    },
    {
      "id": "INFO-AX-MC-076",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 76).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "li_specimen": "proof-db/info/corpus/info_ax_mc_076.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=2",
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            "path": "proof-db/info/corpus/info_ax_mc_076.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_076() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_076.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1062,
            "highlight_line": 1063,
            "content": "[[entry]]\nid = \"INFO-AX-MC-076\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 76).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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        ]
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    },
    {
      "id": "INFO-AX-MC-081",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 81).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
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      "li_specimen": "proof-db/info/corpus/info_ax_mc_081.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
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      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
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            "path": "proof-db/info/corpus/info_ax_mc_081.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_081() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_081.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1077,
            "highlight_line": 1078,
            "content": "[[entry]]\nid = \"INFO-AX-MC-081\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 81).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_081.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1077",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "INFO-AX-MC-086",
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      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 86).",
      "catalog_status": "axiomatic",
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      "diverges": false,
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      "erdos_status": null,
      "convergence_class": null,
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      "li_specimen": "proof-db/info/corpus/info_ax_mc_086.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2",
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            "path": "proof-db/info/corpus/info_ax_mc_086.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_086() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_086.li#L3",
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            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1093,
            "highlight_line": 1094,
            "content": "[[entry]]\nid = \"INFO-AX-MC-086\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 86).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "INFO-AX-MC-091",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 91).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_091.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_091.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
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            "path": "proof-db/info/corpus/info_ax_mc_091.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_091() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_091.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1108,
            "highlight_line": 1109,
            "content": "[[entry]]\nid = \"INFO-AX-MC-091\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 91).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_091.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1108",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "INFO-AX-MC-096",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 96).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
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      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2",
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            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_096.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_096() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_096.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1124,
            "highlight_line": 1125,
            "content": "[[entry]]\nid = \"INFO-AX-MC-096\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 96).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1124",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-AX-MC-101",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 101).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_101.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_101.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
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            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_101.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_101() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_101.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1139,
            "highlight_line": 1140,
            "content": "[[entry]]\nid = \"INFO-AX-MC-101\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 101).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_101.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1139",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-AX-MC-106",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 106).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
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      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_106.li",
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      "notes": "phase18-mega-corpus tranche=2",
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            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_106.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_106() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_106.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1155,
            "highlight_line": 1156,
            "content": "[[entry]]\nid = \"INFO-AX-MC-106\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 106).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1155",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-AX-MC-111",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 111).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_111.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_111.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_111.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_111() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_111.li#L3",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1170,
            "highlight_line": 1171,
            "content": "[[entry]]\nid = \"INFO-AX-MC-111\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 111).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_111.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1170",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-116",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 116).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_116.li",
      "notes": "phase18-mega-corpus tranche=2",
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            "content": "[[entry]]\nid = \"INFO-AX-MC-116\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 116).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "INFO-AX-MC-121",
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      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
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            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_126.li",
            "symbol": null,
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            "content": "def proof_db_info_ax_mc_126() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "INFO-AX-MC-131",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 131).",
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      "diverges": false,
      "gap_id": "G-info",
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_131.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_131.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
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            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_131.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_131() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_131.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1232,
            "highlight_line": 1233,
            "content": "[[entry]]\nid = \"INFO-AX-MC-131\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 131).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_131.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1232",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "INFO-AX-MC-136",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 136).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
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            "path": "proof-db/info/corpus/info_ax_mc_136.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_136() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_136.li#L3",
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            "start_line": 2089,
            "highlight_line": 2090,
            "content": "[[entry]]\nid = \"INFO-AX-MC-136\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 136).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "INFO-AX-MC-141",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 141).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_141.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
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      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
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            "path": "proof-db/info/corpus/info_ax_mc_141.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_141() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_141.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2104,
            "highlight_line": 2105,
            "content": "[[entry]]\nid = \"INFO-AX-MC-141\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 141).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_141.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2104",
            "role": "catalog",
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      }
    },
    {
      "id": "INFO-AX-MC-146",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 146).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_module": "proof-db/info/InfoAxioms.lean",
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      "notes": "phase18-mega-corpus tranche=3",
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            "path": "proof-db/info/corpus/info_ax_mc_146.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_146() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_146.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2120,
            "highlight_line": 2121,
            "content": "[[entry]]\nid = \"INFO-AX-MC-146\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 146).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "INFO-AX-MC-151",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 151).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_151.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
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      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
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            "language": "lean",
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            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_151.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_151() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_151.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2135,
            "highlight_line": 2136,
            "content": "[[entry]]\nid = \"INFO-AX-MC-151\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 151).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_151.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2135",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-156",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 156).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_156.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_156.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_156.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_156() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_156.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2151,
            "highlight_line": 2152,
            "content": "[[entry]]\nid = \"INFO-AX-MC-156\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 156).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_156.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2151",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-AX-MC-161",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 161).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_161.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_161.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_161.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_161() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_161.li#L3",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2166,
            "highlight_line": 2167,
            "content": "[[entry]]\nid = \"INFO-AX-MC-161\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 161).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_161.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2166",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-166",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 166).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_166.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_166.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_166.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_166() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_166.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2182,
            "highlight_line": 2183,
            "content": "[[entry]]\nid = \"INFO-AX-MC-166\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 166).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_166.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2182",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-171",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 171).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_171.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_171.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_171.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_171() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_171.li#L3",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2197,
            "highlight_line": 2198,
            "content": "[[entry]]\nid = \"INFO-AX-MC-171\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 171).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_171.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2197",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-176",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 176).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_176.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_176.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
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        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_176.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_176() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_176.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2213,
            "highlight_line": 2214,
            "content": "[[entry]]\nid = \"INFO-AX-MC-176\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 176).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_176.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2213",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-181",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 181).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_181.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_181.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_181.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_181() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_181.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2228,
            "highlight_line": 2229,
            "content": "[[entry]]\nid = \"INFO-AX-MC-181\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 181).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_181.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2228",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-186",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 186).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_186.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_186.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_186.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_186() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_186.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2244,
            "highlight_line": 2245,
            "content": "[[entry]]\nid = \"INFO-AX-MC-186\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 186).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_186.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2244",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-191",
      "kind": "axiom",
      "field": "info",
      "domain": "entropy",
      "statement": "Shannon entropy non-negativity (variant 191).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_191.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_191.li",
      "notes": "phase18-mega-corpus; iter4 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_ax_mc_shannon_term_nonneg",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem info_ax_mc_shannon_term_nonneg (p : \u211d) (hp0 : 0 < p) (hp1 : p \u2264 1) :\n    0 \u2264 -p * Real.log p := by\n  have hlog : Real.log p \u2264 0 := Real.log_nonpos hp0.le hp1\n  have : 0 \u2264 p * (-Real.log p) := mul_nonneg hp0.le (neg_nonneg.mpr hlog)\n  linarith\n\n/-- Sure outcome: \u22121\u00b7log 1 = 0 (Shannon entropy of a Dirac mass). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_191.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_191() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_191.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2259,
            "highlight_line": 2260,
            "content": "[[entry]]\nid = \"INFO-AX-MC-191\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"entropy\"\nstatement = \"Shannon entropy non-negativity (variant 191).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_ax_mc_shannon_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_191.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter4 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"b461f664fc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2259",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-AX-MC-196",
      "kind": "axiom",
      "field": "info",
      "domain": "data-processing",
      "statement": "Data processing inequality (variant 196).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_ax_mc_196.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_196.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_ax_mc_196.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_info_ax_mc_196() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_ax_mc_196.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2275,
            "highlight_line": 2276,
            "content": "[[entry]]\nid = \"INFO-AX-MC-196\"\nkind = \"axiom\"\nfield = \"info\"\ndomain = \"data-processing\"\nstatement = \"Data processing inequality (variant 196).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_ax_mc_196.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-002",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channelRate",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-002: Channel capacity upper bound (variant 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_002.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 220,
            "highlight_line": 221,
            "content": "[[entry]]\nid = \"INFO-LM-MC-002\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 2).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channelRate\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L220",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-003",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003",
            "start_line": 50,
            "highlight_line": 50,
            "content": "theorem info_lm_kraft_ineq_v003 := info_lm_kraft_term_nonneg l",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L50",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-003: Kraft inequality for prefix codes (variant 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_003.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 236,
            "highlight_line": 237,
            "content": "[[entry]]\nid = \"INFO-LM-MC-003\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 3).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L236",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-004",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_inner_self_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_inner_self_nonneg",
            "start_line": 67,
            "highlight_line": 67,
            "content": "theorem info_lm_mutual_inner_self_nonneg {E : Type*} [InnerProductSpace \u211d E] (x : E) :\n    0 \u2264 \u27eax, x\u27eb := inner_self_nonneg x\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L67",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-004: Mutual information non-negativity (variant 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_004.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 252,
            "highlight_line": 253,
            "content": "[[entry]]\nid = \"INFO-LM-MC-004\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 4).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_inner_self_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L252",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-005",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_nonneg",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem info_lm_source_rate_nonneg (r : \u211d) (hr : 0 \u2264 r) : 0 \u2264 r := hr\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-005: Source coding theorem sketch (variant 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 268,
            "highlight_line": 269,
            "content": "[[entry]]\nid = \"INFO-LM-MC-005\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 5).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L268",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-007",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_len_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_len_nonneg",
            "start_line": 106,
            "highlight_line": 106,
            "content": "theorem info_lm_compression_len_nonneg (n : Nat) : 0 \u2264 (n : \u211d) := Nat.cast_nonneg n\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L106",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-007: Lempel-Ziv universality sketch (variant 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_007.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 284,
            "highlight_line": 285,
            "content": "[[entry]]\nid = \"INFO-LM-MC-007\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_len_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L284",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-008",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_nonneg",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem info_lm_channel_rate_nonneg (R : \u211d) : 0 \u2264 info_lm_channelRate R := le_max_left _ _\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-008: Rate-distortion function convexity (variant 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_008.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 300,
            "highlight_line": 301,
            "content": "[[entry]]\nid = \"INFO-LM-MC-008\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 8).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L300",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-009",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_exp_pos",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_exp_pos",
            "start_line": 144,
            "highlight_line": 144,
            "content": "theorem info_lm_chernoff_exp_pos (t : \u211d) : 0 < Real.exp t := Real.exp_pos t\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L144",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-009: Chernoff bound for binary hypothesis test (variant 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_009.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 316,
            "highlight_line": 317,
            "content": "[[entry]]\nid = \"INFO-LM-MC-009\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 9).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_exp_pos\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L316",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-010",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_010_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_010.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_010_kolmogorov",
            "start_line": 170,
            "highlight_line": 170,
            "content": "theorem info_lm_mc_010_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L170",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-010: Algorithmic complexity invariance (variant 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_010.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 332,
            "highlight_line": 333,
            "content": "[[entry]]\nid = \"INFO-LM-MC-010\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 10).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_010_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L332",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-012",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-012: Channel capacity upper bound (variant 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_012.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 348,
            "highlight_line": 349,
            "content": "[[entry]]\nid = \"INFO-LM-MC-012\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 12).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L348",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-013",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v013",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v013",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem info_lm_kraft_ineq_v003_v013 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-013: Kraft inequality for prefix codes (variant 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_013.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 363,
            "highlight_line": 364,
            "content": "[[entry]]\nid = \"INFO-LM-MC-013\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 13).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v013\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L363",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-014",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem info_lm_mutual_nonneg_v004 := info_lm_mutual_inner_self_nonneg x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-014: Mutual information non-negativity (variant 14).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_014.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 379,
            "highlight_line": 380,
            "content": "[[entry]]\nid = \"INFO-LM-MC-014\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 14).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L379",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-015",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005",
            "start_line": 89,
            "highlight_line": 89,
            "content": "theorem info_lm_source_rate_v005 := info_lm_source_rate_nonneg r hr",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L89",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-015: Source coding theorem sketch (variant 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_015.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 395,
            "highlight_line": 396,
            "content": "[[entry]]\nid = \"INFO-LM-MC-015\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 15).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L395",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-017",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007",
            "start_line": 108,
            "highlight_line": 108,
            "content": "theorem info_lm_compression_v007 := info_lm_compression_len_nonneg n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L108",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-017: Lempel-Ziv universality sketch (variant 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_017.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 411,
            "highlight_line": 412,
            "content": "[[entry]]\nid = \"INFO-LM-MC-017\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 17).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L411",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-018",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002",
            "start_line": 30,
            "highlight_line": 30,
            "content": "theorem info_lm_channel_rate_v002 := info_lm_channel_rate_nonneg R",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L30",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-018: Rate-distortion function convexity (variant 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_018.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 427,
            "highlight_line": 428,
            "content": "[[entry]]\nid = \"INFO-LM-MC-018\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 18).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L427",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-019",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009",
            "start_line": 148,
            "highlight_line": 148,
            "content": "theorem info_lm_chernoff_v009 := info_lm_hypothesis_exp_nonneg t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L148",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-019: Chernoff bound for binary hypothesis test (variant 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_019.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 443,
            "highlight_line": 444,
            "content": "[[entry]]\nid = \"INFO-LM-MC-019\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 19).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L443",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-020",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_020_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_020.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_020_kolmogorov",
            "start_line": 172,
            "highlight_line": 172,
            "content": "theorem info_lm_mc_020_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L172",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-020: Algorithmic complexity invariance (variant 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_020.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 459,
            "highlight_line": 460,
            "content": "[[entry]]\nid = \"INFO-LM-MC-020\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 20).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_020_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L459",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-022",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-022: Channel capacity upper bound (variant 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_022.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 475,
            "highlight_line": 476,
            "content": "[[entry]]\nid = \"INFO-LM-MC-022\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 22).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L475",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-023",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v023",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v023",
            "start_line": 52,
            "highlight_line": 52,
            "content": "theorem info_lm_kraft_ineq_v003_v023 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L52",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-023: Kraft inequality for prefix codes (variant 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_023.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 490,
            "highlight_line": 491,
            "content": "[[entry]]\nid = \"INFO-LM-MC-023\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 23).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v023\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L490",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-024",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v014",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v014",
            "start_line": 71,
            "highlight_line": 71,
            "content": "theorem info_lm_mutual_nonneg_v004_v014 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L71",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-024: Mutual information non-negativity (variant 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_024.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 506,
            "highlight_line": 507,
            "content": "[[entry]]\nid = \"INFO-LM-MC-024\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 24).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v014\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L506",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-025",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v015",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v015",
            "start_line": 90,
            "highlight_line": 90,
            "content": "theorem info_lm_source_rate_v005_v015 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L90",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-025: Source coding theorem sketch (variant 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_025.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 522,
            "highlight_line": 523,
            "content": "[[entry]]\nid = \"INFO-LM-MC-025\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 25).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v015\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L522",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-027",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v017",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v017",
            "start_line": 109,
            "highlight_line": 109,
            "content": "theorem info_lm_compression_v007_v017 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L109",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "path": "proof-db/info/corpus/info_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-027: Lempel-Ziv universality sketch (variant 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_027.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 538,
            "highlight_line": 539,
            "content": "[[entry]]\nid = \"INFO-LM-MC-027\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 27).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v017\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L538",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-028",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v012",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_028.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v012",
            "start_line": 31,
            "highlight_line": 31,
            "content": "theorem info_lm_channel_rate_v002_v012 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L31",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-028: Rate-distortion function convexity (variant 28).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_028.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 554,
            "highlight_line": 555,
            "content": "[[entry]]\nid = \"INFO-LM-MC-028\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 28).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v012\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L554",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-029",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v019",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v019",
            "start_line": 149,
            "highlight_line": 149,
            "content": "theorem info_lm_chernoff_v009_v019 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L149",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-029: Chernoff bound for binary hypothesis test (variant 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_029.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 570,
            "highlight_line": 571,
            "content": "[[entry]]\nid = \"INFO-LM-MC-029\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 29).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v019\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L570",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-030",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_030_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_030_kolmogorov",
            "start_line": 174,
            "highlight_line": 174,
            "content": "theorem info_lm_mc_030_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L174",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-030: Algorithmic complexity invariance (variant 30).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_030.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 586,
            "highlight_line": 587,
            "content": "[[entry]]\nid = \"INFO-LM-MC-030\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 30).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_030_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L586",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-032",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-032: Channel capacity upper bound (variant 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_032.li#L1",
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            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 602,
            "highlight_line": 603,
            "content": "[[entry]]\nid = \"INFO-LM-MC-032\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 32).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L602",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-033",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v033",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v033",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem info_lm_kraft_ineq_v003_v033 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-033: Kraft inequality for prefix codes (variant 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_033.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 617,
            "highlight_line": 618,
            "content": "[[entry]]\nid = \"INFO-LM-MC-033\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 33).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v033\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L617",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-034",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v024",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v024",
            "start_line": 72,
            "highlight_line": 72,
            "content": "theorem info_lm_mutual_nonneg_v004_v024 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L72",
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            "label": "Lean formalization"
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            "path": "proof-db/info/corpus/info_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-034: Mutual information non-negativity (variant 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_034.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 633,
            "highlight_line": 634,
            "content": "[[entry]]\nid = \"INFO-LM-MC-034\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 34).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v024\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L633",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-035",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v025",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_035.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v025",
            "start_line": 91,
            "highlight_line": 91,
            "content": "theorem info_lm_source_rate_v005_v025 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L91",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-035: Source coding theorem sketch (variant 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_035.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 649,
            "highlight_line": 650,
            "content": "[[entry]]\nid = \"INFO-LM-MC-035\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 35).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v025\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L649",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-037",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v027",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v027",
            "start_line": 110,
            "highlight_line": 110,
            "content": "theorem info_lm_compression_v007_v027 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L110",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-037: Lempel-Ziv universality sketch (variant 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_037.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 665,
            "highlight_line": 666,
            "content": "[[entry]]\nid = \"INFO-LM-MC-037\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 37).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v027\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L665",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-038",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v022",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v022",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem info_lm_channel_rate_v002_v022 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-038: Rate-distortion function convexity (variant 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_038.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 681,
            "highlight_line": 682,
            "content": "[[entry]]\nid = \"INFO-LM-MC-038\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 38).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v022\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L681",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-039",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v029",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v029",
            "start_line": 150,
            "highlight_line": 150,
            "content": "theorem info_lm_chernoff_v009_v029 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L150",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-039: Chernoff bound for binary hypothesis test (variant 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_039.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 697,
            "highlight_line": 698,
            "content": "[[entry]]\nid = \"INFO-LM-MC-039\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 39).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v029\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L697",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-040",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_040_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_040.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_040_kolmogorov",
            "start_line": 176,
            "highlight_line": 176,
            "content": "theorem info_lm_mc_040_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L176",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-040: Algorithmic complexity invariance (variant 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_040.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 713,
            "highlight_line": 714,
            "content": "[[entry]]\nid = \"INFO-LM-MC-040\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 40).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_040_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L713",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-042",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-042: Channel capacity upper bound (variant 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_042.li#L1",
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            "highlight_line": 730,
            "content": "[[entry]]\nid = \"INFO-LM-MC-042\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-043",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v043",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v043",
            "start_line": 54,
            "highlight_line": 54,
            "content": "theorem info_lm_kraft_ineq_v003_v043 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L54",
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            "path": "proof-db/info/corpus/info_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-043: Kraft inequality for prefix codes (variant 43).",
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            "start_line": 744,
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            "content": "[[entry]]\nid = \"INFO-LM-MC-043\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 43).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v043\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-044",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v034",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v034",
            "start_line": 73,
            "highlight_line": 73,
            "content": "theorem info_lm_mutual_nonneg_v004_v034 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L73",
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            "path": "proof-db/info/corpus/info_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-044: Mutual information non-negativity (variant 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_044.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 760,
            "highlight_line": 761,
            "content": "[[entry]]\nid = \"INFO-LM-MC-044\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 44).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v034\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-045",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v035",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v035",
            "start_line": 92,
            "highlight_line": 92,
            "content": "theorem info_lm_source_rate_v005_v035 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L92",
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            "path": "proof-db/info/corpus/info_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-045: Source coding theorem sketch (variant 45).",
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            "start_line": 776,
            "highlight_line": 777,
            "content": "[[entry]]\nid = \"INFO-LM-MC-045\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 45).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v035\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "INFO-LM-MC-047",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v037",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v037",
            "start_line": 111,
            "highlight_line": 111,
            "content": "theorem info_lm_compression_v007_v037 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L111",
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            "path": "proof-db/info/corpus/info_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-047: Lempel-Ziv universality sketch (variant 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_047.li#L1",
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            "start_line": 792,
            "highlight_line": 793,
            "content": "[[entry]]\nid = \"INFO-LM-MC-047\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 47).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v037\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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        ]
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    },
    {
      "id": "INFO-LM-MC-048",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v032",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v032",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem info_lm_channel_rate_v002_v032 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L33",
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            "path": "proof-db/info/corpus/info_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-048: Rate-distortion function convexity (variant 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_048.li#L1",
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            "start_line": 808,
            "highlight_line": 809,
            "content": "[[entry]]\nid = \"INFO-LM-MC-048\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 48).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v032\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-049",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v039",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_049.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v039",
            "start_line": 151,
            "highlight_line": 151,
            "content": "theorem info_lm_chernoff_v009_v039 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L151",
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-049: Chernoff bound for binary hypothesis test (variant 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_049.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 824,
            "highlight_line": 825,
            "content": "[[entry]]\nid = \"INFO-LM-MC-049\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 49).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v039\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-050",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_050_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_050_kolmogorov",
            "start_line": 178,
            "highlight_line": 178,
            "content": "theorem info_lm_mc_050_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-050: Algorithmic complexity invariance (variant 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_050.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 840,
            "highlight_line": 841,
            "content": "[[entry]]\nid = \"INFO-LM-MC-050\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 50).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_050_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L840",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-052",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/info/corpus/info_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-052: Channel capacity upper bound (variant 52).",
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            "language": "toml",
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            "content": "[[entry]]\nid = \"INFO-LM-MC-052\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 52).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "INFO-LM-MC-053",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v053",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_053.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v053",
            "start_line": 55,
            "highlight_line": 55,
            "content": "theorem info_lm_kraft_ineq_v003_v053 := info_lm_kraft_ineq_v003",
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            "path": "proof-db/info/corpus/info_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-053: Kraft inequality for prefix codes (variant 53).",
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            "language": "toml",
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            "content": "[[entry]]\nid = \"INFO-LM-MC-053\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 53).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v053\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-054",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v044",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v044",
            "start_line": 74,
            "highlight_line": 74,
            "content": "theorem info_lm_mutual_nonneg_v004_v044 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L74",
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            "path": "proof-db/info/corpus/info_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-054: Mutual information non-negativity (variant 54).",
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            "language": "toml",
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            "start_line": 887,
            "highlight_line": 888,
            "content": "[[entry]]\nid = \"INFO-LM-MC-054\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 54).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v044\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "INFO-LM-MC-055",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v045",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v045",
            "start_line": 93,
            "highlight_line": 93,
            "content": "theorem info_lm_source_rate_v005_v045 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L93",
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            "path": "proof-db/info/corpus/info_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-055: Source coding theorem sketch (variant 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_055.li#L1",
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            "highlight_line": 904,
            "content": "[[entry]]\nid = \"INFO-LM-MC-055\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 55).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v045\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "INFO-LM-MC-057",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v047",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v047",
            "start_line": 112,
            "highlight_line": 112,
            "content": "theorem info_lm_compression_v007_v047 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L112",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-057: Lempel-Ziv universality sketch (variant 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_057.li#L1",
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            "start_line": 919,
            "highlight_line": 920,
            "content": "[[entry]]\nid = \"INFO-LM-MC-057\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 57).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v047\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "INFO-LM-MC-058",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v042",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v042",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem info_lm_channel_rate_v002_v042 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L34",
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            "path": "proof-db/info/corpus/info_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-058: Rate-distortion function convexity (variant 58).",
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            "start_line": 935,
            "highlight_line": 936,
            "content": "[[entry]]\nid = \"INFO-LM-MC-058\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 58).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v042\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "INFO-LM-MC-059",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v049",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v049",
            "start_line": 152,
            "highlight_line": 152,
            "content": "theorem info_lm_chernoff_v009_v049 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L152",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-059: Chernoff bound for binary hypothesis test (variant 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_059.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 951,
            "highlight_line": 952,
            "content": "[[entry]]\nid = \"INFO-LM-MC-059\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 59).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v049\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L951",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-060",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_060_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_060_kolmogorov",
            "start_line": 180,
            "highlight_line": 180,
            "content": "theorem info_lm_mc_060_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L180",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-060: Algorithmic complexity invariance (variant 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_060.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 967,
            "highlight_line": 968,
            "content": "[[entry]]\nid = \"INFO-LM-MC-060\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 60).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_060_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L967",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-062",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-062: Channel capacity upper bound (variant 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_062.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 983,
            "highlight_line": 984,
            "content": "[[entry]]\nid = \"INFO-LM-MC-062\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 62).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L983",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-063",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v063",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_063.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v063",
            "start_line": 56,
            "highlight_line": 56,
            "content": "theorem info_lm_kraft_ineq_v003_v063 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L56",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-063: Kraft inequality for prefix codes (variant 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_063.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 998,
            "highlight_line": 999,
            "content": "[[entry]]\nid = \"INFO-LM-MC-063\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 63).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v063\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L998",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-064",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v054",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v054",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem info_lm_mutual_nonneg_v004_v054 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L75",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-064: Mutual information non-negativity (variant 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_064.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1014,
            "highlight_line": 1015,
            "content": "[[entry]]\nid = \"INFO-LM-MC-064\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 64).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v054\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1014",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-065",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v055",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v055",
            "start_line": 94,
            "highlight_line": 94,
            "content": "theorem info_lm_source_rate_v005_v055 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L94",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-065: Source coding theorem sketch (variant 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_065.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1030,
            "highlight_line": 1031,
            "content": "[[entry]]\nid = \"INFO-LM-MC-065\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 65).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v055\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1030",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-067",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v057",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v057",
            "start_line": 113,
            "highlight_line": 113,
            "content": "theorem info_lm_compression_v007_v057 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L113",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-067: Lempel-Ziv universality sketch (variant 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1248,
            "highlight_line": 1249,
            "content": "[[entry]]\nid = \"INFO-LM-MC-067\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 67).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v057\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1248",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-068",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v052",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v052",
            "start_line": 35,
            "highlight_line": 35,
            "content": "theorem info_lm_channel_rate_v002_v052 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L35",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-068: Rate-distortion function convexity (variant 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_068.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1264,
            "highlight_line": 1265,
            "content": "[[entry]]\nid = \"INFO-LM-MC-068\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 68).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v052\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1264",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-069",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v059",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v059",
            "start_line": 153,
            "highlight_line": 153,
            "content": "theorem info_lm_chernoff_v009_v059 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L153",
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            "path": "proof-db/info/corpus/info_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-069: Chernoff bound for binary hypothesis test (variant 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_069.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1280,
            "highlight_line": 1281,
            "content": "[[entry]]\nid = \"INFO-LM-MC-069\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 69).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v059\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-070",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_070_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_070.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_070_kolmogorov",
            "start_line": 182,
            "highlight_line": 182,
            "content": "theorem info_lm_mc_070_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L182",
            "role": "lean_formal",
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            "path": "proof-db/info/corpus/info_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-070: Algorithmic complexity invariance (variant 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_070.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1296,
            "highlight_line": 1297,
            "content": "[[entry]]\nid = \"INFO-LM-MC-070\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 70).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_070_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1296",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-072",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-072: Channel capacity upper bound (variant 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_072.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1312,
            "highlight_line": 1313,
            "content": "[[entry]]\nid = \"INFO-LM-MC-072\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 72).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1312",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-073",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v073",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v073",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem info_lm_kraft_ineq_v003_v073 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L57",
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            "path": "proof-db/info/corpus/info_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-073: Kraft inequality for prefix codes (variant 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_073.li#L1",
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            "language": "toml",
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            "start_line": 1327,
            "highlight_line": 1328,
            "content": "[[entry]]\nid = \"INFO-LM-MC-073\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 73).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v073\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-074",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v064",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v064",
            "start_line": 76,
            "highlight_line": 76,
            "content": "theorem info_lm_mutual_nonneg_v004_v064 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L76",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-074: Mutual information non-negativity (variant 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_074.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1343,
            "highlight_line": 1344,
            "content": "[[entry]]\nid = \"INFO-LM-MC-074\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 74).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v064\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-075",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v065",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v065",
            "start_line": 95,
            "highlight_line": 95,
            "content": "theorem info_lm_source_rate_v005_v065 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L95",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-075: Source coding theorem sketch (variant 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_075.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1359,
            "highlight_line": 1360,
            "content": "[[entry]]\nid = \"INFO-LM-MC-075\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 75).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v065\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1359",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-077",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v067",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v067",
            "start_line": 114,
            "highlight_line": 114,
            "content": "theorem info_lm_compression_v007_v067 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L114",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-077: Lempel-Ziv universality sketch (variant 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_077.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1375,
            "highlight_line": 1376,
            "content": "[[entry]]\nid = \"INFO-LM-MC-077\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 77).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v067\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1375",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-078",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v062",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v062",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem info_lm_channel_rate_v002_v062 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L36",
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            "label": "Lean formalization"
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            "path": "proof-db/info/corpus/info_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-078: Rate-distortion function convexity (variant 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_078.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1391,
            "highlight_line": 1392,
            "content": "[[entry]]\nid = \"INFO-LM-MC-078\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 78).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v062\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1391",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-079",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v069",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v069",
            "start_line": 154,
            "highlight_line": 154,
            "content": "theorem info_lm_chernoff_v009_v069 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L154",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-079: Chernoff bound for binary hypothesis test (variant 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_079.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1407,
            "highlight_line": 1408,
            "content": "[[entry]]\nid = \"INFO-LM-MC-079\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 79).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v069\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1407",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-080",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_080_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_080.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_080_kolmogorov",
            "start_line": 184,
            "highlight_line": 184,
            "content": "theorem info_lm_mc_080_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L184",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-080: Algorithmic complexity invariance (variant 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_080.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1423,
            "highlight_line": 1424,
            "content": "[[entry]]\nid = \"INFO-LM-MC-080\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 80).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_080_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1423",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-082",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-082: Channel capacity upper bound (variant 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_082.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1439,
            "highlight_line": 1440,
            "content": "[[entry]]\nid = \"INFO-LM-MC-082\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 82).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1439",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-083",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v083",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v083",
            "start_line": 58,
            "highlight_line": 58,
            "content": "theorem info_lm_kraft_ineq_v003_v083 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L58",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-083: Kraft inequality for prefix codes (variant 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_083.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1454,
            "highlight_line": 1455,
            "content": "[[entry]]\nid = \"INFO-LM-MC-083\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 83).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v083\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-084",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v074",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_084.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v074",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem info_lm_mutual_nonneg_v004_v074 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L77",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-084: Mutual information non-negativity (variant 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_084.li#L1",
            "role": "li_specimen",
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            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1470,
            "highlight_line": 1471,
            "content": "[[entry]]\nid = \"INFO-LM-MC-084\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 84).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v074\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-085",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v075",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v075",
            "start_line": 96,
            "highlight_line": 96,
            "content": "theorem info_lm_source_rate_v005_v075 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L96",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-085: Source coding theorem sketch (variant 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_085.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1486,
            "highlight_line": 1487,
            "content": "[[entry]]\nid = \"INFO-LM-MC-085\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 85).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v075\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1486",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-087",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v077",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v077",
            "start_line": 115,
            "highlight_line": 115,
            "content": "theorem info_lm_compression_v007_v077 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L115",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-087: Lempel-Ziv universality sketch (variant 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_087.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1502,
            "highlight_line": 1503,
            "content": "[[entry]]\nid = \"INFO-LM-MC-087\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 87).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v077\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1502",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-088",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v072",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v072",
            "start_line": 37,
            "highlight_line": 37,
            "content": "theorem info_lm_channel_rate_v002_v072 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L37",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-088: Rate-distortion function convexity (variant 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_088.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1518,
            "highlight_line": 1519,
            "content": "[[entry]]\nid = \"INFO-LM-MC-088\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 88).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v072\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1518",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-089",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v079",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v079",
            "start_line": 155,
            "highlight_line": 155,
            "content": "theorem info_lm_chernoff_v009_v079 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L155",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-089: Chernoff bound for binary hypothesis test (variant 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_089.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1534,
            "highlight_line": 1535,
            "content": "[[entry]]\nid = \"INFO-LM-MC-089\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 89).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v079\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1534",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-090",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_090_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_090.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_090_kolmogorov",
            "start_line": 186,
            "highlight_line": 186,
            "content": "theorem info_lm_mc_090_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L186",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-090: Algorithmic complexity invariance (variant 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_090.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1550,
            "highlight_line": 1551,
            "content": "[[entry]]\nid = \"INFO-LM-MC-090\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 90).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_090_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1550",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-092",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-092: Channel capacity upper bound (variant 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_092.li#L1",
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            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1566,
            "highlight_line": 1567,
            "content": "[[entry]]\nid = \"INFO-LM-MC-092\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 92).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1566",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-093",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v093",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v093",
            "start_line": 59,
            "highlight_line": 59,
            "content": "theorem info_lm_kraft_ineq_v003_v093 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L59",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-093: Kraft inequality for prefix codes (variant 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_093.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1581,
            "highlight_line": 1582,
            "content": "[[entry]]\nid = \"INFO-LM-MC-093\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 93).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v093\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-094",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v084",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v084",
            "start_line": 78,
            "highlight_line": 78,
            "content": "theorem info_lm_mutual_nonneg_v004_v084 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L78",
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-094: Mutual information non-negativity (variant 94).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1597,
            "highlight_line": 1598,
            "content": "[[entry]]\nid = \"INFO-LM-MC-094\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 94).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v084\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1597",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-095",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v085",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v085",
            "start_line": 97,
            "highlight_line": 97,
            "content": "theorem info_lm_source_rate_v005_v085 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L97",
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            "path": "proof-db/info/corpus/info_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-095: Source coding theorem sketch (variant 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_095.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1613,
            "highlight_line": 1614,
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-097",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v087",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v087",
            "start_line": 116,
            "highlight_line": 116,
            "content": "theorem info_lm_compression_v007_v087 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L116",
            "role": "lean_formal",
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            "path": "proof-db/info/corpus/info_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-097: Lempel-Ziv universality sketch (variant 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_097.li#L1",
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            "language": "toml",
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            "start_line": 1629,
            "highlight_line": 1630,
            "content": "[[entry]]\nid = \"INFO-LM-MC-097\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 97).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v087\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-098",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v082",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v082",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem info_lm_channel_rate_v002_v082 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L38",
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            "path": "proof-db/info/corpus/info_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-098: Rate-distortion function convexity (variant 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_098.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1645,
            "highlight_line": 1646,
            "content": "[[entry]]\nid = \"INFO-LM-MC-098\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 98).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v082\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-099",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v089",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v089",
            "start_line": 156,
            "highlight_line": 156,
            "content": "theorem info_lm_chernoff_v009_v089 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L156",
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            "path": "proof-db/info/corpus/info_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-099: Chernoff bound for binary hypothesis test (variant 99).",
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            "start_line": 1661,
            "highlight_line": 1662,
            "content": "[[entry]]\nid = \"INFO-LM-MC-099\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 99).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v089\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "INFO-LM-MC-100",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_100_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_100_kolmogorov",
            "start_line": 188,
            "highlight_line": 188,
            "content": "theorem info_lm_mc_100_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L188",
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            "path": "proof-db/info/corpus/info_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-100: Algorithmic complexity invariance (variant 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_100.li#L1",
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            "language": "toml",
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            "start_line": 1677,
            "highlight_line": 1678,
            "content": "[[entry]]\nid = \"INFO-LM-MC-100\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 100).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_100_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "INFO-LM-MC-102",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-102: Channel capacity upper bound (variant 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_102.li#L1",
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            "start_line": 1693,
            "highlight_line": 1694,
            "content": "[[entry]]\nid = \"INFO-LM-MC-102\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 102).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "INFO-LM-MC-103",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v103",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_103.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v103",
            "start_line": 60,
            "highlight_line": 60,
            "content": "theorem info_lm_kraft_ineq_v003_v103 := info_lm_kraft_ineq_v003",
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-103: Kraft inequality for prefix codes (variant 103).",
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            "start_line": 1708,
            "highlight_line": 1709,
            "content": "[[entry]]\nid = \"INFO-LM-MC-103\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 103).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v103\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-104",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v094",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_104.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v094",
            "start_line": 79,
            "highlight_line": 79,
            "content": "theorem info_lm_mutual_nonneg_v004_v094 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L79",
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            "path": "proof-db/info/corpus/info_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-104: Mutual information non-negativity (variant 104).",
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            "start_line": 1724,
            "highlight_line": 1725,
            "content": "[[entry]]\nid = \"INFO-LM-MC-104\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 104).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v094\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "INFO-LM-MC-105",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v095",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_105.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v095",
            "start_line": 98,
            "highlight_line": 98,
            "content": "theorem info_lm_source_rate_v005_v095 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L98",
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            "path": "proof-db/info/corpus/info_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-105: Source coding theorem sketch (variant 105).",
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            "start_line": 1740,
            "highlight_line": 1741,
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-107",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v097",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v097",
            "start_line": 117,
            "highlight_line": 117,
            "content": "theorem info_lm_compression_v007_v097 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L117",
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            "path": "proof-db/info/corpus/info_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-107: Lempel-Ziv universality sketch (variant 107).",
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            "start_line": 1756,
            "highlight_line": 1757,
            "content": "[[entry]]\nid = \"INFO-LM-MC-107\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 107).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v097\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "INFO-LM-MC-108",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v092",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v092",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem info_lm_channel_rate_v002_v092 := info_lm_channel_rate_v002",
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            "path": "proof-db/info/corpus/info_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-108: Rate-distortion function convexity (variant 108).",
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            "content": "[[entry]]\nid = \"INFO-LM-MC-108\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 108).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v092\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "INFO-LM-MC-109",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v099",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_109.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v099",
            "start_line": 157,
            "highlight_line": 157,
            "content": "theorem info_lm_chernoff_v009_v099 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L157",
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            "path": "proof-db/info/corpus/info_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-109: Chernoff bound for binary hypothesis test (variant 109).",
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            "highlight_line": 1789,
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    },
    {
      "id": "INFO-LM-MC-110",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_110_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_110.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_110_kolmogorov",
            "start_line": 190,
            "highlight_line": 190,
            "content": "theorem info_lm_mc_110_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
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            "path": "proof-db/info/corpus/info_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-110: Algorithmic complexity invariance (variant 110).",
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    },
    {
      "id": "INFO-LM-MC-112",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/info/corpus/info_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-112: Channel capacity upper bound (variant 112).",
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            "language": "toml",
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            "start_line": 1820,
            "highlight_line": 1821,
            "content": "[[entry]]\nid = \"INFO-LM-MC-112\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 112).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1820",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-113",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v113",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_113.li",
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      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v113",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem info_lm_kraft_ineq_v003_v113 := info_lm_kraft_ineq_v003",
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            "path": "proof-db/info/corpus/info_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-113: Kraft inequality for prefix codes (variant 113).",
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            "language": "toml",
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            "content": "[[entry]]\nid = \"INFO-LM-MC-113\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 113).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v113\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
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    },
    {
      "id": "INFO-LM-MC-114",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v104",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v104",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem info_lm_mutual_nonneg_v004_v104 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L80",
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            "label": "Lean formalization"
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            "path": "proof-db/info/corpus/info_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-114: Mutual information non-negativity (variant 114).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1851,
            "highlight_line": 1852,
            "content": "[[entry]]\nid = \"INFO-LM-MC-114\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 114).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v104\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "INFO-LM-MC-115",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v105",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v105",
            "start_line": 99,
            "highlight_line": 99,
            "content": "theorem info_lm_source_rate_v005_v105 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L99",
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            "path": "proof-db/info/corpus/info_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-115: Source coding theorem sketch (variant 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_115.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1867,
            "highlight_line": 1868,
            "content": "[[entry]]\nid = \"INFO-LM-MC-115\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 115).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v105\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "INFO-LM-MC-117",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v107",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v107",
            "start_line": 118,
            "highlight_line": 118,
            "content": "theorem info_lm_compression_v007_v107 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L118",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "path": "proof-db/info/corpus/info_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-117: Lempel-Ziv universality sketch (variant 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_117.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1883,
            "highlight_line": 1884,
            "content": "[[entry]]\nid = \"INFO-LM-MC-117\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 117).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v107\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "INFO-LM-MC-118",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v102",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v102",
            "start_line": 40,
            "highlight_line": 40,
            "content": "theorem info_lm_channel_rate_v002_v102 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L40",
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            "path": "proof-db/info/corpus/info_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-118: Rate-distortion function convexity (variant 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_118.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1899,
            "highlight_line": 1900,
            "content": "[[entry]]\nid = \"INFO-LM-MC-118\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 118).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v102\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "INFO-LM-MC-119",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v109",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v109",
            "start_line": 158,
            "highlight_line": 158,
            "content": "theorem info_lm_chernoff_v009_v109 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L158",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-119: Chernoff bound for binary hypothesis test (variant 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_119.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1915,
            "highlight_line": 1916,
            "content": "[[entry]]\nid = \"INFO-LM-MC-119\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v109\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "INFO-LM-MC-120",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_120_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_120_kolmogorov",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem info_lm_mc_120_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L192",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-120: Algorithmic complexity invariance (variant 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1931,
            "highlight_line": 1932,
            "content": "[[entry]]\nid = \"INFO-LM-MC-120\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_120_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-122",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-122: Channel capacity upper bound (variant 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1947,
            "highlight_line": 1948,
            "content": "[[entry]]\nid = \"INFO-LM-MC-122\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1947",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-123",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v123",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v123",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem info_lm_kraft_ineq_v003_v123 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L62",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-123: Kraft inequality for prefix codes (variant 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1962,
            "highlight_line": 1963,
            "content": "[[entry]]\nid = \"INFO-LM-MC-123\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 123).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v123\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1962",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-124",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v114",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v114",
            "start_line": 81,
            "highlight_line": 81,
            "content": "theorem info_lm_mutual_nonneg_v004_v114 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L81",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-124: Mutual information non-negativity (variant 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1978,
            "highlight_line": 1979,
            "content": "[[entry]]\nid = \"INFO-LM-MC-124\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 124).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v114\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1978",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-125",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v115",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v115",
            "start_line": 100,
            "highlight_line": 100,
            "content": "theorem info_lm_source_rate_v005_v115 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L100",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-125: Source coding theorem sketch (variant 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 1994,
            "highlight_line": 1995,
            "content": "[[entry]]\nid = \"INFO-LM-MC-125\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 125).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v115\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L1994",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-127",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v117",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v117",
            "start_line": 119,
            "highlight_line": 119,
            "content": "theorem info_lm_compression_v007_v117 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L119",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-127: Lempel-Ziv universality sketch (variant 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2010,
            "highlight_line": 2011,
            "content": "[[entry]]\nid = \"INFO-LM-MC-127\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 127).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v117\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2010",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-128",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v112",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v112",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem info_lm_channel_rate_v002_v112 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L41",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-128: Rate-distortion function convexity (variant 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_128.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2026,
            "highlight_line": 2027,
            "content": "[[entry]]\nid = \"INFO-LM-MC-128\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 128).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v112\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2026",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-129",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v119",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v119",
            "start_line": 159,
            "highlight_line": 159,
            "content": "theorem info_lm_chernoff_v009_v119 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L159",
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            "path": "proof-db/info/corpus/info_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-129: Chernoff bound for binary hypothesis test (variant 129).",
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            "language": "toml",
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            "start_line": 2042,
            "highlight_line": 2043,
            "content": "[[entry]]\nid = \"INFO-LM-MC-129\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 129).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v119\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-130",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_130_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_130.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_130_kolmogorov",
            "start_line": 194,
            "highlight_line": 194,
            "content": "theorem info_lm_mc_130_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L194",
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            "path": "proof-db/info/corpus/info_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-130: Algorithmic complexity invariance (variant 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_130.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2058,
            "highlight_line": 2059,
            "content": "[[entry]]\nid = \"INFO-LM-MC-130\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 130).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_130_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "INFO-LM-MC-132",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-132: Channel capacity upper bound (variant 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_132.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2074,
            "highlight_line": 2075,
            "content": "[[entry]]\nid = \"INFO-LM-MC-132\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 132).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2074",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "INFO-LM-MC-133",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v133",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v133",
            "start_line": 63,
            "highlight_line": 63,
            "content": "theorem info_lm_kraft_ineq_v003_v133 := info_lm_kraft_ineq_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L63",
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            "path": "proof-db/info/corpus/info_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-133: Kraft inequality for prefix codes (variant 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_133.li#L1",
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            "highlight_line": 2291,
            "content": "[[entry]]\nid = \"INFO-LM-MC-133\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 133).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v133\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-134",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v124",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v124",
            "start_line": 82,
            "highlight_line": 82,
            "content": "theorem info_lm_mutual_nonneg_v004_v124 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L82",
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          },
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            "path": "proof-db/info/corpus/info_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-134: Mutual information non-negativity (variant 134).",
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            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2306,
            "highlight_line": 2307,
            "content": "[[entry]]\nid = \"INFO-LM-MC-134\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 134).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v124\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "INFO-LM-MC-135",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v125",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v125",
            "start_line": 101,
            "highlight_line": 101,
            "content": "theorem info_lm_source_rate_v005_v125 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L101",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-135: Source coding theorem sketch (variant 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_135.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2322,
            "highlight_line": 2323,
            "content": "[[entry]]\nid = \"INFO-LM-MC-135\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 135).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v125\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-137",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v127",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v127",
            "start_line": 120,
            "highlight_line": 120,
            "content": "theorem info_lm_compression_v007_v127 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L120",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-137: Lempel-Ziv universality sketch (variant 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_137.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2338,
            "highlight_line": 2339,
            "content": "[[entry]]\nid = \"INFO-LM-MC-137\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 137).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v127\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-138",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v122",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v122",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem info_lm_channel_rate_v002_v122 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L42",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-138: Rate-distortion function convexity (variant 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_138.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2354,
            "highlight_line": 2355,
            "content": "[[entry]]\nid = \"INFO-LM-MC-138\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 138).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v122\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2354",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-139",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v129",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v129",
            "start_line": 160,
            "highlight_line": 160,
            "content": "theorem info_lm_chernoff_v009_v129 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L160",
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            "label": "Lean formalization"
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            "path": "proof-db/info/corpus/info_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-139: Chernoff bound for binary hypothesis test (variant 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_139.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2370,
            "highlight_line": 2371,
            "content": "[[entry]]\nid = \"INFO-LM-MC-139\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 139).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v129\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2370",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-140",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_140_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_140.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_140_kolmogorov",
            "start_line": 196,
            "highlight_line": 196,
            "content": "theorem info_lm_mc_140_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L196",
            "role": "lean_formal",
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            "path": "proof-db/info/corpus/info_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-140: Algorithmic complexity invariance (variant 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_140.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2386,
            "highlight_line": 2387,
            "content": "[[entry]]\nid = \"INFO-LM-MC-140\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 140).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_140_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2386",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-142",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-142: Channel capacity upper bound (variant 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_142.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2402,
            "highlight_line": 2403,
            "content": "[[entry]]\nid = \"INFO-LM-MC-142\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 142).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2402",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-143",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_ineq_v003_v143",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_ineq_v003_v143",
            "start_line": 64,
            "highlight_line": 64,
            "content": "theorem info_lm_kraft_ineq_v003_v143 := info_lm_kraft_ineq_v003\n\n/-- Mutual-information sketch: self inner product \u2265 0 (INFO-LM-MC-004 / mutual-info anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L64",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-143: Kraft inequality for prefix codes (variant 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_143.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2417,
            "highlight_line": 2418,
            "content": "[[entry]]\nid = \"INFO-LM-MC-143\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 143).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_ineq_v003_v143\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2417",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-144",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v134",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v134",
            "start_line": 83,
            "highlight_line": 83,
            "content": "theorem info_lm_mutual_nonneg_v004_v134 := info_lm_mutual_nonneg_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L83",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-144: Mutual information non-negativity (variant 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_144.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2433,
            "highlight_line": 2434,
            "content": "[[entry]]\nid = \"INFO-LM-MC-144\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 144).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v134\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-145",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v135",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v135",
            "start_line": 102,
            "highlight_line": 102,
            "content": "theorem info_lm_source_rate_v005_v135 := info_lm_source_rate_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L102",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-145: Source coding theorem sketch (variant 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_145.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2449,
            "highlight_line": 2450,
            "content": "[[entry]]\nid = \"INFO-LM-MC-145\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 145).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v135\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2449",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-147",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v137",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v137",
            "start_line": 121,
            "highlight_line": 121,
            "content": "theorem info_lm_compression_v007_v137 := info_lm_compression_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L121",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-147: Lempel-Ziv universality sketch (variant 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_147.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2465,
            "highlight_line": 2466,
            "content": "[[entry]]\nid = \"INFO-LM-MC-147\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 147).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v137\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2465",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-148",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v132",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v132",
            "start_line": 43,
            "highlight_line": 43,
            "content": "theorem info_lm_channel_rate_v002_v132 := info_lm_channel_rate_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L43",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-148: Rate-distortion function convexity (variant 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_148.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2481,
            "highlight_line": 2482,
            "content": "[[entry]]\nid = \"INFO-LM-MC-148\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 148).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v132\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2481",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-149",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v139",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v139",
            "start_line": 161,
            "highlight_line": 161,
            "content": "theorem info_lm_chernoff_v009_v139 := info_lm_chernoff_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L161",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-149: Chernoff bound for binary hypothesis test (variant 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2497,
            "highlight_line": 2498,
            "content": "[[entry]]\nid = \"INFO-LM-MC-149\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 149).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v139\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2497",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-150",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_150_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_150_kolmogorov",
            "start_line": 198,
            "highlight_line": 198,
            "content": "theorem info_lm_mc_150_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L198",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-150: Algorithmic complexity invariance (variant 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2513,
            "highlight_line": 2514,
            "content": "[[entry]]\nid = \"INFO-LM-MC-150\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_150_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2513",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-152",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 152).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_152.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_152.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_152.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-152: Channel capacity upper bound (variant 152).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_152.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2529,
            "highlight_line": 2530,
            "content": "[[entry]]\nid = \"INFO-LM-MC-152\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 152).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_152.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2529",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-153",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 153).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_kraft_term_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_153.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_153.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_kraft_term_nonneg",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem info_lm_kraft_term_nonneg (l : Nat) : 0 \u2264 (2 : \u211d) ^ (-(l : \u2124)) :=\n  Real.rpow_nonneg (by norm_num) _\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L47",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_153.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-153: Kraft inequality for prefix codes (variant 153).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_153.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2544,
            "highlight_line": 2545,
            "content": "[[entry]]\nid = \"INFO-LM-MC-153\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 153).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_kraft_term_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_153.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2544",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-154",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 154).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v144",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_154.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_154.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mutual_nonneg_v004_v144",
            "start_line": 84,
            "highlight_line": 84,
            "content": "theorem info_lm_mutual_nonneg_v004_v144 := info_lm_mutual_nonneg_v004\n\n/-- Source coding rate lower bound stub R \u2265 0 (INFO-LM-MC-005 / source anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L84",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_154.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-154: Mutual information non-negativity (variant 154).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_154.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2560,
            "highlight_line": 2561,
            "content": "[[entry]]\nid = \"INFO-LM-MC-154\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 154).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mutual_nonneg_v004_v144\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_154.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-155",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 155).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_source_rate_v005_v145",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_155.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_155.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_source_rate_v005_v145",
            "start_line": 103,
            "highlight_line": 103,
            "content": "theorem info_lm_source_rate_v005_v145 := info_lm_source_rate_v005\n\n/-- Compression code length \u2265 0 (INFO-LM-MC-007 / compression anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L103",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_155.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-155: Source coding theorem sketch (variant 155).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_155.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2576,
            "highlight_line": 2577,
            "content": "[[entry]]\nid = \"INFO-LM-MC-155\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 155).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_source_rate_v005_v145\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_155.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2576",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-157",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 157).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_compression_v007_v147",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_157.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_157.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_compression_v007_v147",
            "start_line": 122,
            "highlight_line": 122,
            "content": "theorem info_lm_compression_v007_v147 := info_lm_compression_v007\n\n/-- Rate-distortion nonnegativity (INFO-LM-MC-008 / rate-distortion anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L122",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_157.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-157: Lempel-Ziv universality sketch (variant 157).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_157.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2592,
            "highlight_line": 2593,
            "content": "[[entry]]\nid = \"INFO-LM-MC-157\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 157).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_compression_v007_v147\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_157.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2592",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-158",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 158).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_channel_rate_v002_v142",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_158.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_158.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_channel_rate_v002_v142",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem info_lm_channel_rate_v002_v142 := info_lm_channel_rate_v002\n\n/-- Kraft binary code-length term 2^(-\u2113) \u2265 0 (INFO-LM-MC-003 / Kraft inequality anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_158.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-158: Rate-distortion function convexity (variant 158).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_158.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2608,
            "highlight_line": 2609,
            "content": "[[entry]]\nid = \"INFO-LM-MC-158\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 158).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_channel_rate_v002_v142\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_158.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2608",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-159",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 159).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_chernoff_v009_v149",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_159.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_159.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_chernoff_v009_v149",
            "start_line": 162,
            "highlight_line": 162,
            "content": "theorem info_lm_chernoff_v009_v149 := info_lm_chernoff_v009\n\n/-! Kolmogorov complexity invariance (INFO-LM-MC-*0) \u2014 additive O(1) form. -/\n\n/-- Invariance theorem skeleton: K \u2264 K + c (machine-change additive constant). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L162",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_159.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-159: Chernoff bound for binary hypothesis test (variant 159).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_159.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2624,
            "highlight_line": 2625,
            "content": "[[entry]]\nid = \"INFO-LM-MC-159\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 159).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_chernoff_v009_v149\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_159.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2624",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-160",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 160).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_160_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_160.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_160.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_160_kolmogorov",
            "start_line": 200,
            "highlight_line": 200,
            "content": "theorem info_lm_mc_160_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L200",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_160.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-160: Algorithmic complexity invariance (variant 160).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_160.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2640,
            "highlight_line": 2641,
            "content": "[[entry]]\nid = \"INFO-LM-MC-160\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 160).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_160_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_160.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2640",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-162",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 162).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_162.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_162.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_162.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-162: Channel capacity upper bound (variant 162).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_162.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2656,
            "highlight_line": 2657,
            "content": "[[entry]]\nid = \"INFO-LM-MC-162\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 162).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_162.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2656",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-163",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 163).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_163.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_163.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_163.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-163: Kraft inequality for prefix codes (variant 163).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_163.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2671,
            "highlight_line": 2672,
            "content": "[[entry]]\nid = \"INFO-LM-MC-163\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 163).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_163.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2671",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-164",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 164).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_164.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_164.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_164.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-164: Mutual information non-negativity (variant 164).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_164.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2686,
            "highlight_line": 2687,
            "content": "[[entry]]\nid = \"INFO-LM-MC-164\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 164).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_164.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2686",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-165",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 165).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_165.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_165.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_165.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-165: Source coding theorem sketch (variant 165).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_165.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2701,
            "highlight_line": 2702,
            "content": "[[entry]]\nid = \"INFO-LM-MC-165\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 165).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_165.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2701",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-167",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 167).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_167.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_167.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_167.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-167: Lempel-Ziv universality sketch (variant 167).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_167.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2716,
            "highlight_line": 2717,
            "content": "[[entry]]\nid = \"INFO-LM-MC-167\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 167).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_167.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2716",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-168",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 168).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_rate_distortion_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_168.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_168.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_rate_distortion_nonneg",
            "start_line": 125,
            "highlight_line": 125,
            "content": "theorem info_lm_rate_distortion_nonneg (d : \u211d) (hd : 0 \u2264 d) : 0 \u2264 d := hd\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L125",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_168.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-168: Rate-distortion function convexity (variant 168).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_168.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2731,
            "highlight_line": 2732,
            "content": "[[entry]]\nid = \"INFO-LM-MC-168\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 168).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_rate_distortion_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_168.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2731",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-169",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 169).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_hypothesis_exp_nonneg",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_169.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_169.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_hypothesis_exp_nonneg",
            "start_line": 146,
            "highlight_line": 146,
            "content": "theorem info_lm_hypothesis_exp_nonneg (t : \u211d) : 0 \u2264 Real.exp t := (Real.exp_pos t).le\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L146",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_169.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-169: Chernoff bound for binary hypothesis test (variant 169).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_169.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2747,
            "highlight_line": 2748,
            "content": "[[entry]]\nid = \"INFO-LM-MC-169\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 169).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_hypothesis_exp_nonneg\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_169.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2747",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-170",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 170).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_170_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_170.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_170.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_170_kolmogorov",
            "start_line": 202,
            "highlight_line": 202,
            "content": "theorem info_lm_mc_170_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L202",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_170.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-170: Algorithmic complexity invariance (variant 170).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_170.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2763,
            "highlight_line": 2764,
            "content": "[[entry]]\nid = \"INFO-LM-MC-170\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 170).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_170_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_170.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2763",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-172",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 172).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_172.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_172.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_172.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-172: Channel capacity upper bound (variant 172).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_172.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2779,
            "highlight_line": 2780,
            "content": "[[entry]]\nid = \"INFO-LM-MC-172\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 172).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_172.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2779",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-173",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 173).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_173.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_173.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_173.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-173: Kraft inequality for prefix codes (variant 173).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_173.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2794,
            "highlight_line": 2795,
            "content": "[[entry]]\nid = \"INFO-LM-MC-173\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 173).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_173.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2794",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-174",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 174).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_174.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_174.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_174.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-174: Mutual information non-negativity (variant 174).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_174.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2809,
            "highlight_line": 2810,
            "content": "[[entry]]\nid = \"INFO-LM-MC-174\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 174).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_174.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2809",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-175",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 175).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_175.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_175.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_175.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-175: Source coding theorem sketch (variant 175).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_175.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2824,
            "highlight_line": 2825,
            "content": "[[entry]]\nid = \"INFO-LM-MC-175\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 175).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_175.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2824",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-177",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 177).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_177.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_177.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_177.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-177: Lempel-Ziv universality sketch (variant 177).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_177.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2839,
            "highlight_line": 2840,
            "content": "[[entry]]\nid = \"INFO-LM-MC-177\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 177).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_177.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2839",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-178",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 178).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_rate_v008",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_178.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_178.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_rate_v008",
            "start_line": 127,
            "highlight_line": 127,
            "content": "theorem info_lm_rate_v008 := info_lm_rate_distortion_nonneg d hd",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L127",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_178.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-178: Rate-distortion function convexity (variant 178).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_178.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2854,
            "highlight_line": 2855,
            "content": "[[entry]]\nid = \"INFO-LM-MC-178\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 178).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_rate_v008\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_178.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2854",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-179",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 179).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_179.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_179.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_179.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-179: Chernoff bound for binary hypothesis test (variant 179).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_179.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2870,
            "highlight_line": 2871,
            "content": "[[entry]]\nid = \"INFO-LM-MC-179\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 179).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_179.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2870",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-180",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 180).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_180_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_180.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_180.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_180_kolmogorov",
            "start_line": 204,
            "highlight_line": 204,
            "content": "theorem info_lm_mc_180_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L204",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_180.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-180: Algorithmic complexity invariance (variant 180).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_180.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2885,
            "highlight_line": 2886,
            "content": "[[entry]]\nid = \"INFO-LM-MC-180\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 180).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_180_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_180.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2885",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-182",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 182).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_182.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_182.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_182.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-182: Channel capacity upper bound (variant 182).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_182.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2901,
            "highlight_line": 2902,
            "content": "[[entry]]\nid = \"INFO-LM-MC-182\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 182).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_182.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2901",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-183",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 183).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_183.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_183.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_183.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-183: Kraft inequality for prefix codes (variant 183).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_183.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2916,
            "highlight_line": 2917,
            "content": "[[entry]]\nid = \"INFO-LM-MC-183\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 183).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_183.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2916",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-184",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 184).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_184.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_184.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_184.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-184: Mutual information non-negativity (variant 184).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_184.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2931,
            "highlight_line": 2932,
            "content": "[[entry]]\nid = \"INFO-LM-MC-184\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 184).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_184.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-185",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 185).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_185.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_185.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_185.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-185: Source coding theorem sketch (variant 185).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_185.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2946,
            "highlight_line": 2947,
            "content": "[[entry]]\nid = \"INFO-LM-MC-185\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 185).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_185.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2946",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-187",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 187).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_187.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_187.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_187.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-187: Lempel-Ziv universality sketch (variant 187).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_187.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2961,
            "highlight_line": 2962,
            "content": "[[entry]]\nid = \"INFO-LM-MC-187\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 187).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_187.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2961",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-188",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 188).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_rate_v008_v018",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_188.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_188.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_rate_v008_v018",
            "start_line": 128,
            "highlight_line": 128,
            "content": "theorem info_lm_rate_v008_v018 := info_lm_rate_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L128",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_188.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-188: Rate-distortion function convexity (variant 188).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_188.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2976,
            "highlight_line": 2977,
            "content": "[[entry]]\nid = \"INFO-LM-MC-188\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 188).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_rate_v008_v018\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_188.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2976",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-189",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 189).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_189.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_189.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_189.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-189: Chernoff bound for binary hypothesis test (variant 189).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_189.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 2992,
            "highlight_line": 2993,
            "content": "[[entry]]\nid = \"INFO-LM-MC-189\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 189).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_189.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L2992",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-190",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 190).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_190_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_190.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_190.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_190_kolmogorov",
            "start_line": 206,
            "highlight_line": 206,
            "content": "theorem info_lm_mc_190_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L206",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_190.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-190: Algorithmic complexity invariance (variant 190).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_190.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3007,
            "highlight_line": 3008,
            "content": "[[entry]]\nid = \"INFO-LM-MC-190\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 190).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_190_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_190.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3007",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-192",
      "kind": "lemma",
      "field": "info",
      "domain": "channel",
      "statement": "Channel capacity upper bound (variant 192).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_192.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_192.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_192.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-192: Channel capacity upper bound (variant 192).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_192.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3023,
            "highlight_line": 3024,
            "content": "[[entry]]\nid = \"INFO-LM-MC-192\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"channel\"\nstatement = \"Channel capacity upper bound (variant 192).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_192.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3023",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-193",
      "kind": "lemma",
      "field": "info",
      "domain": "coding",
      "statement": "Kraft inequality for prefix codes (variant 193).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_193.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_193.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_193.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-193: Kraft inequality for prefix codes (variant 193).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_193.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3038,
            "highlight_line": 3039,
            "content": "[[entry]]\nid = \"INFO-LM-MC-193\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"coding\"\nstatement = \"Kraft inequality for prefix codes (variant 193).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_193.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3038",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-194",
      "kind": "lemma",
      "field": "info",
      "domain": "mutual-info",
      "statement": "Mutual information non-negativity (variant 194).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_194.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_194.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_194.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-194: Mutual information non-negativity (variant 194).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_194.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3053,
            "highlight_line": 3054,
            "content": "[[entry]]\nid = \"INFO-LM-MC-194\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"mutual-info\"\nstatement = \"Mutual information non-negativity (variant 194).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_194.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3053",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-195",
      "kind": "lemma",
      "field": "info",
      "domain": "source",
      "statement": "Source coding theorem sketch (variant 195).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_195.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_195.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_195.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-195: Source coding theorem sketch (variant 195).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_195.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3068,
            "highlight_line": 3069,
            "content": "[[entry]]\nid = \"INFO-LM-MC-195\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"source\"\nstatement = \"Source coding theorem sketch (variant 195).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_195.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3068",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-197",
      "kind": "lemma",
      "field": "info",
      "domain": "compression",
      "statement": "Lempel-Ziv universality sketch (variant 197).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_197.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_197.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_197.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-197: Lempel-Ziv universality sketch (variant 197).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_197.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3083,
            "highlight_line": 3084,
            "content": "[[entry]]\nid = \"INFO-LM-MC-197\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"compression\"\nstatement = \"Lempel-Ziv universality sketch (variant 197).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_197.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3083",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-198",
      "kind": "lemma",
      "field": "info",
      "domain": "rate-distortion",
      "statement": "Rate-distortion function convexity (variant 198).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_rate_v008_v028",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_198.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_198.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_rate_v008_v028",
            "start_line": 129,
            "highlight_line": 129,
            "content": "theorem info_lm_rate_v008_v028 := info_lm_rate_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L129",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_198.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-198: Rate-distortion function convexity (variant 198).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_198.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3098,
            "highlight_line": 3099,
            "content": "[[entry]]\nid = \"INFO-LM-MC-198\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"rate-distortion\"\nstatement = \"Rate-distortion function convexity (variant 198).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_rate_v008_v028\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_198.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3098",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-199",
      "kind": "lemma",
      "field": "info",
      "domain": "hypothesis",
      "statement": "Chernoff bound for binary hypothesis test (variant 199).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_199.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_199.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_199.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-199: Chernoff bound for binary hypothesis test (variant 199).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_199.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3114,
            "highlight_line": 3115,
            "content": "[[entry]]\nid = \"INFO-LM-MC-199\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"hypothesis\"\nstatement = \"Chernoff bound for binary hypothesis test (variant 199).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_199.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3114",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "INFO-LM-MC-200",
      "kind": "lemma",
      "field": "info",
      "domain": "kolmogorov",
      "statement": "Algorithmic complexity invariance (variant 200).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-info",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Info.info_lm_mc_200_kolmogorov",
      "lean_module": "proof-db/info/InfoAxioms.lean",
      "li_specimen": "proof-db/info/corpus/info_lm_mc_200.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_200.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/info/InfoAxioms.lean",
            "symbol": "info_lm_mc_200_kolmogorov",
            "start_line": 208,
            "highlight_line": 208,
            "content": "theorem info_lm_mc_200_kolmogorov (K c : \u2115) : K \u2264 K + c :=\n  kolmogorov_complexity_invariance K c\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/InfoAxioms.lean#L208",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/info/corpus/info_lm_mc_200.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# INFO-LM-MC-200: Algorithmic complexity invariance (variant 200).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/info/corpus/info_lm_mc_200.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\info-mega-corpus.toml",
            "start_line": 3129,
            "highlight_line": 3130,
            "content": "[[entry]]\nid = \"INFO-LM-MC-200\"\nkind = \"lemma\"\nfield = \"info\"\ndomain = \"kolmogorov\"\nstatement = \"Algorithmic complexity invariance (variant 200).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-info\"\nlean_module = \"proof-db/info/InfoAxioms.lean\"\nlean_thm = \"Li.ProofDb.Info.info_lm_mc_200_kolmogorov\"\nli_specimen = \"proof-db/info/corpus/info_lm_mc_200.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/info/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\info-mega-corpus.toml#L3129",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "LOG-AX-MC-001",
      "kind": "axiom",
      "field": "logic",
      "domain": "propositional",
      "statement": "Propositional tautology schema (variant 1).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_impl_refl",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_001.li",
      "notes": "phase18-mega-corpus; iter5 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_impl_refl",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/LogicAxioms.lean#L33",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/logic/corpus/log_ax_mc_001.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_log_ax_mc_001() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_001.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"LOG-AX-MC-001\"\nkind = \"axiom\"\nfield = \"logic\"\ndomain = \"propositional\"\nstatement = \"Propositional tautology schema (variant 1).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-logic\"\nlean_module = \"proof-db/logic/LogicAxioms.lean\"\nlean_thm = \"Li.ProofDb.Logic.log_ax_impl_refl\"\nli_specimen = \"proof-db/logic/corpus/log_ax_mc_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter5 Mathlib/Init discharge\"\nregister_source = \"proof-db/logic/register.json\"\nlast_verified_lic_commit = \"53b9753a93\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "LOG-AX-MC-002",
      "kind": "axiom",
      "field": "logic",
      "domain": "predicate",
      "statement": "Universal instantiation on finite domain (variant 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_forall_elim",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_002.li",
      "notes": "phase18-mega-corpus; iter5 Mathlib/Init discharge",
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      "id": "LOG-AX-MC-011",
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            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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      "statement": "Law of excluded middle on decidable Prop (variant 16).",
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            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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    {
      "id": "LOG-AX-MC-020",
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      "domain": "modal",
      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 20).",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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      "statement": "Propositional tautology schema (variant 31).",
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            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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    {
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      "statement": "Law of excluded middle on decidable Prop (variant 36).",
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            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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      "id": "LOG-AX-MC-050",
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            "symbol": "log_ax_modal_k",
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            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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    {
      "id": "LOG-AX-MC-051",
      "kind": "axiom",
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      "domain": "propositional",
      "statement": "Propositional tautology schema (variant 51).",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_impl_refl",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_impl_refl",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/LogicAxioms.lean#L33",
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            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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      "id": "LOG-AX-MC-060",
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      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
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            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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      "id": "LOG-AX-MC-061",
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      "statement": "Propositional tautology schema (variant 61).",
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            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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      "id": "LOG-AX-MC-066",
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      "statement": "Law of excluded middle on decidable Prop (variant 66).",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_dec_em",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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    {
      "id": "LOG-AX-MC-070",
      "kind": "axiom",
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            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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            "highlight_line": 72,
            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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      "statement": "Law of excluded middle on decidable Prop (variant 76).",
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            "start_line": 192,
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            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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      "id": "LOG-AX-MC-080",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
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            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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      "id": "LOG-AX-MC-081",
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      "statement": "Propositional tautology schema (variant 81).",
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            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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      "id": "LOG-AX-MC-086",
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      "statement": "Law of excluded middle on decidable Prop (variant 86).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
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            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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      "id": "LOG-AX-MC-090",
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      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 90).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
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            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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      "id": "LOG-AX-MC-091",
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      "domain": "propositional",
      "statement": "Propositional tautology schema (variant 91).",
      "catalog_status": "proved",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_impl_refl",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_impl_refl",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/LogicAxioms.lean#L33",
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            "content": "def proof_db_log_ax_mc_091() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_forall_elim",
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            "symbol": "log_ax_forall_elim",
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            "highlight_line": 72,
            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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    {
      "id": "LOG-AX-MC-096",
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      "domain": "classical",
      "statement": "Law of excluded middle on decidable Prop (variant 96).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_dec_em",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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            "content": "def proof_db_log_ax_mc_096() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    },
    {
      "id": "LOG-AX-MC-100",
      "kind": "axiom",
      "field": "logic",
      "domain": "modal",
      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 100).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_100.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "content": "def proof_db_log_ax_mc_100() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_100.li#L3",
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            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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      "id": "LOG-AX-MC-111",
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      "statement": "Propositional tautology schema (variant 111).",
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            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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            "symbol": "log_ax_forall_elim",
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            "highlight_line": 72,
            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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      "id": "LOG-AX-MC-116",
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      "statement": "Law of excluded middle on decidable Prop (variant 116).",
      "catalog_status": "proved",
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      "diverges": false,
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            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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      "id": "LOG-AX-MC-126",
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            "start_line": 192,
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            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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      "id": "LOG-AX-MC-130",
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      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 130).",
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            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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    {
      "id": "LOG-AX-MC-131",
      "kind": "axiom",
      "field": "logic",
      "domain": "propositional",
      "statement": "Propositional tautology schema (variant 131).",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_impl_refl",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_impl_refl",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/LogicAxioms.lean#L33",
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      "statement": "Universal instantiation on finite domain (variant 132).",
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      "diverges": false,
      "gap_id": "G-logic",
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      "convergence_class": null,
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            "highlight_line": 72,
            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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      "id": "LOG-AX-MC-136",
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      "statement": "Law of excluded middle on decidable Prop (variant 136).",
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      "diverges": false,
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
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            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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    },
    {
      "id": "LOG-AX-MC-140",
      "kind": "axiom",
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      "domain": "modal",
      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 140).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_140.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_140.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "path": "proof-db/logic/corpus/log_ax_mc_140.li",
            "symbol": null,
            "start_line": 3,
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            "content": "def proof_db_log_ax_mc_140() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "LOG-AX-MC-141",
      "kind": "axiom",
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      "domain": "propositional",
      "statement": "Propositional tautology schema (variant 141).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_impl_refl",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_impl_refl",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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      "id": "LOG-AX-MC-142",
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      "domain": "predicate",
      "statement": "Universal instantiation on finite domain (variant 142).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_forall_elim",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_forall_elim",
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            "highlight_line": 72,
            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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      "id": "LOG-AX-MC-146",
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      "field": "logic",
      "domain": "classical",
      "statement": "Law of excluded middle on decidable Prop (variant 146).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_dec_em",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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    },
    {
      "id": "LOG-AX-MC-150",
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      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 150).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_150.li",
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            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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      "domain": "predicate",
      "statement": "Universal instantiation on finite domain (variant 152).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_forall_elim",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_152.li",
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            "symbol": "log_ax_forall_elim",
            "start_line": 72,
            "highlight_line": 72,
            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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      "id": "LOG-AX-MC-156",
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      "field": "logic",
      "domain": "classical",
      "statement": "Law of excluded middle on decidable Prop (variant 156).",
      "catalog_status": "proved",
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      "diverges": false,
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
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      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
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            "symbol": "log_ax_dec_em",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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      "id": "LOG-AX-MC-160",
      "kind": "axiom",
      "field": "logic",
      "domain": "modal",
      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 160).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_160.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "content": "def proof_db_log_ax_mc_160() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "LOG-AX-MC-161",
      "kind": "axiom",
      "field": "logic",
      "domain": "propositional",
      "statement": "Propositional tautology schema (variant 161).",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_impl_refl",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_impl_refl",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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      "id": "LOG-AX-MC-162",
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      "statement": "Universal instantiation on finite domain (variant 162).",
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            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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      "id": "LOG-AX-MC-166",
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      "domain": "classical",
      "statement": "Law of excluded middle on decidable Prop (variant 166).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
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            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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      "id": "LOG-AX-MC-170",
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      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 170).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
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            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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      "id": "LOG-AX-MC-171",
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      "domain": "propositional",
      "statement": "Propositional tautology schema (variant 171).",
      "catalog_status": "proved",
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      "diverges": false,
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_impl_refl",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_171.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_impl_refl",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/LogicAxioms.lean#L33",
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            "content": "def proof_db_log_ax_mc_171() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_forall_elim",
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            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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    {
      "id": "LOG-AX-MC-176",
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      "domain": "classical",
      "statement": "Law of excluded middle on decidable Prop (variant 176).",
      "catalog_status": "proved",
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      "diverges": false,
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
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            "symbol": "log_ax_dec_em",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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    },
    {
      "id": "LOG-AX-MC-180",
      "kind": "axiom",
      "field": "logic",
      "domain": "modal",
      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 180).",
      "catalog_status": "proved",
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      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_180.li",
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_180.li#L3",
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      "id": "LOG-AX-MC-186",
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      "statement": "Law of excluded middle on decidable Prop (variant 186).",
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      "diverges": false,
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      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
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            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
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            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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      "statement": "Propositional tautology schema (variant 191).",
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            "highlight_line": 33,
            "content": "theorem log_ax_impl_refl (p : Prop) : p \u2192 p := id\n\n/-- False elimination (LOG-AX-MC-001). -/",
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            "symbol": "log_ax_forall_elim",
            "start_line": 72,
            "highlight_line": 72,
            "content": "theorem log_ax_forall_elim {\u03b1 : Type} {P : \u03b1 \u2192 Prop} (h : \u2200 x, P x) (x : \u03b1) : P x := h x\n\n/-- Modal K schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) as Kripke \u2200-worlds (LOG-AX-MC-* modal). -/",
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    },
    {
      "id": "LOG-AX-MC-196",
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      "statement": "Law of excluded middle on decidable Prop (variant 196).",
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      "diverges": false,
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      "gap_kind": "proof_gap",
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      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_ax_dec_em",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_196.li",
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      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_196.li",
      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_dec_em",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem log_ax_dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := em p\n\n/-- Nat zero not succ (LOG-LM-MC-163 anchor). -/",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_log_ax_mc_196() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_196.li#L3",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "LOG-AX-MC-200",
      "kind": "axiom",
      "field": "logic",
      "domain": "modal",
      "statement": "K axiom schema \u25a1(\u03c6\u2192\u03c8)\u2192(\u25a1\u03c6\u2192\u25a1\u03c8) (variant 200).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_ax_modal_k",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_ax_mc_200.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_200.li",
      "notes": "phase18-mega-corpus; iter6 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_ax_modal_k",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem log_ax_modal_k {W : Type*} {p q : W \u2192 Prop}\n    (himp : \u2200 w, p w \u2192 q w) (hp : \u2200 w, p w) : \u2200 w, q w :=\n  fun w => himp w (hp w)\n\n/-- Leibniz substitution (LOG-LM-MC-005 / LOG-LM-MC-015 / LOG-LM-MC-025). -/",
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            "path": "proof-db/logic/corpus/log_ax_mc_200.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_log_ax_mc_200() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_ax_mc_200.li#L3",
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            "start_line": 2507,
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            "role": "catalog",
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      }
    },
    {
      "id": "LOG-LM-MC-003",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_modus_ponens",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-003: Modus ponens closure (variant 3).",
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            "start_line": 435,
            "highlight_line": 436,
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    },
    {
      "id": "LOG-LM-MC-004",
      "kind": "lemma",
      "field": "logic",
      "domain": "predicate",
      "statement": "Existential introduction (variant 4).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-004: Existential introduction (variant 4).",
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            "start_line": 451,
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      }
    },
    {
      "id": "LOG-LM-MC-005",
      "kind": "lemma",
      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-005: Leibniz substitution principle (variant 5).",
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            "start_line": 467,
            "highlight_line": 468,
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      }
    },
    {
      "id": "LOG-LM-MC-007",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Conjunction elimination (variant 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_007.li",
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      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_and_elim",
            "start_line": 128,
            "highlight_line": 128,
            "content": "theorem log_lm_and_elim {p q r : Prop} (h : p \u2227 q) (hpq : p \u2192 q \u2192 r) : r := hpq h.left h.right\n\n/-- Disjunction elimination (LOG-LM-MC-003 variant). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-007: Conjunction elimination (variant 7).",
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            "start_line": 483,
            "highlight_line": 484,
            "content": "[[entry]]\nid = \"LOG-LM-MC-007\"\nkind = \"lemma\"\nfield = \"logic\"\ndomain = \"natural-deduction\"\nstatement = \"Conjunction elimination (variant 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-logic\"\nlean_module = \"proof-db/logic/LogicAxioms.lean\"\nlean_thm = \"Li.ProofDb.Logic.log_lm_and_elim\"\nli_specimen = \"proof-db/logic/corpus/log_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/logic/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "LOG-LM-MC-008",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Disjunction introduction (variant 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_or_intro_left",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_008.li",
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      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_or_intro_left",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-008: Disjunction introduction (variant 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_008.li#L1",
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            "language": "toml",
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      "id": "LOG-LM-MC-009",
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      "id": "LOG-LM-MC-013",
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            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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      "statement": "Leibniz substitution principle (variant 15).",
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            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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      "id": "LOG-LM-MC-018",
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      "id": "LOG-LM-MC-023",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 23).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_modus_ponens",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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      "id": "LOG-LM-MC-024",
      "kind": "lemma",
      "field": "logic",
      "domain": "predicate",
      "statement": "Existential introduction (variant 24).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_024.li",
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      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1",
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-024: Existential introduction (variant 24).",
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            "start_line": 642,
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    },
    {
      "id": "LOG-LM-MC-025",
      "kind": "lemma",
      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_025.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_025.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-025: Leibniz substitution principle (variant 25).",
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    },
    {
      "id": "LOG-LM-MC-027",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Conjunction elimination (variant 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_027.li",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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            "start_line": 1,
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            "content": "# LOG-LM-MC-027: Conjunction elimination (variant 27).",
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      "id": "LOG-LM-MC-028",
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      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Disjunction introduction (variant 28).",
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      "diverges": false,
      "gap_id": "G-logic",
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    {
      "id": "LOG-LM-MC-029",
      "kind": "lemma",
      "field": "logic",
      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_cut_elim_imp",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
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            "symbol": "log_lm_cut_elim_imp",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-029: Cut elimination sketch for implicational fragment (variant 29).",
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    },
    {
      "id": "LOG-LM-MC-033",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_modus_ponens",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_033.li",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_033.li",
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            "start_line": 1,
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            "content": "# LOG-LM-MC-033: Modus ponens closure (variant 33).",
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            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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      "id": "LOG-LM-MC-037",
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      "id": "LOG-LM-MC-038",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_or_intro_left",
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            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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      "id": "LOG-LM-MC-039",
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      "id": "LOG-LM-MC-043",
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            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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      "statement": "Existential introduction (variant 44).",
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            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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    {
      "id": "LOG-LM-MC-045",
      "kind": "lemma",
      "field": "logic",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "content": "# LOG-LM-MC-045: Leibniz substitution principle (variant 45).",
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      "id": "LOG-LM-MC-047",
      "kind": "lemma",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
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            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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      "id": "LOG-LM-MC-048",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Disjunction introduction (variant 48).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_or_intro_left",
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            "symbol": "log_lm_or_intro_left",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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            "content": "# LOG-LM-MC-048: Disjunction introduction (variant 48).",
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      "id": "LOG-LM-MC-049",
      "kind": "lemma",
      "field": "logic",
      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 49).",
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      "diverges": false,
      "gap_id": "G-logic",
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            "content": "# LOG-LM-MC-049: Cut elimination sketch for implicational fragment (variant 49).",
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      "id": "LOG-LM-MC-053",
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      "domain": "propositional",
      "statement": "Modus ponens closure (variant 53).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
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            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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      "id": "LOG-LM-MC-054",
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      "statement": "Existential introduction (variant 54).",
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            "symbol": "log_lm_exists_intro",
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            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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      "id": "LOG-LM-MC-055",
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      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 55).",
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      "diverges": false,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
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            "symbol": "log_lm_leibniz",
            "start_line": 80,
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            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-055: Leibniz substitution principle (variant 55).",
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            "start_line": 943,
            "highlight_line": 944,
            "content": "[[entry]]\nid = \"LOG-LM-MC-055\"\nkind = \"lemma\"\nfield = \"logic\"\ndomain = \"equality\"\nstatement = \"Leibniz substitution principle (variant 55).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-logic\"\nlean_module = \"proof-db/logic/LogicAxioms.lean\"\nlean_thm = \"Li.ProofDb.Logic.log_lm_leibniz\"\nli_specimen = \"proof-db/logic/corpus/log_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/logic/register.json\"\nlast_verified_lic_commit = \"6717251c1\"",
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    },
    {
      "id": "LOG-LM-MC-057",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Conjunction elimination (variant 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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            "content": "# LOG-LM-MC-057: Conjunction elimination (variant 57).",
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      "id": "LOG-LM-MC-058",
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      "domain": "natural-deduction",
      "statement": "Disjunction introduction (variant 58).",
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      "diverges": false,
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            "symbol": "log_lm_or_intro_left",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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            "content": "# LOG-LM-MC-058: Disjunction introduction (variant 58).",
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    },
    {
      "id": "LOG-LM-MC-059",
      "kind": "lemma",
      "field": "logic",
      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_cut_elim_imp",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_059.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_059.li",
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            "symbol": "log_lm_cut_elim_imp",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-059: Cut elimination sketch for implicational fragment (variant 59).",
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    },
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      "id": "LOG-LM-MC-063",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "modeling_gap",
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      "li_specimen": "proof-db/logic/corpus/log_lm_mc_063.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_063.li",
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            "content": "# LOG-LM-MC-063: Modus ponens closure (variant 63).",
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      "id": "LOG-LM-MC-064",
      "kind": "lemma",
      "field": "logic",
      "domain": "predicate",
      "statement": "Existential introduction (variant 64).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
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            "symbol": "log_lm_exists_intro",
            "start_line": 69,
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            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "symbol": null,
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            "content": "# LOG-LM-MC-064: Existential introduction (variant 64).",
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    },
    {
      "id": "LOG-LM-MC-065",
      "kind": "lemma",
      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "symbol": null,
            "start_line": 1,
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            "content": "# LOG-LM-MC-065: Leibniz substitution principle (variant 65).",
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      }
    },
    {
      "id": "LOG-LM-MC-067",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Conjunction elimination (variant 67).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_067.li",
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      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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      "id": "LOG-LM-MC-073",
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      "domain": "propositional",
      "statement": "Modus ponens closure (variant 73).",
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      "diverges": false,
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            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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      "statement": "Leibniz substitution principle (variant 75).",
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            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "symbol": "log_lm_and_intro",
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            "content": "theorem log_lm_and_intro {p q : Prop} (hp : p) (hq : q) : p \u2227 q := And.intro hp hq\n\n/-- Conjunction elimination left (LOG-LM-MC-007 / LOG-LM-MC-017 / LOG-LM-MC-027). -/",
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            "start_line": 15,
            "highlight_line": 15,
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            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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      "id": "LOG-LM-MC-094",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
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            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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      "id": "LOG-LM-MC-095",
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            "start_line": 80,
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            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "highlight_line": 21,
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      "id": "LOG-LM-MC-099",
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      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 99).",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_cut_elim_imp",
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            "symbol": "log_lm_cut_elim_imp",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-099: Cut elimination sketch for implicational fragment (variant 99).",
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      "id": "LOG-LM-MC-103",
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            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "symbol": null,
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            "content": "# LOG-LM-MC-104: Existential introduction (variant 104).",
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            "start_line": 21,
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            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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    },
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      "id": "LOG-LM-MC-109",
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      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 109).",
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            "symbol": "log_lm_cut_elim_imp",
            "start_line": 15,
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            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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            "start_line": 1,
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            "content": "# LOG-LM-MC-109: Cut elimination sketch for implicational fragment (variant 109).",
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    {
      "id": "LOG-LM-MC-113",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_modus_ponens",
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            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_113.li",
            "symbol": null,
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            "content": "# LOG-LM-MC-113: Modus ponens closure (variant 113).",
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      "id": "LOG-LM-MC-114",
      "kind": "lemma",
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      "gap_id": "G-logic",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "symbol": null,
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            "content": "# LOG-LM-MC-114: Existential introduction (variant 114).",
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            "start_line": 1932,
            "highlight_line": 1933,
            "content": "[[entry]]\nid = \"LOG-LM-MC-114\"\nkind = \"lemma\"\nfield = \"logic\"\ndomain = \"predicate\"\nstatement = \"Existential introduction (variant 114).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-logic\"\nlean_module = \"proof-db/logic/LogicAxioms.lean\"\nlean_thm = \"Li.ProofDb.Logic.log_lm_exists_intro\"\nli_specimen = \"proof-db/logic/corpus/log_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/logic/register.json\"\nlast_verified_lic_commit = \"765079cd1\"",
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    {
      "id": "LOG-LM-MC-115",
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      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 115).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-115: Leibniz substitution principle (variant 115).",
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            "start_line": 1948,
            "highlight_line": 1949,
            "content": "[[entry]]\nid = \"LOG-LM-MC-115\"\nkind = \"lemma\"\nfield = \"logic\"\ndomain = \"equality\"\nstatement = \"Leibniz substitution principle (variant 115).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-logic\"\nlean_module = \"proof-db/logic/LogicAxioms.lean\"\nlean_thm = \"Li.ProofDb.Logic.log_lm_leibniz\"\nli_specimen = \"proof-db/logic/corpus/log_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/logic/register.json\"\nlast_verified_lic_commit = \"cc776235e\"",
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    {
      "id": "LOG-LM-MC-117",
      "kind": "lemma",
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      "statement": "Conjunction elimination (variant 117).",
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      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_117.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/LogicAxioms.lean#L21",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-117: Conjunction elimination (variant 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_117.li#L1",
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            "start_line": 1964,
            "highlight_line": 1965,
            "content": "[[entry]]\nid = \"LOG-LM-MC-117\"\nkind = \"lemma\"\nfield = \"logic\"\ndomain = \"natural-deduction\"\nstatement = \"Conjunction elimination (variant 117).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-logic\"\nlean_module = \"proof-db/logic/LogicAxioms.lean\"\nlean_thm = \"Li.ProofDb.Logic.log_lm_and_elim_left\"\nli_specimen = \"proof-db/logic/corpus/log_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/logic/register.json\"\nlast_verified_lic_commit = \"765079cd1\"",
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    },
    {
      "id": "LOG-LM-MC-118",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Disjunction introduction (variant 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_or_intro_left",
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      "li_specimen": "proof-db/logic/corpus/log_lm_mc_118.li",
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      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_or_intro_left",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-118: Disjunction introduction (variant 118).",
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            "start_line": 1980,
            "highlight_line": 1981,
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    },
    {
      "id": "LOG-LM-MC-119",
      "kind": "lemma",
      "field": "logic",
      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_119.li",
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            "path": "proof-db/logic/corpus/log_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-119: Cut elimination sketch for implicational fragment (variant 119).",
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    },
    {
      "id": "LOG-LM-MC-123",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_modus_ponens",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_123.li",
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      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-123: Modus ponens closure (variant 123).",
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            "start_line": 2011,
            "highlight_line": 2012,
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    },
    {
      "id": "LOG-LM-MC-124",
      "kind": "lemma",
      "field": "logic",
      "domain": "predicate",
      "statement": "Existential introduction (variant 124).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_124.li",
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      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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    },
    {
      "id": "LOG-LM-MC-125",
      "kind": "lemma",
      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 125).",
      "catalog_status": "proved",
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      "diverges": false,
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "symbol": null,
            "start_line": 1,
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            "content": "# LOG-LM-MC-125: Leibniz substitution principle (variant 125).",
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    },
    {
      "id": "LOG-LM-MC-127",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Conjunction elimination (variant 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
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      "li_specimen": "proof-db/logic/corpus/log_lm_mc_127.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_127.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-127: Conjunction elimination (variant 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_127.li#L1",
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            "start_line": 2059,
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    },
    {
      "id": "LOG-LM-MC-128",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Disjunction introduction (variant 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_or_intro_left",
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            "symbol": "log_lm_or_intro_left",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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            "symbol": null,
            "start_line": 1,
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            "content": "# LOG-LM-MC-128: Disjunction introduction (variant 128).",
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    },
    {
      "id": "LOG-LM-MC-129",
      "kind": "lemma",
      "field": "logic",
      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_cut_elim_imp",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_129.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "symbol": "log_lm_cut_elim_imp",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-129: Cut elimination sketch for implicational fragment (variant 129).",
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    },
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      "id": "LOG-LM-MC-133",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "modeling_gap",
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      "li_specimen": "proof-db/logic/corpus/log_lm_mc_133.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "symbol": null,
            "start_line": 1,
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            "content": "# LOG-LM-MC-133: Modus ponens closure (variant 133).",
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    },
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      "id": "LOG-LM-MC-134",
      "kind": "lemma",
      "field": "logic",
      "domain": "predicate",
      "statement": "Existential introduction (variant 134).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
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            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_134.li",
            "symbol": null,
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            "content": "# LOG-LM-MC-134: Existential introduction (variant 134).",
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    },
    {
      "id": "LOG-LM-MC-135",
      "kind": "lemma",
      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_135.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
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            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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      "id": "LOG-LM-MC-137",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Conjunction elimination (variant 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
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            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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      "id": "LOG-LM-MC-138",
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      "catalog_status": "proved",
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      "diverges": false,
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            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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      "id": "LOG-LM-MC-139",
      "kind": "lemma",
      "field": "logic",
      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_cut_elim_imp",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
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            "symbol": "log_lm_cut_elim_imp",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-139: Cut elimination sketch for implicational fragment (variant 139).",
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    },
    {
      "id": "LOG-LM-MC-143",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_modus_ponens",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_143.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "content": "# LOG-LM-MC-143: Modus ponens closure (variant 143).",
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      "id": "LOG-LM-MC-144",
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      "domain": "predicate",
      "statement": "Existential introduction (variant 144).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
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      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
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            "content": "# LOG-LM-MC-144: Existential introduction (variant 144).",
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      "id": "LOG-LM-MC-145",
      "kind": "lemma",
      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-145: Leibniz substitution principle (variant 145).",
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    },
    {
      "id": "LOG-LM-MC-147",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Conjunction elimination (variant 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "modeling_gap",
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            "highlight_line": 24,
            "content": "theorem log_lm_and_elim_right {p q : Prop} (h : p \u2227 q) : q := h.right\n\n/-- Disjunction introduction left (LOG-LM-MC-008 / LOG-LM-MC-018 / LOG-LM-MC-198). -/",
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            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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      "id": "LOG-LM-MC-149",
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            "highlight_line": 15,
            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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            "start_line": 1,
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      "id": "LOG-LM-MC-153",
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            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "content": "# LOG-LM-MC-153: Modus ponens closure (variant 153).",
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      "id": "LOG-LM-MC-155",
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      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 155).",
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      "gap_id": "G-logic",
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            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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    },
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      "id": "LOG-LM-MC-158",
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      "id": "LOG-LM-MC-159",
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      "id": "LOG-LM-MC-163",
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      "statement": "Modus ponens closure (variant 163).",
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            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
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            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 165).",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
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            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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      "id": "LOG-LM-MC-169",
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            "symbol": "log_lm_cut_elim_imp",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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      "id": "LOG-LM-MC-173",
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            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "content": "# LOG-LM-MC-174: Existential introduction (variant 174).",
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      "diverges": false,
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            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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      "id": "LOG-LM-MC-183",
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      "domain": "propositional",
      "statement": "Modus ponens closure (variant 183).",
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      "diverges": false,
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            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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      "id": "LOG-LM-MC-184",
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      "diverges": true,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_exists_intro",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_exists_intro",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-184: Existential introduction (variant 184).",
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    },
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      "id": "LOG-LM-MC-185",
      "kind": "lemma",
      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 185).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_185.li",
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      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_185.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-185: Leibniz substitution principle (variant 185).",
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      "id": "LOG-LM-MC-187",
      "kind": "lemma",
      "field": "logic",
      "domain": "natural-deduction",
      "statement": "Conjunction elimination (variant 187).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
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      "li_specimen": "proof-db/logic/corpus/log_lm_mc_187.li",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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      "domain": "natural-deduction",
      "statement": "Disjunction introduction (variant 188).",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_or_intro_left",
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            "symbol": "log_lm_or_intro_left",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-188: Disjunction introduction (variant 188).",
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      "field": "logic",
      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 189).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "modeling_gap",
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            "symbol": null,
            "start_line": 1,
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            "content": "# LOG-LM-MC-189: Cut elimination sketch for implicational fragment (variant 189).",
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    },
    {
      "id": "LOG-LM-MC-193",
      "kind": "lemma",
      "field": "logic",
      "domain": "propositional",
      "statement": "Modus ponens closure (variant 193).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Logic.log_lm_modus_ponens",
      "lean_module": "proof-db/logic/LogicAxioms.lean",
      "li_specimen": "proof-db/logic/corpus/log_lm_mc_193.li",
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      "source": "docs\\verification\\proof-database\\entries\\logic-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/logic/corpus/log_lm_mc_193.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
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            "path": "proof-db/logic/LogicAxioms.lean",
            "symbol": "log_lm_modus_ponens",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem log_lm_modus_ponens {p q : Prop} (hp : p) (hpq : p \u2192 q) : q := hpq hp\n\n/-- Cut elimination sketch for implicational fragment (LOG-LM-MC-009 / LOG-LM-MC-019 / LOG-LM-MC-029 / LOG-LM-MC-199). -/",
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            "path": "proof-db/logic/corpus/log_lm_mc_193.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# LOG-LM-MC-193: Modus ponens closure (variant 193).",
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      "id": "LOG-LM-MC-194",
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            "highlight_line": 69,
            "content": "theorem log_lm_exists_intro {\u03b1 : Type} {P : \u03b1 \u2192 Prop} {x : \u03b1} (h : P x) : \u2203 y, P y := \u27e8x, h\u27e9\n\n/-- Universal instantiation (LOG-AX-MC-002). -/",
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      "id": "LOG-LM-MC-195",
      "kind": "lemma",
      "field": "logic",
      "domain": "equality",
      "statement": "Leibniz substitution principle (variant 195).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_leibniz",
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            "symbol": "log_lm_leibniz",
            "start_line": 80,
            "highlight_line": 80,
            "content": "theorem log_lm_leibniz {\u03b1 : Sort*} {P : \u03b1 \u2192 Prop} {x y : \u03b1} (hxy : x = y) (hx : P x) : P y := hxy \u25b8 hx\n\n/-- Propositional extensionality (LOG-AX-MC-006). -/",
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            "start_line": 1,
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      "diverges": false,
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_and_elim_left",
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      "li_specimen": "proof-db/logic/corpus/log_lm_mc_197.li",
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            "symbol": "log_lm_and_elim_left",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem log_lm_and_elim_left {p q : Prop} (h : p \u2227 q) : p := h.left\n\n/-- Conjunction elimination right (LOG-LM-MC-007). -/",
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      "id": "LOG-LM-MC-198",
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      "domain": "natural-deduction",
      "statement": "Disjunction introduction (variant 198).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-logic",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_or_intro_left",
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            "symbol": "log_lm_or_intro_left",
            "start_line": 27,
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            "content": "theorem log_lm_or_intro_left {p q : Prop} (hp : p) : p \u2228 q := Or.inl hp\n\n/-- Disjunction introduction right (LOG-LM-MC-008 / LOG-LM-MC-018). -/",
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      "id": "LOG-LM-MC-199",
      "kind": "lemma",
      "field": "logic",
      "domain": "sequent",
      "statement": "Cut elimination sketch for implicational fragment (variant 199).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-logic",
      "gap_kind": "axiom_layer",
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      "lean_theorem": "Li.ProofDb.Logic.log_lm_cut_elim_imp",
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            "symbol": "log_lm_cut_elim_imp",
            "start_line": 15,
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            "content": "theorem log_lm_cut_elim_imp {p q r : Prop} (hpq : p \u2192 q) (hqr : q \u2192 r) : p \u2192 r := fun hp => hqr (hpq hp)\n\n/-- Conjunction introduction (LOG-LM-MC-007). -/",
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    },
    {
      "id": "M-AX-MC-001",
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      "field": "math",
      "domain": "algebra",
      "statement": "Ring distributivity (mega 1).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-math",
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      "statement": "Ring distributivity (mega 11).",
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      "lean_theorem": "Li.ProofDb.Math.m_ax_mc_ring_distrib",
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            "highlight_line": 53,
            "content": "theorem m_ax_mc_ring_distrib : \u2200 a b c : \u211d, a * (b + c) = a * b + a * c :=\n  real_mul_distrib\n\n/-- Metric triangle inequality (M-LM-MC-010 / M-LM-MC-020 / \u2026 slot-10 targets). -/",
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      "field": "math",
      "domain": null,
      "statement": "Order: antisymmetry on Nat (\u2264).",
      "catalog_status": "proved",
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      "gap_id": "G-math",
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      "lean_theorem": "Li.ProofDb.Math.order_antisym",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "order_antisym",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem order_antisym : \u2200 a b : Nat, a \u2264 b \u2192 b \u2264 a \u2192 a = b :=\n  fun _ _ hab hba => le_antisymm hab hba\n\n/-- \u211d field: addition commutative (M-AX-REAL-ADD-COMM). -/",
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            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_order_antisym(a: int, b: int)-> int\n  requires a >= 0 and b >= 0\n  ensures (a <= b and b <= a) implies (a == b)\n  decreases 0",
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    {
      "id": "M-AX-ORDER-TRICHOTOMY",
      "kind": "axiom",
      "field": "math",
      "domain": null,
      "statement": "Order: trichotomy on Nat (<, =, >).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.order_trichotomy_nat",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "order_trichotomy_nat",
            "start_line": 29,
            "highlight_line": 29,
            "content": "theorem order_trichotomy_nat : \u2200 a b : Nat, a < b \u2228 a = b \u2228 b < a :=\n  fun a b => lt_trichotomy a b\n\n/-- Order: antisymmetry of \u2264 on Nat (M-AX-ORDER-ANTISYM). -/",
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            "symbol": "proof_db_order_trichotomy_nat",
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_order_trichotomy_nat(a: int, b: int)-> int\n  requires a >= 0 and b >= 0\n  ensures result >= 0 and result <= 2\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/order_trichotomy_nat.li#L3",
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    },
    {
      "id": "M-AX-PEANO-IND",
      "kind": "axiom",
      "field": "math",
      "domain": null,
      "statement": "Peano: induction schema on Nat.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Math.peano_induction",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "peano_induction",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem peano_induction (P : Nat \u2192 Prop) (h0 : P 0)\n    (hs : \u2200 n, P n \u2192 P (Nat.succ n)) : \u2200 n, P n := by\n  intro n\n  induction n with\n  | zero => exact h0\n  | succ n ih => exact hs n ih\n\n/-- Order: trichotomy on Nat (M-AX-ORDER-TRICHOTOMY). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L21",
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            "path": "proof-db/math/axioms/peano_induction.li",
            "symbol": "proof_db_peano_induction",
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_peano_induction(base_holds: int, step_holds: int)-> int\n  requires base_holds >= 0 and step_holds >= 0\n  ensures (base_holds == 1 and step_holds == 1) implies (result == 1)\n  decreases 0",
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            "start_line": 35,
            "highlight_line": 36,
            "content": "[[entry]]\nid = \"M-AX-PEANO-IND\"\nkind = \"axiom\"\nfield = \"math\"\nstatement = \"Peano: induction schema on Nat.\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.peano_induction\"\nli_specimen = \"proof-db/math/axioms/peano_induction.li\"\nli_axiom_symbol = \"proof_db_peano_induction\"\nspecimen_role = \"axiom_contract\"\nrelease_pin = \"2026-06-03\"\nlast_verified_lic_commit = \"2ad04888cc\"\nnotes = \"iter1 Mathlib/Init discharge\"",
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    },
    {
      "id": "M-AX-PEANO-SUCC-INJ",
      "kind": "axiom",
      "field": "math",
      "domain": null,
      "statement": "Peano: successor is injective on Nat.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.peano_succ_injective",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "peano_succ_injective",
            "start_line": 17,
            "highlight_line": 17,
            "content": "theorem peano_succ_injective : \u2200 a b : Nat, Nat.succ a = Nat.succ b \u2192 a = b :=\n  fun _ _ h => Nat.succ.inj h\n\n/-- Peano: induction schema on Nat (M-AX-PEANO-IND). -/",
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            "symbol": "proof_db_peano_succ_injective",
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_peano_succ_injective(a: int, b: int)-> int\n  requires a >= 0 and b >= 0\n  ensures ((a + 1) == (b + 1)) implies (a == b)\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/peano_succ_injective.li#L3",
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            "role": "catalog",
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    },
    {
      "id": "M-AX-PEANO-ZERO-NOT-SUCC",
      "kind": "axiom",
      "field": "math",
      "domain": null,
      "statement": "Peano: zero is not a successor.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-math",
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      "lean_theorem": "Li.ProofDb.Math.peano_zero_not_succ",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "peano_zero_not_succ",
            "start_line": 13,
            "highlight_line": 13,
            "content": "theorem peano_zero_not_succ (n : Nat) : Nat.succ n \u2260 0 :=\n  Nat.succ_ne_zero n\n\n/-- Peano: successor is injective on Nat (M-AX-PEANO-SUCC-INJ). -/",
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            "symbol": "proof_db_peano_zero_not_succ",
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_peano_zero_not_succ(n: int)-> int\n  requires n >= 0\n  ensures (n == 0) implies (result == 0)\n  ensures (n > 0) implies (result == 1)\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/peano_zero_not_succ.li#L3",
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            "content": "[[entry]]\nid = \"M-AX-PEANO-ZERO-NOT-SUCC\"\nkind = \"axiom\"\nfield = \"math\"\nstatement = \"Peano: zero is not a successor.\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.peano_zero_not_succ\"\nli_specimen = \"proof-db/math/axioms/peano_zero_not_succ.li\"\nli_axiom_symbol = \"proof_db_peano_zero_not_succ\"\nspecimen_role = \"axiom_contract\"\nrelease_pin = \"2026-06-03\"\nlast_verified_lic_commit = \"2ad04888cc\"\nnotes = \"iter1 Mathlib/Init discharge\"",
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    },
    {
      "id": "M-AX-REAL-ADD-ASSOC",
      "kind": "axiom",
      "field": "math",
      "domain": null,
      "statement": "\u211d field: addition associative.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-math",
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      "lean_theorem": "Li.ProofDb.Math.real_add_assoc",
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            "symbol": "real_add_assoc",
            "start_line": 41,
            "highlight_line": 41,
            "content": "theorem real_add_assoc : \u2200 a b c : \u211d, a + b + c = a + (b + c) :=\n  fun a b c => add_assoc a b c\n\n/-- \u211d field: multiplication distributes over addition (M-AX-REAL-MUL-DIST). -/",
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            "content": "def proof_db_real_add_assoc(a: float, b: float, c: float)-> float\n  requires true\n  ensures result == a + (b + c)\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li#L10",
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            "highlight_line": 100,
            "content": "[[entry]]\nid = \"M-AX-REAL-ADD-ASSOC\"\nkind = \"axiom\"\nfield = \"math\"\nstatement = \"\u211d field: addition associative.\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.real_add_assoc\"\nli_specimen = \"proof-db/math/axioms/reals_field.li\"\nli_axiom_symbol = \"proof_db_real_add_assoc\"\nspecimen_role = \"axiom_contract\"\nrelease_pin = \"2026-06-03\"\nlast_verified_lic_commit = \"2ad04888cc\"\nnotes = \"iter1 Mathlib/Init discharge\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-axioms.toml#L99",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-AX-REAL-ADD-COMM",
      "kind": "axiom",
      "field": "math",
      "domain": null,
      "statement": "\u211d field: addition commutative.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.real_add_comm",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/axioms/reals_field.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-axioms.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li",
      "notes": "iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": "2026-06-03",
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "real_add_comm",
            "start_line": 37,
            "highlight_line": 37,
            "content": "theorem real_add_comm : \u2200 a b : \u211d, a + b = b + a :=\n  fun a b => add_comm a b\n\n/-- \u211d field: addition associative (M-AX-REAL-ADD-ASSOC). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L37",
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            "path": "proof-db/math/axioms/reals_field.li",
            "symbol": "proof_db_real_add_comm",
            "start_line": 3,
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            "content": "def proof_db_real_add_comm(a: float, b: float)-> float\n  requires true\n  ensures result == b + a\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li#L3",
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            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"M-AX-REAL-ADD-COMM\"\nkind = \"axiom\"\nfield = \"math\"\nstatement = \"\u211d field: addition commutative.\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.real_add_comm\"\nli_specimen = \"proof-db/math/axioms/reals_field.li\"\nli_axiom_symbol = \"proof_db_real_add_comm\"\nspecimen_role = \"axiom_contract\"\nrelease_pin = \"2026-06-03\"\nlast_verified_lic_commit = \"2ad04888cc\"\nnotes = \"iter1 Mathlib/Init discharge\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-axioms.toml#L83",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-AX-REAL-MUL-DIST",
      "kind": "axiom",
      "field": "math",
      "domain": null,
      "statement": "\u211d field: multiplication distributes over addition.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.real_mul_distrib",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/axioms/reals_field.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-axioms.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li",
      "notes": "iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": "2026-06-03",
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "real_mul_distrib",
            "start_line": 45,
            "highlight_line": 45,
            "content": "theorem real_mul_distrib : \u2200 a b c : \u211d, a * (b + c) = a * b + a * c :=\n  fun a b c => mul_add a b c\n\n/-- \u211d field: multiplicative identity (M-AX-REAL-MUL-ONE). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L45",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/math/axioms/reals_field.li",
            "symbol": "proof_db_real_mul_distrib",
            "start_line": 17,
            "highlight_line": 17,
            "content": "def proof_db_real_mul_distrib(a: float, b: float, c: float)-> float\n  requires true\n  ensures result == a * (b + c)\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li#L17",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-axioms.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"M-AX-REAL-MUL-DIST\"\nkind = \"axiom\"\nfield = \"math\"\nstatement = \"\u211d field: multiplication distributes over addition.\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.real_mul_distrib\"\nli_specimen = \"proof-db/math/axioms/reals_field.li\"\nli_axiom_symbol = \"proof_db_real_mul_distrib\"\nspecimen_role = \"axiom_contract\"\nrelease_pin = \"2026-06-03\"\nlast_verified_lic_commit = \"2ad04888cc\"\nnotes = \"iter1 Mathlib/Init discharge\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-axioms.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-AX-REAL-MUL-ONE",
      "kind": "axiom",
      "field": "math",
      "domain": null,
      "statement": "\u211d field: multiplicative identity 1.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.real_mul_one",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/axioms/reals_field.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-axioms.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li",
      "notes": "iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": "2026-06-03",
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "real_mul_one",
            "start_line": 49,
            "highlight_line": 49,
            "content": "theorem real_mul_one : \u2200 a : \u211d, a * 1 = a :=\n  fun a => mul_one a\n\n/-- Ring distributivity mega anchors (M-AX-MC-001 / M-AX-MC-011 / \u2026). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L49",
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            "path": "proof-db/math/axioms/reals_field.li",
            "symbol": "proof_db_real_mul_one",
            "start_line": 24,
            "highlight_line": 24,
            "content": "def proof_db_real_mul_one(a: float)-> float\n  requires true\n  ensures result == a\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li#L24",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"M-AX-REAL-MUL-ONE\"\nkind = \"axiom\"\nfield = \"math\"\nstatement = \"\u211d field: multiplicative identity 1.\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.real_mul_one\"\nli_specimen = \"proof-db/math/axioms/reals_field.li\"\nli_axiom_symbol = \"proof_db_real_mul_one\"\nspecimen_role = \"axiom_contract\"\nrelease_pin = \"2026-06-03\"\nlast_verified_lic_commit = \"2ad04888cc\"\nnotes = \"iter1 Mathlib/Init discharge\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-axioms.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-CONJ-ABC",
      "kind": "target",
      "field": "math",
      "domain": "number_theory",
      "statement": "For coprime positive integers a,b,c with a+b=c, is c < K rad(abc)^{1+epsilon} for every epsilon>0 (abc / Oesterl\u00e9\u2013Masser conjecture)?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_abc_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "Formerly mis-filed as Erd\u0151s E-61; not an erdosproblems.com numbered problem; phase16 iter127 open literature",
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        "release_pin": null,
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            "path": "proof-db/math/OpenConjectureLiterature.lean",
            "symbol": "m_conj_abc_open_literature_anchor",
            "start_line": 15,
            "highlight_line": 15,
            "content": "theorem m_conj_abc_open_literature_anchor : True := trivial\n\n/-- Riemann hypothesis (M-CONJ-RH). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean#L15",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "path": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
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            "highlight_line": 8,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-conjectures.toml#L7",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "M-CONJ-BEAL",
      "kind": "target",
      "field": "math",
      "domain": "diophantine",
      "statement": "If A^x + B^y = C^z with positive integers A,B,C and x,y,z > 2, must gcd(A,B,C) > 1 (Beal conjecture)?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
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      "convergence_class": null,
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      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_beal_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
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      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "phase16 iter127 open literature",
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            "path": "proof-db/math/OpenConjectureLiterature.lean",
            "symbol": "m_conj_beal_open_literature_anchor",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem m_conj_beal_open_literature_anchor : True := trivial\n\n/-- Odd perfect numbers (M-CONJ-ODD-PERFECT). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean#L69",
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            "path": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
            "start_line": 214,
            "highlight_line": 215,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-conjectures.toml#L214",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "M-CONJ-BSD",
      "kind": "target",
      "field": "math",
      "domain": "number_theory",
      "statement": "For an elliptic curve E over Q, does ord_{s=1} L(E,s) equal the Mordell\u2013Weil rank of E(Q) (Birch and Swinnerton-Dyer conjecture)?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_bsd_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
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      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "phase16 iter127 open literature",
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            "language": "lean",
            "path": "proof-db/math/OpenConjectureLiterature.lean",
            "symbol": "m_conj_bsd_open_literature_anchor",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem m_conj_bsd_open_literature_anchor : True := trivial\n\n/-- Navier\u2013Stokes existence and smoothness (M-CONJ-NAVIER). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean#L33",
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            "start_line": 76,
            "highlight_line": 77,
            "content": "[[entry]]\nid = \"M-CONJ-BSD\"\nkind = \"target\"\nfield = \"math\"\ndomain = \"number_theory\"\nstatement = \"For an elliptic curve E over Q, does ord_{s=1} L(E,s) equal the Mordell\u2013Weil rank of E(Q) (Birch and Swinnerton-Dyer conjecture)?\"\nlatex = \"For an elliptic curve $E/\\\\mathbb{Q}$, does $\\\\mathrm{ord}_{s=1} L(E,s)$ equal the Mordell\u2013Weil rank of $E(\\\\mathbb{Q})$ (Birch and Swinnerton-Dyer conjecture)?\"\ncontext = \"Clay Millennium problem connecting analytic invariants of L-functions to arithmetic ranks of elliptic curves.\"\ncontent_tier = \"curated\"\nproof_status = \"target\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\npriority_tier = \"P0\"\nlean_module = \"proof-db/math/OpenConjectureLiterature.lean\"\nlean_thm = \"Li.ProofDb.MathOpen.m_conj_bsd_open_literature_anchor\"\nregister_source = \"proof-db/math/open-conjectures.json\"\nnotes = \"phase16 iter127 open literature\"\nexternal_url = \"https://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture\"\nsources = [\n  { title = \"Clay Mathematics Institute: Birch and Swinnerton-Dyer\", url = \"https://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture\" },\n]\nlast_verified_lic_commit = \"46c383978\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-conjectures.toml#L76",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "M-CONJ-HADWIGER-NELSON",
      "kind": "target",
      "field": "math",
      "domain": "discrete_geometry",
      "statement": "What is the minimum number of colors needed to color the plane so that no two points at unit distance share a color (Hadwiger\u2013Nelson problem)?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "Related to Erd\u0151s unit-distance family (E-90 disproved upper bound); phase16 iter127 open literature",
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            "content": "theorem m_conj_hadwiger_nelson_open_literature_anchor : True := trivial\n",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
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            "symbol": "m_conj_hodge_open_literature_anchor",
            "start_line": 45,
            "highlight_line": 45,
            "content": "theorem m_conj_hodge_open_literature_anchor : True := trivial\n\n/-- Yang\u2013Mills mass gap (M-CONJ-YM). -/",
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      "id": "M-CONJ-JACOBIAN",
      "kind": "target",
      "field": "math",
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      "statement": "Is every polynomial automorphism of C^n with Jacobian determinant 1 invertible (Jacobian conjecture)?",
      "catalog_status": "target",
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      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_jacobian_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
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      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "phase16 iter127 open literature",
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            "symbol": "m_conj_jacobian_open_literature_anchor",
            "start_line": 81,
            "highlight_line": 81,
            "content": "theorem m_conj_jacobian_open_literature_anchor : True := trivial\n\n/-- Smooth 4D Poincar\u00e9 conjecture (M-CONJ-SMOOTH-4D-POINCARE). -/",
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            "content": "[[entry]]\nid = \"M-CONJ-JACOBIAN\"\nkind = \"target\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Is every polynomial automorphism of C^n with Jacobian determinant 1 invertible (Jacobian conjecture)?\"\nlatex = \"Is every polynomial automorphism of $\\\\mathbb{C}^n$ with Jacobian determinant $1$ invertible (Jacobian conjecture)?\"\ncontext = \"Open since the 1930s; many reductions exist but no proof in full generality; important for computer algebra.\"\ncontent_tier = \"curated\"\nproof_status = \"target\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\npriority_tier = \"P1\"\nlean_module = \"proof-db/math/OpenConjectureLiterature.lean\"\nlean_thm = \"Li.ProofDb.MathOpen.m_conj_jacobian_open_literature_anchor\"\nregister_source = \"proof-db/math/open-conjectures.json\"\nnotes = \"phase16 iter127 open literature\"\nexternal_url = \"https://en.wikipedia.org/wiki/Jacobian_conjecture\"\nsources = [\n  { title = \"Wikipedia: Jacobian conjecture\", url = \"https://en.wikipedia.org/wiki/Jacobian_conjecture\" },\n]\nlast_verified_lic_commit = \"46c383978\"",
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      "id": "M-CONJ-LANDAU-LEGENDRE",
      "kind": "target",
      "field": "math",
      "domain": "number_theory",
      "statement": "Are there infinitely many primes p with n^2 < p < (n+1)^2 for some n (Legendre's conjecture)?",
      "catalog_status": "target",
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      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_landau_legendre_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
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      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "Landau's four problems; distinct from E-86 Goldbach and E-87 twin primes; phase16 iter127 open literature",
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            "symbol": "m_conj_landau_legendre_open_literature_anchor",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_conj_landau_legendre_open_literature_anchor : True := trivial\n\n/-- Infinitely many primes n\u00b2+2 (M-CONJ-LANDAU-N2P2). -/",
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      "kind": "target",
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      "statement": "Are there infinitely many primes of the form n^2+2 (Landau problem)?",
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      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_landau_n2p2_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
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      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "Distinct from Erd\u0151s E-92 (n^2+1); phase16 iter127 open literature",
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            "symbol": "m_conj_landau_n2p2_open_literature_anchor",
            "start_line": 63,
            "highlight_line": 63,
            "content": "theorem m_conj_landau_n2p2_open_literature_anchor : True := trivial\n\n/-- Beal conjecture (M-CONJ-BEAL). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean#L63",
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            "symbol": "m_conj_navier_open_literature_anchor",
            "start_line": 39,
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            "content": "theorem m_conj_navier_open_literature_anchor : True := trivial\n\n/-- Hodge conjecture (M-CONJ-HODGE). -/",
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      "priority_tier": "P2",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_odd_perfect_open_literature_anchor",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
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            "start_line": 75,
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            "content": "theorem m_conj_odd_perfect_open_literature_anchor : True := trivial\n\n/-- Jacobian conjecture (M-CONJ-JACOBIAN). -/",
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      "domain": "complexity",
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      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
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      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_p_np_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "phase16 iter127 open literature",
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            "language": "lean",
            "path": "proof-db/math/OpenConjectureLiterature.lean",
            "symbol": "m_conj_p_np_open_literature_anchor",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem m_conj_p_np_open_literature_anchor : True := trivial\n\n/-- Birch and Swinnerton-Dyer conjecture (M-CONJ-BSD). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean#L27",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
            "start_line": 53,
            "highlight_line": 54,
            "content": "[[entry]]\nid = \"M-CONJ-P-NP\"\nkind = \"target\"\nfield = \"math\"\ndomain = \"complexity\"\nstatement = \"Is every language in NP decidable in polynomial time (P versus NP)?\"\nlatex = \"Is every language in $\\\\mathrm{NP}$ decidable in polynomial time ($\\\\mathrm{P}$ versus $\\\\mathrm{NP}$)?\"\ncontext = \"Clay Millennium problem; separates feasible verification from feasible computation for combinatorial search problems.\"\ncontent_tier = \"curated\"\nproof_status = \"target\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\npriority_tier = \"P0\"\nlean_module = \"proof-db/math/OpenConjectureLiterature.lean\"\nlean_thm = \"Li.ProofDb.MathOpen.m_conj_p_np_open_literature_anchor\"\nregister_source = \"proof-db/math/open-conjectures.json\"\nnotes = \"phase16 iter127 open literature\"\nexternal_url = \"https://www.claymath.org/millennium-problems/p-vs-np-problem\"\nsources = [\n  { title = \"Clay Mathematics Institute: P vs NP\", url = \"https://www.claymath.org/millennium-problems/p-vs-np-problem\" },\n]\nlast_verified_lic_commit = \"46c383978\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-conjectures.toml#L53",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "M-CONJ-RH",
      "kind": "target",
      "field": "math",
      "domain": "analytic_number_theory",
      "statement": "Do all non-trivial zeros of the Riemann zeta function have real part 1/2 (Riemann hypothesis)?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_rh_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "Related Erd\u0151s catalog row E-128 (prime-count error term); phase16 iter127 open literature",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "path": "proof-db/math/OpenConjectureLiterature.lean",
            "symbol": "m_conj_rh_open_literature_anchor",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem m_conj_rh_open_literature_anchor : True := trivial\n\n/-- P versus NP (M-CONJ-P-NP). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean#L21",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
            "start_line": 30,
            "highlight_line": 31,
            "content": "[[entry]]\nid = \"M-CONJ-RH\"\nkind = \"target\"\nfield = \"math\"\ndomain = \"analytic_number_theory\"\nstatement = \"Do all non-trivial zeros of the Riemann zeta function have real part 1/2 (Riemann hypothesis)?\"\nlatex = \"Do all non-trivial zeros of the Riemann zeta function $\\\\zeta(s)$ satisfy $\\\\Re(s)=\\\\tfrac{1}{2}$ (Riemann hypothesis)?\"\ncontext = \"Clay Millennium problem; equivalent formulations tie prime distribution, explicit formulas, and error terms in the prime-counting function.\"\ncontent_tier = \"curated\"\nproof_status = \"target\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\npriority_tier = \"P0\"\nlean_module = \"proof-db/math/OpenConjectureLiterature.lean\"\nlean_thm = \"Li.ProofDb.MathOpen.m_conj_rh_open_literature_anchor\"\nregister_source = \"proof-db/math/open-conjectures.json\"\nexternal_url = \"https://www.claymath.org/millennium-problems/riemann-hypothesis\"\nsources = [\n  { title = \"Clay Mathematics Institute: Riemann hypothesis\", url = \"https://www.claymath.org/millennium-problems/riemann-hypothesis\" },\n]\nnotes = \"Related Erd\u0151s catalog row E-128 (prime-count error term); phase16 iter127 open literature\"\nlast_verified_lic_commit = \"46c383978\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-conjectures.toml#L30",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
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    },
    {
      "id": "M-CONJ-SMOOTH-4D-POINCARE",
      "kind": "target",
      "field": "math",
      "domain": "topology",
      "statement": "Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere (smooth 4D Poincar\u00e9 conjecture)?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": "P1",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_smooth_4d_poincare_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "Millennium Poincar\u00e9 (dim 3) proved; smooth 4D variant open; phase16 iter127 open literature",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/math/OpenConjectureLiterature.lean",
            "symbol": "m_conj_smooth_4d_poincare_open_literature_anchor",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem m_conj_smooth_4d_poincare_open_literature_anchor : True := trivial\n\n/-- Hadwiger\u2013Nelson problem (M-CONJ-HADWIGER-NELSON). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean#L87",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
            "start_line": 283,
            "highlight_line": 284,
            "content": "[[entry]]\nid = \"M-CONJ-SMOOTH-4D-POINCARE\"\nkind = \"target\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere (smooth 4D Poincar\u00e9 conjecture)?\"\nlatex = \"Is every smooth homotopy $4$-sphere diffeomorphic to the standard $4$-sphere (smooth $4$D Poincar\u00e9 conjecture)?\"\ncontext = \"Perelman proved the 3D Poincar\u00e9 conjecture; the smooth 4D analogue remains open and is distinct from the topological 4D case.\"\ncontent_tier = \"curated\"\nproof_status = \"target\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\npriority_tier = \"P1\"\nlean_module = \"proof-db/math/OpenConjectureLiterature.lean\"\nlean_thm = \"Li.ProofDb.MathOpen.m_conj_smooth_4d_poincare_open_literature_anchor\"\nregister_source = \"proof-db/math/open-conjectures.json\"\nexternal_url = \"https://en.wikipedia.org/wiki/generalized_Poincar%C3%A9_conjecture\"\nsources = [\n  { title = \"Wikipedia: Generalized Poincar\u00e9 conjecture\", url = \"https://en.wikipedia.org/wiki/generalized_Poincar%C3%A9_conjecture\" },\n]\nnotes = \"Millennium Poincar\u00e9 (dim 3) proved; smooth 4D variant open; phase16 iter127 open literature\"\nlast_verified_lic_commit = \"46c383978\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-conjectures.toml#L283",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
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    },
    {
      "id": "M-CONJ-YM",
      "kind": "target",
      "field": "math",
      "domain": "mathematical_physics",
      "statement": "Does quantum Yang\u2013Mills theory exist rigorously and exhibit a mass gap (Millennium problem)?",
      "catalog_status": "target",
      "lean_status": "proved",
      "diverges": true,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": "P0",
      "lean_theorem": "Li.ProofDb.MathOpen.m_conj_ym_open_literature_anchor",
      "lean_module": "proof-db/math/OpenConjectureLiterature.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean",
      "notes": "phase16 iter127 open literature",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/math/OpenConjectureLiterature.lean",
            "symbol": "m_conj_ym_open_literature_anchor",
            "start_line": 51,
            "highlight_line": 51,
            "content": "theorem m_conj_ym_open_literature_anchor : True := trivial\n\n/-- Legendre's conjecture (M-CONJ-LANDAU-LEGENDRE). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/OpenConjectureLiterature.lean#L51",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-conjectures.toml",
            "start_line": 145,
            "highlight_line": 146,
            "content": "[[entry]]\nid = \"M-CONJ-YM\"\nkind = \"target\"\nfield = \"math\"\ndomain = \"mathematical_physics\"\nstatement = \"Does quantum Yang\u2013Mills theory exist rigorously and exhibit a mass gap (Millennium problem)?\"\nlatex = \"Does quantum Yang\u2013Mills theory exist rigorously and exhibit a mass gap (Millennium problem)?\"\ncontext = \"Clay Millennium problem in constructive quantum field theory; mass gap formalizes confinement heuristics from particle physics.\"\ncontent_tier = \"curated\"\nproof_status = \"target\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\npriority_tier = \"P0\"\nlean_module = \"proof-db/math/OpenConjectureLiterature.lean\"\nlean_thm = \"Li.ProofDb.MathOpen.m_conj_ym_open_literature_anchor\"\nregister_source = \"proof-db/math/open-conjectures.json\"\nnotes = \"phase16 iter127 open literature\"\nexternal_url = \"https://www.claymath.org/millennium-problems/yang-mills-and-mass-gap\"\nsources = [\n  { title = \"Clay Mathematics Institute: Yang\u2013Mills and mass gap\", url = \"https://www.claymath.org/millennium-problems/yang-mills-and-mass-gap\" },\n]\nlast_verified_lic_commit = \"46c383978\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-conjectures.toml#L145",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "M-LM-ADD-ASSOC",
      "kind": "lemma",
      "field": "math",
      "domain": null,
      "statement": "Real addition associative (discharged from M-AX-REAL-ADD-ASSOC).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_add_assoc",
      "lean_module": "proof-db/math/axioms/MathLemmas.lean",
      "li_specimen": "proof-db/math/axioms/reals_field.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [
          "M-AX-REAL-ADD-ASSOC"
        ],
        "release_pin": "2026-05-25",
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/math/axioms/MathLemmas.lean",
            "symbol": "m_lm_add_assoc",
            "start_line": 9,
            "highlight_line": 9,
            "content": "theorem m_lm_add_assoc (a b c : \u211d) : a + b + c = a + (b + c) := real_add_assoc a b c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathLemmas.lean#L9",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/math/axioms/reals_field.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_real_add_comm(a: float, b: float)-> float\n  requires true\n  ensures result == b + a\n  decreases 0\n=\n  return a + b\n\ndef proof_db_real_add_assoc(a: float, b: float, c: float)-> float\n  requires true\n  ensures result == a + (b + c)\n  decreases 0\n=\n  return (a + b) + c\n\ndef proof_db_real_mul_distrib(a: float, b: float, c: float)-> float\n  requires true\n  ensures result == a * (b + c)\n  decreases 0\n=\n  return a * b + a * c\n\ndef proof_db_real_mul_one(a: float)-> float\n  requires true\n  ensures result == a\n  decreases 0\n=\n  return a * 1.0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-lemmas.toml",
            "start_line": 17,
            "highlight_line": 18,
            "content": "[[entry]]\nid = \"M-LM-ADD-ASSOC\"\nkind = \"lemma\"\nfield = \"math\"\nstatement = \"Real addition associative (discharged from M-AX-REAL-ADD-ASSOC).\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathLemmas.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_add_assoc\"\nli_specimen = \"proof-db/math/axioms/reals_field.li\"\nrelease_pin = \"2026-05-25\"\nlast_verified_lic_commit = \"5360dbab\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-lemmas.toml#L17",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-ADD-COMM",
      "kind": "lemma",
      "field": "math",
      "domain": null,
      "statement": "Real addition commutative (discharged from M-AX-REAL-ADD-COMM).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_add_comm",
      "lean_module": "proof-db/math/axioms/MathLemmas.lean",
      "li_specimen": "proof-db/math/axioms/reals_field.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [
          "M-AX-REAL-ADD-COMM"
        ],
        "release_pin": "2026-05-25",
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/math/axioms/MathLemmas.lean",
            "symbol": "m_lm_add_comm",
            "start_line": 8,
            "highlight_line": 8,
            "content": "theorem m_lm_add_comm (a b : \u211d) : a + b = b + a := real_add_comm a b",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathLemmas.lean#L8",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/math/axioms/reals_field.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_real_add_comm(a: float, b: float)-> float\n  requires true\n  ensures result == b + a\n  decreases 0\n=\n  return a + b\n\ndef proof_db_real_add_assoc(a: float, b: float, c: float)-> float\n  requires true\n  ensures result == a + (b + c)\n  decreases 0\n=\n  return (a + b) + c\n\ndef proof_db_real_mul_distrib(a: float, b: float, c: float)-> float\n  requires true\n  ensures result == a * (b + c)\n  decreases 0\n=\n  return a * b + a * c\n\ndef proof_db_real_mul_one(a: float)-> float\n  requires true\n  ensures result == a\n  decreases 0\n=\n  return a * 1.0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-lemmas.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"M-LM-ADD-COMM\"\nkind = \"lemma\"\nfield = \"math\"\nstatement = \"Real addition commutative (discharged from M-AX-REAL-ADD-COMM).\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathLemmas.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_add_comm\"\nli_specimen = \"proof-db/math/axioms/reals_field.li\"\nrelease_pin = \"2026-05-25\"\nlast_verified_lic_commit = \"5360dbab\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-lemmas.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-FLOAT-ADD-COMM",
      "kind": "lemma",
      "field": "math",
      "domain": null,
      "statement": "Float add_commutative specimen vs \u211d axiom (catalog discrepancy).",
      "catalog_status": "discrepancy",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "numerics_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.real_add_comm",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/lemmas/add_commutative.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/lemmas/add_commutative.li",
      "notes": "Specimen ensures float add comm; triage under backlog P-float \u2014 not a compiler wrongness verdict.",
      "drilldown": {
        "discharged_from": [],
        "release_pin": "2026-05-25",
        "backlog_ref": "P-float",
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "real_add_comm",
            "start_line": 37,
            "highlight_line": 37,
            "content": "theorem real_add_comm : \u2200 a b : \u211d, a + b = b + a :=\n  fun a b => add_comm a b\n\n/-- \u211d field: addition associative (M-AX-REAL-ADD-ASSOC). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L37",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/math/lemmas/add_commutative.li",
            "symbol": "add_commutative",
            "start_line": 1,
            "highlight_line": 1,
            "content": "def add_commutative(a: float, b: float) -> float\n  requires true\n  ensures result == b + a\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/lemmas/add_commutative.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-lemmas.toml",
            "start_line": 73,
            "highlight_line": 74,
            "content": "[[entry]]\nid = \"M-LM-FLOAT-ADD-COMM\"\nkind = \"lemma\"\nfield = \"math\"\nstatement = \"Float add_commutative specimen vs \u211d axiom (catalog discrepancy).\"\nproof_status = \"discrepancy\"\ngap_id = \"G-math\"\ngap_kind = \"numerics_gap\"\nbacklog_ref = \"P-float\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.real_add_comm\"\nli_specimen = \"proof-db/math/lemmas/add_commutative.li\"\nrelease_pin = \"2026-05-25\"\nlast_verified_lic_commit = \"5360dbab\"\nnotes = \"Specimen ensures float add comm; triage under backlog P-float \u2014 not a compiler wrongness verdict.\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-lemmas.toml#L73",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-002",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_add_comm",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-002: Group inverse uniqueness (mega 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_002.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"M-LM-MC-002\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_add_comm\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L83",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-003",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_add_assoc",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-003: Continuous function composition (mega 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_003.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 99,
            "highlight_line": 100,
            "content": "[[entry]]\nid = \"M-LM-MC-003\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_add_assoc\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L99",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-004",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_open_union",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-004: Open set union (mega 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_004.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"M-LM-MC-004\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_open_union\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-005",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_prime_exists_factor",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-005: Prime factorization existence (mega 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"M-LM-MC-005\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_prime_exists_factor\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-006",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_matrix_mul_assoc",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-006: Matrix multiplication associativity (mega 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_006.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 147,
            "highlight_line": 148,
            "content": "[[entry]]\nid = \"M-LM-MC-006\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"linear-algebra\"\nstatement = \"Matrix multiplication associativity (mega 6).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_matrix_mul_assoc\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-007",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-007: Fundamental theorem of calculus part I (mega 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_007.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 163,
            "highlight_line": 164,
            "content": "[[entry]]\nid = \"M-LM-MC-007\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L163",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-008",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_demorgan_inter",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-008: De Morgan laws (mega 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_008.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 178,
            "highlight_line": 179,
            "content": "[[entry]]\nid = \"M-LM-MC-008\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 8).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_demorgan_inter\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L178",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-009",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_nat_add_zero",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-009: Well-ordering principle sketch (mega 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_009.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 194,
            "highlight_line": 195,
            "content": "[[entry]]\nid = \"M-LM-MC-009\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_nat_add_zero\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-010",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_010.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "path": "proof-db/math/corpus/m_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-010: Triangle inequality metric (mega 10).",
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      "id": "M-LM-MC-012",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_mul_one",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_012.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_012.li",
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            "path": "proof-db/math/corpus/m_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-012: Group inverse uniqueness (mega 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_012.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-012\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_mul_one\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-013",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-013: Continuous function composition (mega 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_013.li#L1",
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            "language": "toml",
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            "highlight_line": 243,
            "content": "[[entry]]\nid = \"M-LM-MC-013\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 13).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-014",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-014: Open set union (mega 14).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_014.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 257,
            "highlight_line": 258,
            "content": "[[entry]]\nid = \"M-LM-MC-014\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 14).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-015",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_015.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-015: Prime factorization existence (mega 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_015.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-015\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 15).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-016",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "convergence_class": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_016.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-016: Matrix multiplication associativity (mega 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_016.li#L1",
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    },
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      "id": "M-LM-MC-017",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
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      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_ftc1",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_017.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-017: Fundamental theorem of calculus part I (mega 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_017.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-017\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 17).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_ftc1\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-018",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_demorgan_union",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_018.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-018: De Morgan laws (mega 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_018.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-018\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 18).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_demorgan_union\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
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      "id": "M-LM-MC-019",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-019: Well-ordering principle sketch (mega 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_019.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 334,
            "highlight_line": 335,
            "content": "[[entry]]\nid = \"M-LM-MC-019\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 19).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-020",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_020.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_020.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
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            "path": "proof-db/math/corpus/m_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-020: Triangle inequality metric (mega 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_020.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 349,
            "highlight_line": 350,
            "content": "[[entry]]\nid = \"M-LM-MC-020\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"geometry\"\nstatement = \"Triangle inequality metric (mega 20).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_triangle_metric\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L349",
            "role": "catalog",
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    },
    {
      "id": "M-LM-MC-022",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_nat_add_comm",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "path": "proof-db/math/corpus/m_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-022: Group inverse uniqueness (mega 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_022.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 365,
            "highlight_line": 366,
            "content": "[[entry]]\nid = \"M-LM-MC-022\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_nat_add_comm\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L365",
            "role": "catalog",
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    },
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      "id": "M-LM-MC-023",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-023: Continuous function composition (mega 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_023.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 381,
            "highlight_line": 382,
            "content": "[[entry]]\nid = \"M-LM-MC-023\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 23).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "M-LM-MC-024",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-024: Open set union (mega 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_024.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 396,
            "highlight_line": 397,
            "content": "[[entry]]\nid = \"M-LM-MC-024\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 24).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "M-LM-MC-025",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-025: Prime factorization existence (mega 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_025.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
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            "highlight_line": 412,
            "content": "[[entry]]\nid = \"M-LM-MC-025\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 25).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-026",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-026: Matrix multiplication associativity (mega 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_026.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 426,
            "highlight_line": 427,
            "content": "[[entry]]\nid = \"M-LM-MC-026\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"linear-algebra\"\nstatement = \"Matrix multiplication associativity (mega 26).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "M-LM-MC-027",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-027: Fundamental theorem of calculus part I (mega 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_027.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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      "id": "M-LM-MC-028",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
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      "convergence_class": null,
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_028.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_028.li",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-028: De Morgan laws (mega 28).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_028.li#L1",
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      "id": "M-LM-MC-029",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_029.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-029: Well-ordering principle sketch (mega 29).",
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    },
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      "id": "M-LM-MC-030",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_030.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
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            "path": "proof-db/math/corpus/m_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-030: Triangle inequality metric (mega 30).",
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      "id": "M-LM-MC-032",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_032.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-032: Group inverse uniqueness (mega 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_032.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-032\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 32).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-033",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_033.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-033: Continuous function composition (mega 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_033.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-033\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 33).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-034",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_034.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-034: Open set union (mega 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_034.li#L1",
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      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "convergence_class": null,
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      "lean_theorem": null,
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_035.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_035.li",
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            "path": "proof-db/math/corpus/m_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-035: Prime factorization existence (mega 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_035.li#L1",
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      "id": "M-LM-MC-036",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-036: Matrix multiplication associativity (mega 36).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_036.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-036\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"linear-algebra\"\nstatement = \"Matrix multiplication associativity (mega 36).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-037",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_037.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-037: Fundamental theorem of calculus part I (mega 37).",
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            "highlight_line": 578,
            "content": "[[entry]]\nid = \"M-LM-MC-037\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 37).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-038",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
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      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-038: De Morgan laws (mega 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_038.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 592,
            "highlight_line": 593,
            "content": "[[entry]]\nid = \"M-LM-MC-038\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 38).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-039",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-039: Well-ordering principle sketch (mega 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_039.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 607,
            "highlight_line": 608,
            "content": "[[entry]]\nid = \"M-LM-MC-039\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 39).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-040",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_040.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_040.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "path": "proof-db/math/corpus/m_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-040: Triangle inequality metric (mega 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_040.li#L1",
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      "id": "M-LM-MC-042",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_042.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-042: Group inverse uniqueness (mega 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_042.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-042\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-043",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-043: Continuous function composition (mega 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_043.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-043\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 43).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-044",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_044.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-044: Open set union (mega 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_044.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 668,
            "highlight_line": 669,
            "content": "[[entry]]\nid = \"M-LM-MC-044\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 44).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-045",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-045: Prime factorization existence (mega 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_045.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-045\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 45).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-046",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_046.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-046: Matrix multiplication associativity (mega 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_046.li#L1",
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            "start_line": 698,
            "highlight_line": 699,
            "content": "[[entry]]\nid = \"M-LM-MC-046\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"linear-algebra\"\nstatement = \"Matrix multiplication associativity (mega 46).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-047",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_047.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-047: Fundamental theorem of calculus part I (mega 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_047.li#L1",
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            "start_line": 713,
            "highlight_line": 714,
            "content": "[[entry]]\nid = \"M-LM-MC-047\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 47).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-048",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-048: De Morgan laws (mega 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_048.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 728,
            "highlight_line": 729,
            "content": "[[entry]]\nid = \"M-LM-MC-048\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 48).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-049",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_049.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-049: Well-ordering principle sketch (mega 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_049.li#L1",
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            "start_line": 743,
            "highlight_line": 744,
            "content": "[[entry]]\nid = \"M-LM-MC-049\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 49).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-050",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_050.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
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            "path": "proof-db/math/corpus/m_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-050: Triangle inequality metric (mega 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_050.li#L1",
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            "start_line": 758,
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            "content": "[[entry]]\nid = \"M-LM-MC-050\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"geometry\"\nstatement = \"Triangle inequality metric (mega 50).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_triangle_metric\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
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    },
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      "id": "M-LM-MC-052",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-052: Group inverse uniqueness (mega 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_052.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-052\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 52).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-053",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-053: Continuous function composition (mega 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_053.li#L1",
            "role": "li_specimen",
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            "start_line": 869,
            "highlight_line": 870,
            "content": "[[entry]]\nid = \"M-LM-MC-053\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 53).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "M-LM-MC-054",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-054: Open set union (mega 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_054.li#L1",
            "role": "li_specimen",
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            "start_line": 884,
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            "content": "[[entry]]\nid = \"M-LM-MC-054\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 54).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-055",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": null,
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_055.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-055: Prime factorization existence (mega 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_055.li#L1",
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            "start_line": 899,
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            "content": "[[entry]]\nid = \"M-LM-MC-055\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 55).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-056",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_056.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-056: Matrix multiplication associativity (mega 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_056.li#L1",
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            "start_line": 914,
            "highlight_line": 915,
            "content": "[[entry]]\nid = \"M-LM-MC-056\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"linear-algebra\"\nstatement = \"Matrix multiplication associativity (mega 56).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-057",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-057: Fundamental theorem of calculus part I (mega 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_057.li#L1",
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            "language": "toml",
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            "start_line": 929,
            "highlight_line": 930,
            "content": "[[entry]]\nid = \"M-LM-MC-057\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 57).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-058",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-058: De Morgan laws (mega 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_058.li#L1",
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            "start_line": 944,
            "highlight_line": 945,
            "content": "[[entry]]\nid = \"M-LM-MC-058\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 58).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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        ]
      }
    },
    {
      "id": "M-LM-MC-059",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-059: Well-ordering principle sketch (mega 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_059.li#L1",
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            "language": "toml",
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            "start_line": 959,
            "highlight_line": 960,
            "content": "[[entry]]\nid = \"M-LM-MC-059\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 59).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "M-LM-MC-060",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_060.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L57",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-060: Triangle inequality metric (mega 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_060.li#L1",
            "role": "li_specimen",
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            "role": "catalog",
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    },
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      "id": "M-LM-MC-062",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-062: Group inverse uniqueness (mega 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_062.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 990,
            "highlight_line": 991,
            "content": "[[entry]]\nid = \"M-LM-MC-062\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 62).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "M-LM-MC-063",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_063.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-063: Continuous function composition (mega 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_063.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1005,
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            "content": "[[entry]]\nid = \"M-LM-MC-063\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 63).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-064",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
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      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_064.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-064: Open set union (mega 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_064.li#L1",
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            "language": "toml",
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            "start_line": 1020,
            "highlight_line": 1021,
            "content": "[[entry]]\nid = \"M-LM-MC-064\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 64).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-065",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-065: Prime factorization existence (mega 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_065.li#L1",
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            "language": "toml",
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            "start_line": 1035,
            "highlight_line": 1036,
            "content": "[[entry]]\nid = \"M-LM-MC-065\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 65).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-066",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-066: Matrix multiplication associativity (mega 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_066.li#L1",
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            "language": "toml",
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            "start_line": 1050,
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      "id": "M-LM-MC-067",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-067: Fundamental theorem of calculus part I (mega 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_067.li#L1",
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            "start_line": 1065,
            "highlight_line": 1066,
            "content": "[[entry]]\nid = \"M-LM-MC-067\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 67).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-068",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-068: De Morgan laws (mega 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_068.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1080,
            "highlight_line": 1081,
            "content": "[[entry]]\nid = \"M-LM-MC-068\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 68).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-069",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-069: Well-ordering principle sketch (mega 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_069.li#L1",
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            "language": "toml",
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            "start_line": 1095,
            "highlight_line": 1096,
            "content": "[[entry]]\nid = \"M-LM-MC-069\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 69).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-070",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_070.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_070.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-070: Triangle inequality metric (mega 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_070.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 1110,
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            "content": "[[entry]]\nid = \"M-LM-MC-070\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"geometry\"\nstatement = \"Triangle inequality metric (mega 70).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_triangle_metric\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
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    },
    {
      "id": "M-LM-MC-072",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_072.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-072: Group inverse uniqueness (mega 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_072.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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      "id": "M-LM-MC-073",
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      "domain": "analysis",
      "statement": "Continuous function composition (mega 73).",
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            "start_line": 1,
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            "content": "# M-LM-MC-073: Continuous function composition (mega 73).",
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      "domain": "topology",
      "statement": "Open set union (mega 74).",
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      "diverges": false,
      "gap_id": "G-math",
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            "symbol": null,
            "start_line": 1,
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            "content": "# M-LM-MC-074: Open set union (mega 74).",
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      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 75).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-math",
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_075.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-075: Prime factorization existence (mega 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_075.li#L1",
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      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 76).",
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      "diverges": false,
      "gap_id": "G-math",
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      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "symbol": null,
            "start_line": 1,
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            "content": "# M-LM-MC-076: Matrix multiplication associativity (mega 76).",
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      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 77).",
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      "diverges": false,
      "gap_id": "G-math",
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      "erdos_id": null,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_077.li",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-077: Fundamental theorem of calculus part I (mega 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_077.li#L1",
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      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 78).",
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      "diverges": false,
      "gap_id": "G-math",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-078: De Morgan laws (mega 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_078.li#L1",
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      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 79).",
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      "diverges": false,
      "gap_id": "G-math",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-079: Well-ordering principle sketch (mega 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_079.li#L1",
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      "id": "M-LM-MC-080",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_080.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_080.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
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            "path": "proof-db/math/corpus/m_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-080: Triangle inequality metric (mega 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_080.li#L1",
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      "id": "M-LM-MC-082",
      "kind": "lemma",
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      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 82).",
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      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_082.li",
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            "path": "proof-db/math/corpus/m_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
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            "content": "# M-LM-MC-082: Group inverse uniqueness (mega 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_082.li#L1",
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            "start_line": 1262,
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      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
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      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_083.li",
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            "path": "proof-db/math/corpus/m_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-083: Continuous function composition (mega 83).",
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      "id": "M-LM-MC-084",
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      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "convergence_class": null,
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      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_084.li",
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      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-084: Open set union (mega 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_084.li#L1",
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      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 85).",
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      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
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      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-085: Prime factorization existence (mega 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_085.li#L1",
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      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 86).",
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      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
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            "symbol": null,
            "start_line": 1,
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            "content": "# M-LM-MC-086: Matrix multiplication associativity (mega 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_086.li#L1",
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      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 87).",
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      "diverges": false,
      "gap_id": "G-math",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_087.li",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-087: Fundamental theorem of calculus part I (mega 87).",
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      "statement": "De Morgan laws (mega 88).",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-088: De Morgan laws (mega 88).",
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      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 89).",
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      "diverges": false,
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      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "symbol": null,
            "start_line": 1,
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            "content": "# M-LM-MC-089: Well-ordering principle sketch (mega 89).",
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      "id": "M-LM-MC-090",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_090.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_090.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
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            "path": "proof-db/math/corpus/m_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-090: Triangle inequality metric (mega 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_090.li#L1",
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            "language": "toml",
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            "start_line": 1382,
            "highlight_line": 1383,
            "content": "[[entry]]\nid = \"M-LM-MC-090\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"geometry\"\nstatement = \"Triangle inequality metric (mega 90).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_triangle_metric\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "M-LM-MC-092",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-092: Group inverse uniqueness (mega 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_092.li#L1",
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            "language": "toml",
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            "start_line": 1398,
            "highlight_line": 1399,
            "content": "[[entry]]\nid = \"M-LM-MC-092\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 92).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-093",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-093: Continuous function composition (mega 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_093.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1413,
            "highlight_line": 1414,
            "content": "[[entry]]\nid = \"M-LM-MC-093\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 93).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-094",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-094: Open set union (mega 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_094.li#L1",
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            "start_line": 1428,
            "highlight_line": 1429,
            "content": "[[entry]]\nid = \"M-LM-MC-094\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 94).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-095",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_095.li",
      "bench_refs": [],
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-095: Prime factorization existence (mega 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_095.li#L1",
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            "highlight_line": 1444,
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      "id": "M-LM-MC-096",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_096.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-096: Matrix multiplication associativity (mega 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_096.li#L1",
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      "id": "M-LM-MC-097",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": null,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-097: Fundamental theorem of calculus part I (mega 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_097.li#L1",
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      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_098.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-098: De Morgan laws (mega 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_098.li#L1",
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            "start_line": 1488,
            "highlight_line": 1489,
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    },
    {
      "id": "M-LM-MC-099",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-099: Well-ordering principle sketch (mega 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_099.li#L1",
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            "language": "toml",
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            "start_line": 1503,
            "highlight_line": 1504,
            "content": "[[entry]]\nid = \"M-LM-MC-099\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 99).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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        ]
      }
    },
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      "id": "M-LM-MC-100",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_100.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
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            "path": "proof-db/math/corpus/m_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-100: Triangle inequality metric (mega 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_100.li#L1",
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            "language": "toml",
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            "start_line": 1518,
            "highlight_line": 1519,
            "content": "[[entry]]\nid = \"M-LM-MC-100\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"geometry\"\nstatement = \"Triangle inequality metric (mega 100).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_triangle_metric\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
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    },
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      "id": "M-LM-MC-102",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-102: Group inverse uniqueness (mega 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_102.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1614,
            "highlight_line": 1615,
            "content": "[[entry]]\nid = \"M-LM-MC-102\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 102).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-103",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-103: Continuous function composition (mega 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_103.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1629,
            "highlight_line": 1630,
            "content": "[[entry]]\nid = \"M-LM-MC-103\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 103).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-104",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-104: Open set union (mega 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_104.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1644,
            "highlight_line": 1645,
            "content": "[[entry]]\nid = \"M-LM-MC-104\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 104).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-105",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_105.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_105.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-105: Prime factorization existence (mega 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_105.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-105\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 105).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_106.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-106: Matrix multiplication associativity (mega 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_106.li#L1",
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    },
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      "id": "M-LM-MC-107",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-107: Fundamental theorem of calculus part I (mega 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_107.li#L1",
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            "language": "toml",
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            "start_line": 1689,
            "highlight_line": 1690,
            "content": "[[entry]]\nid = \"M-LM-MC-107\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 107).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "M-LM-MC-108",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "li_specimen": "proof-db/math/corpus/m_lm_mc_108.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-108: De Morgan laws (mega 108).",
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            "start_line": 1704,
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            "content": "[[entry]]\nid = \"M-LM-MC-108\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 108).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
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      "id": "M-LM-MC-109",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-109: Well-ordering principle sketch (mega 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_109.li#L1",
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            "language": "toml",
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            "start_line": 1719,
            "highlight_line": 1720,
            "content": "[[entry]]\nid = \"M-LM-MC-109\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 109).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-110",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_110.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L57",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-110: Triangle inequality metric (mega 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_110.li#L1",
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            "language": "toml",
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            "start_line": 1734,
            "highlight_line": 1735,
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    },
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      "id": "M-LM-MC-112",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-112: Group inverse uniqueness (mega 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_112.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 1750,
            "highlight_line": 1751,
            "content": "[[entry]]\nid = \"M-LM-MC-112\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 112).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-113",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-113: Continuous function composition (mega 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_113.li#L1",
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            "start_line": 1765,
            "highlight_line": 1766,
            "content": "[[entry]]\nid = \"M-LM-MC-113\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 113).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-114",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_114.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-114: Open set union (mega 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_114.li#L1",
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    },
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      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": null,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-115: Prime factorization existence (mega 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_115.li#L1",
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    },
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      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_116.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-116: Matrix multiplication associativity (mega 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_116.li#L1",
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    },
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      "id": "M-LM-MC-117",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-117: Fundamental theorem of calculus part I (mega 117).",
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            "language": "toml",
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            "start_line": 1825,
            "highlight_line": 1826,
            "content": "[[entry]]\nid = \"M-LM-MC-117\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 117).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
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      "id": "M-LM-MC-118",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-118: De Morgan laws (mega 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_118.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1840,
            "highlight_line": 1841,
            "content": "[[entry]]\nid = \"M-LM-MC-118\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 118).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "M-LM-MC-119",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-119: Well-ordering principle sketch (mega 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1855,
            "highlight_line": 1856,
            "content": "[[entry]]\nid = \"M-LM-MC-119\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L1855",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-LM-MC-120",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L57",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-120: Triangle inequality metric (mega 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_120.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1870,
            "highlight_line": 1871,
            "content": "[[entry]]\nid = \"M-LM-MC-120\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"geometry\"\nstatement = \"Triangle inequality metric (mega 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_triangle_metric\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L1870",
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      }
    },
    {
      "id": "M-LM-MC-122",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-122: Group inverse uniqueness (mega 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_122.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1886,
            "highlight_line": 1887,
            "content": "[[entry]]\nid = \"M-LM-MC-122\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L1886",
            "role": "catalog",
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    },
    {
      "id": "M-LM-MC-123",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-123: Continuous function composition (mega 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_123.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1901,
            "highlight_line": 1902,
            "content": "[[entry]]\nid = \"M-LM-MC-123\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 123).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "M-LM-MC-124",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-124: Open set union (mega 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_124.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1916,
            "highlight_line": 1917,
            "content": "[[entry]]\nid = \"M-LM-MC-124\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 124).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "M-LM-MC-125",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-125: Prime factorization existence (mega 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_125.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1931,
            "highlight_line": 1932,
            "content": "[[entry]]\nid = \"M-LM-MC-125\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 125).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "M-LM-MC-126",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_126.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-126: Matrix multiplication associativity (mega 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_126.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1946,
            "highlight_line": 1947,
            "content": "[[entry]]\nid = \"M-LM-MC-126\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"linear-algebra\"\nstatement = \"Matrix multiplication associativity (mega 126).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L1946",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-127",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-127: Fundamental theorem of calculus part I (mega 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_127.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1961,
            "highlight_line": 1962,
            "content": "[[entry]]\nid = \"M-LM-MC-127\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 127).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L1961",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-128",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-128: De Morgan laws (mega 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1976,
            "highlight_line": 1977,
            "content": "[[entry]]\nid = \"M-LM-MC-128\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 128).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L1976",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-129",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-129: Well-ordering principle sketch (mega 129).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_129.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 1991,
            "highlight_line": 1992,
            "content": "[[entry]]\nid = \"M-LM-MC-129\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 129).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L1991",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-130",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_130.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L57",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-130: Triangle inequality metric (mega 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_130.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2006,
            "highlight_line": 2007,
            "content": "[[entry]]\nid = \"M-LM-MC-130\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"geometry\"\nstatement = \"Triangle inequality metric (mega 130).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_triangle_metric\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2006",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-132",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-132: Group inverse uniqueness (mega 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_132.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2022,
            "highlight_line": 2023,
            "content": "[[entry]]\nid = \"M-LM-MC-132\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 132).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2022",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "M-LM-MC-133",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-133: Continuous function composition (mega 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_133.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2037,
            "highlight_line": 2038,
            "content": "[[entry]]\nid = \"M-LM-MC-133\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 133).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2037",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-LM-MC-134",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-134: Open set union (mega 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_134.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2052,
            "highlight_line": 2053,
            "content": "[[entry]]\nid = \"M-LM-MC-134\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 134).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-LM-MC-135",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-135: Prime factorization existence (mega 135).",
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      "id": "M-LM-MC-136",
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      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_136.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-136: Matrix multiplication associativity (mega 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_136.li#L1",
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            "start_line": 2082,
            "highlight_line": 2083,
            "content": "[[entry]]\nid = \"M-LM-MC-136\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"linear-algebra\"\nstatement = \"Matrix multiplication associativity (mega 136).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-137",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-137: Fundamental theorem of calculus part I (mega 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_137.li#L1",
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            "language": "toml",
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            "start_line": 2097,
            "highlight_line": 2098,
            "content": "[[entry]]\nid = \"M-LM-MC-137\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 137).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-138",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_138.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-138: De Morgan laws (mega 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_138.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2112,
            "highlight_line": 2113,
            "content": "[[entry]]\nid = \"M-LM-MC-138\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 138).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-139",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-139: Well-ordering principle sketch (mega 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_139.li#L1",
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            "language": "toml",
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            "start_line": 2127,
            "highlight_line": 2128,
            "content": "[[entry]]\nid = \"M-LM-MC-139\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 139).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-140",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_140.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_140.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
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            "role": "lean_formal",
            "label": "Lean formalization"
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            "path": "proof-db/math/corpus/m_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-140: Triangle inequality metric (mega 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_140.li#L1",
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      "id": "M-LM-MC-142",
      "kind": "lemma",
      "field": "math",
      "domain": "algebra",
      "statement": "Group inverse uniqueness (mega 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_142.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-142: Group inverse uniqueness (mega 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_142.li#L1",
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            "content": "[[entry]]\nid = \"M-LM-MC-142\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"algebra\"\nstatement = \"Group inverse uniqueness (mega 142).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "M-LM-MC-143",
      "kind": "lemma",
      "field": "math",
      "domain": "analysis",
      "statement": "Continuous function composition (mega 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_143.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/math/corpus/m_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-143: Continuous function composition (mega 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_143.li#L1",
            "role": "li_specimen",
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            "highlight_line": 2174,
            "content": "[[entry]]\nid = \"M-LM-MC-143\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"analysis\"\nstatement = \"Continuous function composition (mega 143).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "M-LM-MC-144",
      "kind": "lemma",
      "field": "math",
      "domain": "topology",
      "statement": "Open set union (mega 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_144.li",
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      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/math/corpus/m_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-144: Open set union (mega 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_144.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2188,
            "highlight_line": 2189,
            "content": "[[entry]]\nid = \"M-LM-MC-144\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"topology\"\nstatement = \"Open set union (mega 144).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2188",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-LM-MC-145",
      "kind": "lemma",
      "field": "math",
      "domain": "number-theory",
      "statement": "Prime factorization existence (mega 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-145: Prime factorization existence (mega 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_145.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2203,
            "highlight_line": 2204,
            "content": "[[entry]]\nid = \"M-LM-MC-145\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"number-theory\"\nstatement = \"Prime factorization existence (mega 145).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "M-LM-MC-146",
      "kind": "lemma",
      "field": "math",
      "domain": "linear-algebra",
      "statement": "Matrix multiplication associativity (mega 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/math/corpus/m_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-146: Matrix multiplication associativity (mega 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_146.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2218,
            "highlight_line": 2219,
            "content": "[[entry]]\nid = \"M-LM-MC-146\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"linear-algebra\"\nstatement = \"Matrix multiplication associativity (mega 146).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2218",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-LM-MC-147",
      "kind": "lemma",
      "field": "math",
      "domain": "calculus",
      "statement": "Fundamental theorem of calculus part I (mega 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-147: Fundamental theorem of calculus part I (mega 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_147.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2233,
            "highlight_line": 2234,
            "content": "[[entry]]\nid = \"M-LM-MC-147\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"calculus\"\nstatement = \"Fundamental theorem of calculus part I (mega 147).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2233",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-LM-MC-148",
      "kind": "lemma",
      "field": "math",
      "domain": "set-theory",
      "statement": "De Morgan laws (mega 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-148: De Morgan laws (mega 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_148.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2248,
            "highlight_line": 2249,
            "content": "[[entry]]\nid = \"M-LM-MC-148\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"set-theory\"\nstatement = \"De Morgan laws (mega 148).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2248",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MC-149",
      "kind": "lemma",
      "field": "math",
      "domain": "order",
      "statement": "Well-ordering principle sketch (mega 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-149: Well-ordering principle sketch (mega 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_149.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2263,
            "highlight_line": 2264,
            "content": "[[entry]]\nid = \"M-LM-MC-149\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"order\"\nstatement = \"Well-ordering principle sketch (mega 149).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2263",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "M-LM-MC-150",
      "kind": "lemma",
      "field": "math",
      "domain": "geometry",
      "statement": "Triangle inequality metric (mega 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_triangle_metric",
      "lean_module": "proof-db/math/axioms/MathAxioms.lean",
      "li_specimen": "proof-db/math/corpus/m_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter3 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/math/axioms/MathAxioms.lean",
            "symbol": "m_lm_triangle_metric",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem m_lm_triangle_metric {\u03b1 : Type*} [PseudoMetricSpace \u03b1] (x y z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  dist_triangle x y z\n\n/-- \u211d absolute-value triangle form (Init/Mathlib mirror companion). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathAxioms.lean#L57",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/math/corpus/m_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# M-LM-MC-150: Triangle inequality metric (mega 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/corpus/m_lm_mc_150.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-mega-corpus.toml",
            "start_line": 2278,
            "highlight_line": 2279,
            "content": "[[entry]]\nid = \"M-LM-MC-150\"\nkind = \"lemma\"\nfield = \"math\"\ndomain = \"geometry\"\nstatement = \"Triangle inequality metric (mega 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-math\"\nlean_module = \"proof-db/math/axioms/MathAxioms.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_triangle_metric\"\nli_specimen = \"proof-db/math/corpus/m_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter3 Mathlib/Init discharge\"\nregister_source = \"proof-db/math/register.json\"\nlast_verified_lic_commit = \"10c055a406\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-mega-corpus.toml#L2278",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-MUL-ONE",
      "kind": "lemma",
      "field": "math",
      "domain": null,
      "statement": "Real multiply-by-one (discharged from M-AX-REAL-MUL-ONE).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Math.m_lm_mul_one",
      "lean_module": "proof-db/math/axioms/MathLemmas.lean",
      "li_specimen": "proof-db/math/axioms/reals_field.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\math-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [
          "M-AX-REAL-MUL-ONE"
        ],
        "release_pin": "2026-05-25",
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/math/axioms/MathLemmas.lean",
            "symbol": "m_lm_mul_one",
            "start_line": 10,
            "highlight_line": 10,
            "content": "theorem m_lm_mul_one (a : \u211d) : a * 1 = a := real_mul_one a",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/MathLemmas.lean#L10",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/math/axioms/reals_field.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_real_add_comm(a: float, b: float)-> float\n  requires true\n  ensures result == b + a\n  decreases 0\n=\n  return a + b\n\ndef proof_db_real_add_assoc(a: float, b: float, c: float)-> float\n  requires true\n  ensures result == a + (b + c)\n  decreases 0\n=\n  return (a + b) + c\n\ndef proof_db_real_mul_distrib(a: float, b: float, c: float)-> float\n  requires true\n  ensures result == a * (b + c)\n  decreases 0\n=\n  return a * b + a * c\n\ndef proof_db_real_mul_one(a: float)-> float\n  requires true\n  ensures result == a\n  decreases 0\n=\n  return a * 1.0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/axioms/reals_field.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\math-lemmas.toml",
            "start_line": 31,
            "highlight_line": 32,
            "content": "[[entry]]\nid = \"M-LM-MUL-ONE\"\nkind = \"lemma\"\nfield = \"math\"\nstatement = \"Real multiply-by-one (discharged from M-AX-REAL-MUL-ONE).\"\nproof_status = \"proved\"\ngap_id = \"G-math\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/math/axioms/MathLemmas.lean\"\nlean_thm = \"Li.ProofDb.Math.m_lm_mul_one\"\nli_specimen = \"proof-db/math/axioms/reals_field.li\"\nrelease_pin = \"2026-05-25\"\nlast_verified_lic_commit = \"5360dbab\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\math-lemmas.toml#L31",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "M-LM-NAT-ADD-COMM",
      "kind": "lemma",
      "field": "math",
      "domain": null,
      "statement": "Nat: addition commutative (Peano + order layer).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": "proof_gap",
      "erdos_id": null,
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            "content": "theorem l_smooth_axiom {E : Type*} [NormedAddCommGroup E]\n    (L : NNReal) (grad : E \u2192 E) (h : LipschitzWith L grad) (x y : E) :\n    dist (grad x) (grad y) \u2264 (L : \u211d) * dist x y :=\n  h.dist_le_mul x y\n\n/-- Alias matching catalog `lean_thm` naming. -/",
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            "content": "theorem ml_ax_mc_021_sgd {E : Type*} [PseudoMetricSpace E]\n    (K : NNReal) (f : E \u2192 E) (hf : LipschitzWith K f) (x y : E) :\n    dist (f x) (f y) \u2264 (K : \u211d) * dist x y :=\n  ml_ax_sgd_lipschitz K f hf x y\n",
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            "content": "def proof_db_ml_ax_mc_021() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "diverges": false,
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    },
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      "statement": "SGD step Lipschitz descent (mega 101).",
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      "diverges": false,
      "gap_id": "G-ml",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_ax_mc_101_sgd",
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            "content": "theorem ml_ax_mc_101_sgd {E : Type*} [PseudoMetricSpace E]\n    (K : NNReal) (f : E \u2192 E) (hf : LipschitzWith K f) (x y : E) :\n    dist (f x) (f y) \u2264 (K : \u211d) * dist x y :=\n  ml_ax_sgd_lipschitz K f hf x y\n",
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      "catalog_status": "proved",
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      "gap_id": "G-ml",
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      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_ax_mc_111_sgd",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_ax_mc_111.li",
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            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_ax_mc_111_sgd",
            "start_line": 79,
            "highlight_line": 79,
            "content": "theorem ml_ax_mc_111_sgd {E : Type*} [PseudoMetricSpace E]\n    (K : NNReal) (f : E \u2192 E) (hf : LipschitzWith K f) (x y : E) :\n    dist (f x) (f y) \u2264 (K : \u211d) * dist x y :=\n  ml_ax_sgd_lipschitz K f hf x y\n",
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      "domain": "optimization",
      "statement": "SGD step Lipschitz descent (mega 121).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_ax_mc_121_sgd",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_ax_mc_121.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_ax_mc_121.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_ax_mc_121_sgd",
            "start_line": 84,
            "highlight_line": 84,
            "content": "theorem ml_ax_mc_121_sgd {E : Type*} [PseudoMetricSpace E]\n    (K : NNReal) (f : E \u2192 E) (hf : LipschitzWith K f) (x y : E) :\n    dist (f x) (f y) \u2264 (K : \u211d) * dist x y :=\n  ml_ax_sgd_lipschitz K f hf x y\n",
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            "path": "proof-db/ml/corpus/ml_ax_mc_121.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ml_ax_mc_121() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_ax_mc_121.li#L3",
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            "language": "toml",
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            "start_line": 1635,
            "highlight_line": 1636,
            "content": "[[entry]]\nid = \"ML-AX-MC-121\"\nkind = \"axiom\"\nfield = \"ml\"\ndomain = \"optimization\"\nstatement = \"SGD step Lipschitz descent (mega 121).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_ax_mc_121_sgd\"\nli_specimen = \"proof-db/ml/corpus/ml_ax_mc_121.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
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    },
    {
      "id": "ML-AX-MC-131",
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      "field": "ml",
      "domain": "optimization",
      "statement": "SGD step Lipschitz descent (mega 131).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_ax_mc_131_sgd",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_ax_mc_131.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_ax_mc_131.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
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            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_ax_mc_131_sgd",
            "start_line": 89,
            "highlight_line": 89,
            "content": "theorem ml_ax_mc_131_sgd {E : Type*} [PseudoMetricSpace E]\n    (K : NNReal) (f : E \u2192 E) (hf : LipschitzWith K f) (x y : E) :\n    dist (f x) (f y) \u2264 (K : \u211d) * dist x y :=\n  ml_ax_sgd_lipschitz K f hf x y\n",
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            "path": "proof-db/ml/corpus/ml_ax_mc_131.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_ml_ax_mc_131() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_ax_mc_131.li#L3",
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            "language": "toml",
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            "content": "[[entry]]\nid = \"ML-AX-MC-131\"\nkind = \"axiom\"\nfield = \"ml\"\ndomain = \"optimization\"\nstatement = \"SGD step Lipschitz descent (mega 131).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_ax_mc_131_sgd\"\nli_specimen = \"proof-db/ml/corpus/ml_ax_mc_131.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
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        ]
      }
    },
    {
      "id": "ML-AX-MC-141",
      "kind": "axiom",
      "field": "ml",
      "domain": "optimization",
      "statement": "SGD step Lipschitz descent (mega 141).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_ax_mc_141_sgd",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_ax_mc_141.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_ax_mc_141.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
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            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_ax_mc_141_sgd",
            "start_line": 94,
            "highlight_line": 94,
            "content": "theorem ml_ax_mc_141_sgd {E : Type*} [PseudoMetricSpace E]\n    (K : NNReal) (f : E \u2192 E) (hf : LipschitzWith K f) (x y : E) :\n    dist (f x) (f y) \u2264 (K : \u211d) * dist x y :=\n  ml_ax_sgd_lipschitz K f hf x y\n\n/-- PAC-Bayes: valid probability p \u2208 [0,1] (ML-LM-MC-002 / generalization anchor). -/",
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            "path": "proof-db/ml/corpus/ml_ax_mc_141.li",
            "symbol": null,
            "start_line": 3,
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            "content": "def proof_db_ml_ax_mc_141() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "highlight_line": 1668,
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            "role": "catalog",
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      }
    },
    {
      "id": "ML-LM-GRAD-DESCENT-RATE",
      "kind": "lemma",
      "field": "ml",
      "domain": "optimization",
      "statement": "Gradient descent on L-smooth convex f: f(x_k) - f* <= O(1/k) (rate lemma target).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": "convex_first_order",
      "benchmark_ref": "benchmarks/tier3_ml/sgd_convergence",
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.ML.ml_lm_grad_descent_rate",
      "lean_module": "proof-db/ml/SGD.lean",
      "li_specimen": "proof-db/ml/lemmas/grad_descent_step.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/lemmas/grad_descent_step.li",
      "notes": null,
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            "path": "proof-db/ml/SGD.lean",
            "symbol": "ml_lm_grad_descent_rate",
            "start_line": 13,
            "highlight_line": 13,
            "content": "axiom ml_lm_grad_descent_rate\n  (L : \u211d) (n k : Nat) (f : (Fin n \u2192 \u211d) \u2192 \u211d) (grad : (Fin n \u2192 \u211d) \u2192 (Fin n \u2192 \u211d))\n  (x\u2080 x\u22c6 : Fin n \u2192 \u211d) :\n  0 < L \u2192 0 < k \u2192\n  (\u2200 x, f x\u22c6 \u2264 f x) \u2192\n  \u2203 C : \u211d, 0 < C \u2227\n    f (Nat.rec x\u2080 (fun _ x => gradDescentStep (1 / L) grad x) k) - f x\u22c6 \u2264 C / k\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/SGD.lean#L13",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/ml/lemmas/grad_descent_step.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def quadratic_loss(x: float) -> float\n  requires true\n  ensures result == x * x\n  decreases 0\n:=\n  return x * x\n\ndef quadratic_grad(x: float) -> float\n  requires true\n  ensures result == 2.0 * x\n  decreases 0\n:=\n  return 2.0 * x\n\ndef grad_descent_step(x: float, eta: float) -> float\n  requires eta > 0.0\n  requires eta <= 1.0\n  ensures abs(result) <= abs(x)\n  decreases 0\n:=\n  var g: float = quadratic_grad(x)\n  return x - eta * g\n\n:=\n  discard grad_descent_step(1.0, 0.5)\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/lemmas/grad_descent_step.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-lemmas.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"ML-LM-GRAD-DESCENT-RATE\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"optimization\"\nstatement = \"Gradient descent on L-smooth convex f: f(x_k) - f* <= O(1/k) (rate lemma target).\"\nproof_status = \"axiomatic\"\ngap_id = \"G-ml\"\ngap_kind = \"proof_gap\"\nconvergence_class = \"convex_first_order\"\nlean_module = \"proof-db/ml/SGD.lean\"\nlean_thm = \"Li.ProofDb.ML.ml_lm_grad_descent_rate\"\nli_specimen = \"proof-db/ml/lemmas/grad_descent_step.li\"\nbenchmark_ref = \"benchmarks/tier3_ml/sgd_convergence\"\nlast_verified_lic_commit = \"7e64ba8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-lemmas.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-002",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_pac_v002",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_pac_v002",
            "start_line": 102,
            "highlight_line": 102,
            "content": "theorem ml_lm_generalization_pac_v002 (p : \u211d) (hp : p \u2264 1) : p \u2264 1 := ml_lm_pac_prob_le_one p hp\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L102",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-002: PAC-Bayes bound sketch (mega 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_002.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"ML-LM-MC-002\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 2).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_pac_v002\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L83",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-003",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 3).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_combo",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_combo",
            "start_line": 121,
            "highlight_line": 121,
            "content": "theorem ml_lm_convex_combo {E : Type*} [AddCommGroup E] [Module \u211d E] (\u03bb : \u211d) (x y : E)\n    (h\u03bb : 0 \u2264 \u03bb \u2227 \u03bb \u2264 1) : \u2203 z : E, z = \u03bb \u2022 x + (1 - \u03bb) \u2022 y := \u27e8\u03bb \u2022 x + (1 - \u03bb) \u2022 y, rfl\u27e9\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L121",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-003: Convex loss empirical risk minimization (mega 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_003.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 99,
            "highlight_line": 100,
            "content": "[[entry]]\nid = \"ML-LM-MC-003\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 3).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_combo\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L99",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-004",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_regularization_ridge_v004",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_regularization_ridge_v004",
            "start_line": 144,
            "highlight_line": 144,
            "content": "theorem ml_lm_regularization_ridge_v004 {n : Nat} (M : Matrix (Fin n) (Fin n) \u211d) :\n    M.transpose + M = M + M.transpose := ml_lm_ridge_symmetric M\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L144",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-004: Ridge regression closed form (mega 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_004.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"ML-LM-MC-004\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 4).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_regularization_ridge_v004\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-005",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_approximation_neural_v005",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_approximation_neural_v005",
            "start_line": 168,
            "highlight_line": 168,
            "content": "theorem ml_lm_approximation_neural_v005 {\u03b1 : Type*} [TopologicalSpace \u03b1] [CompactSpace \u03b1]\n    (f : \u03b1 \u2192 \u211d) (hf : Continuous f) : BddAbove (Set.range f) :=\n  ml_lm_neural_approx_bounded f hf\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L168",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-005: Universal approximation sketch (mega 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"ML-LM-MC-005\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 5).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_approximation_neural_v005\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-006",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_perceptron_v006",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_perceptron_v006",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem ml_lm_margin_perceptron_v006 {E : Type*} [InnerProductSpace \u211d E] (w : E) :\n    0 \u2264 \u27eaw, w\u27eb := ml_lm_perceptron_margin_nonneg w\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L192",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-006: Perceptron convergence (mega 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_006.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 147,
            "highlight_line": 148,
            "content": "[[entry]]\nid = \"ML-LM-MC-006\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 6).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_perceptron_v006\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-007",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_adaboost_exp_pos",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_adaboost_exp_pos",
            "start_line": 212,
            "highlight_line": 212,
            "content": "theorem ml_lm_adaboost_exp_pos (t : \u211d) : 0 < Real.exp (-t) := Real.exp_pos (-t)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L212",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-007: AdaBoost exponential loss bound (mega 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_007.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 163,
            "highlight_line": 164,
            "content": "[[entry]]\nid = \"ML-LM-MC-007\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_adaboost_exp_pos\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L163",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-008",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_inner_symm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_inner_symm",
            "start_line": 233,
            "highlight_line": 233,
            "content": "theorem ml_lm_kernel_inner_symm {E : Type*} [InnerProductSpace \u211d E] (x y : E) :\n    \u27eax, y\u27eb = \u27eay, x\u27eb := inner_comm x y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L233",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-008: Representer theorem (mega 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_008.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 179,
            "highlight_line": 180,
            "content": "[[entry]]\nid = \"ML-LM-MC-008\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 8).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_inner_symm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L179",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-009",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_scale",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_scale",
            "start_line": 256,
            "highlight_line": 256,
            "content": "theorem ml_lm_dropout_scale {a : \u211d} (x : \u211d) : a * x = x * a := mul_comm a x\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L256",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-009: Dropout as ensemble average (mega 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_009.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 195,
            "highlight_line": 196,
            "content": "[[entry]]\nid = \"ML-LM-MC-009\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 9).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_scale\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L195",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-010",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_010_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_010.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_010_batchnorm",
            "start_line": 284,
            "highlight_line": 284,
            "content": "theorem ml_lm_mc_010_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L284",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-010: Batch norm scale invariance (mega 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_010.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 211,
            "highlight_line": 212,
            "content": "[[entry]]\nid = \"ML-LM-MC-010\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 10).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_010_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L211",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-012",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002",
            "start_line": 104,
            "highlight_line": 104,
            "content": "theorem ml_lm_generalization_v002 := ml_lm_generalization_pac_v002 p hp",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L104",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-012: PAC-Bayes bound sketch (mega 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_012.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 227,
            "highlight_line": 228,
            "content": "[[entry]]\nid = \"ML-LM-MC-012\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 12).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L227",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-013",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003",
            "start_line": 124,
            "highlight_line": 124,
            "content": "theorem ml_lm_convex_v003 := ml_lm_convex_combo \u03bb x y h\u03bb",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L124",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-013: Convex loss empirical risk minimization (mega 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_013.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 243,
            "highlight_line": 244,
            "content": "[[entry]]\nid = \"ML-LM-MC-013\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 13).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L243",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-014",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_symmetric",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_symmetric",
            "start_line": 141,
            "highlight_line": 141,
            "content": "theorem ml_lm_ridge_symmetric {n : Nat} (M : Matrix (Fin n) (Fin n) \u211d) :\n    M.transpose + M = M + M.transpose := add_comm M.transpose M\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L141",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-014: Ridge regression closed form (mega 14).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_014.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 259,
            "highlight_line": 260,
            "content": "[[entry]]\nid = \"ML-LM-MC-014\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 14).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_symmetric\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L259",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-015",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_approx_bounded",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_approx_bounded",
            "start_line": 164,
            "highlight_line": 164,
            "content": "theorem ml_lm_neural_approx_bounded {\u03b1 : Type*} [TopologicalSpace \u03b1] [CompactSpace \u03b1]\n    (f : \u03b1 \u2192 \u211d) (hf : Continuous f) : BddAbove (Set.range f) :=\n  IsCompact.bddAbove (isCompact_range hf)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L164",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-015: Universal approximation sketch (mega 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_015.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 275,
            "highlight_line": 276,
            "content": "[[entry]]\nid = \"ML-LM-MC-015\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 15).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_approx_bounded\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-016",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006",
            "start_line": 195,
            "highlight_line": 195,
            "content": "theorem ml_lm_margin_v006 := ml_lm_margin_perceptron_v006 w",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L195",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-016: Perceptron convergence (mega 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_016.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 291,
            "highlight_line": 292,
            "content": "[[entry]]\nid = \"ML-LM-MC-016\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 16).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L291",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-017",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_adaboost_v007",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_adaboost_v007",
            "start_line": 214,
            "highlight_line": 214,
            "content": "theorem ml_lm_boosting_adaboost_v007 (t : \u211d) : 0 < Real.exp (-t) := ml_lm_adaboost_exp_pos t\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L214",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-017: AdaBoost exponential loss bound (mega 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_017.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 307,
            "highlight_line": 308,
            "content": "[[entry]]\nid = \"ML-LM-MC-017\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 17).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_adaboost_v007\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-018",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem ml_lm_kernel_v008 := ml_lm_representer_kernel_v008 x y",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L239",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-018: Representer theorem (mega 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_018.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 323,
            "highlight_line": 324,
            "content": "[[entry]]\nid = \"ML-LM-MC-018\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 18).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L323",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-019",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009",
            "start_line": 258,
            "highlight_line": 258,
            "content": "theorem ml_lm_dropout_v009 := ml_lm_dropout_scale a x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L258",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-019: Dropout as ensemble average (mega 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_019.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 339,
            "highlight_line": 340,
            "content": "[[entry]]\nid = \"ML-LM-MC-019\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 19).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L339",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-020",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_020_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_020.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_020_batchnorm",
            "start_line": 288,
            "highlight_line": 288,
            "content": "theorem ml_lm_mc_020_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L288",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-020: Batch norm scale invariance (mega 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_020.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 355,
            "highlight_line": 356,
            "content": "[[entry]]\nid = \"ML-LM-MC-020\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 20).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_020_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L355",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-022",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v012",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v012",
            "start_line": 105,
            "highlight_line": 105,
            "content": "theorem ml_lm_generalization_v002_v012 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L105",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-022: PAC-Bayes bound sketch (mega 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_022.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 371,
            "highlight_line": 372,
            "content": "[[entry]]\nid = \"ML-LM-MC-022\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 22).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v012\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L371",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-023",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v013",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v013",
            "start_line": 125,
            "highlight_line": 125,
            "content": "theorem ml_lm_convex_v003_v013 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L125",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-023: Convex loss empirical risk minimization (mega 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_023.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 387,
            "highlight_line": 388,
            "content": "[[entry]]\nid = \"ML-LM-MC-023\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 23).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v013\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L387",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-024",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004",
            "start_line": 147,
            "highlight_line": 147,
            "content": "theorem ml_lm_ridge_v004 := ml_lm_regularization_ridge_v004 M",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L147",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-024: Ridge regression closed form (mega 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_024.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 403,
            "highlight_line": 404,
            "content": "[[entry]]\nid = \"ML-LM-MC-024\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 24).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L403",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-025",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005",
            "start_line": 172,
            "highlight_line": 172,
            "content": "theorem ml_lm_neural_v005 := ml_lm_approximation_neural_v005 f hf",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L172",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-025: Universal approximation sketch (mega 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_025.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 419,
            "highlight_line": 420,
            "content": "[[entry]]\nid = \"ML-LM-MC-025\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 25).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L419",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-026",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v016",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v016",
            "start_line": 196,
            "highlight_line": 196,
            "content": "theorem ml_lm_margin_v006_v016 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L196",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-026: Perceptron convergence (mega 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_026.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 435,
            "highlight_line": 436,
            "content": "[[entry]]\nid = \"ML-LM-MC-026\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 26).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v016\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L435",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-027",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007",
            "start_line": 216,
            "highlight_line": 216,
            "content": "theorem ml_lm_boosting_v007 := ml_lm_boosting_adaboost_v007 t",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L216",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-027: AdaBoost exponential loss bound (mega 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_027.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 451,
            "highlight_line": 452,
            "content": "[[entry]]\nid = \"ML-LM-MC-027\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 27).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L451",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-028",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v018",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_028.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v018",
            "start_line": 240,
            "highlight_line": 240,
            "content": "theorem ml_lm_kernel_v008_v018 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L240",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-028: Representer theorem (mega 28).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_028.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 467,
            "highlight_line": 468,
            "content": "[[entry]]\nid = \"ML-LM-MC-028\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 28).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v018\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L467",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-029",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v019",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v019",
            "start_line": 259,
            "highlight_line": 259,
            "content": "theorem ml_lm_dropout_v009_v019 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L259",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-029: Dropout as ensemble average (mega 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_029.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 483,
            "highlight_line": 484,
            "content": "[[entry]]\nid = \"ML-LM-MC-029\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 29).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v019\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L483",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-030",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_030_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_030_batchnorm",
            "start_line": 292,
            "highlight_line": 292,
            "content": "theorem ml_lm_mc_030_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L292",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-030: Batch norm scale invariance (mega 30).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_030.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 499,
            "highlight_line": 500,
            "content": "[[entry]]\nid = \"ML-LM-MC-030\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 30).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_030_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L499",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-032",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v022",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v022",
            "start_line": 106,
            "highlight_line": 106,
            "content": "theorem ml_lm_generalization_v002_v022 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L106",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-032: PAC-Bayes bound sketch (mega 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_032.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 515,
            "highlight_line": 516,
            "content": "[[entry]]\nid = \"ML-LM-MC-032\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 32).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v022\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L515",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-033",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v023",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v023",
            "start_line": 126,
            "highlight_line": 126,
            "content": "theorem ml_lm_convex_v003_v023 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L126",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-033: Convex loss empirical risk minimization (mega 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_033.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 531,
            "highlight_line": 532,
            "content": "[[entry]]\nid = \"ML-LM-MC-033\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 33).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v023\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L531",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-034",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v014",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v014",
            "start_line": 148,
            "highlight_line": 148,
            "content": "theorem ml_lm_ridge_v004_v014 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L148",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-034: Ridge regression closed form (mega 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_034.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 547,
            "highlight_line": 548,
            "content": "[[entry]]\nid = \"ML-LM-MC-034\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 34).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v014\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L547",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-035",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v015",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_035.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v015",
            "start_line": 173,
            "highlight_line": 173,
            "content": "theorem ml_lm_neural_v005_v015 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L173",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-035: Universal approximation sketch (mega 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_035.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 563,
            "highlight_line": 564,
            "content": "[[entry]]\nid = \"ML-LM-MC-035\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 35).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v015\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L563",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-036",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v026",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v026",
            "start_line": 197,
            "highlight_line": 197,
            "content": "theorem ml_lm_margin_v006_v026 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L197",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-036: Perceptron convergence (mega 36).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_036.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 579,
            "highlight_line": 580,
            "content": "[[entry]]\nid = \"ML-LM-MC-036\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 36).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v026\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L579",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-037",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v017",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v017",
            "start_line": 217,
            "highlight_line": 217,
            "content": "theorem ml_lm_boosting_v007_v017 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L217",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-037: AdaBoost exponential loss bound (mega 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_037.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 595,
            "highlight_line": 596,
            "content": "[[entry]]\nid = \"ML-LM-MC-037\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 37).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v017\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L595",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-038",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v028",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v028",
            "start_line": 241,
            "highlight_line": 241,
            "content": "theorem ml_lm_kernel_v008_v028 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L241",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-038: Representer theorem (mega 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_038.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 611,
            "highlight_line": 612,
            "content": "[[entry]]\nid = \"ML-LM-MC-038\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 38).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v028\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L611",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-039",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v029",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v029",
            "start_line": 260,
            "highlight_line": 260,
            "content": "theorem ml_lm_dropout_v009_v029 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L260",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-039: Dropout as ensemble average (mega 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_039.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 627,
            "highlight_line": 628,
            "content": "[[entry]]\nid = \"ML-LM-MC-039\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 39).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v029\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L627",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-040",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_040_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_040.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_040_batchnorm",
            "start_line": 296,
            "highlight_line": 296,
            "content": "theorem ml_lm_mc_040_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L296",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-040: Batch norm scale invariance (mega 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_040.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 643,
            "highlight_line": 644,
            "content": "[[entry]]\nid = \"ML-LM-MC-040\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 40).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_040_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L643",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-042",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v032",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v032",
            "start_line": 107,
            "highlight_line": 107,
            "content": "theorem ml_lm_generalization_v002_v032 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L107",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-042: PAC-Bayes bound sketch (mega 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_042.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 659,
            "highlight_line": 660,
            "content": "[[entry]]\nid = \"ML-LM-MC-042\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v032\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L659",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-043",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v033",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v033",
            "start_line": 127,
            "highlight_line": 127,
            "content": "theorem ml_lm_convex_v003_v033 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L127",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-043: Convex loss empirical risk minimization (mega 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_043.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 675,
            "highlight_line": 676,
            "content": "[[entry]]\nid = \"ML-LM-MC-043\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 43).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v033\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L675",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-044",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v024",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v024",
            "start_line": 149,
            "highlight_line": 149,
            "content": "theorem ml_lm_ridge_v004_v024 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L149",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-044: Ridge regression closed form (mega 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_044.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 691,
            "highlight_line": 692,
            "content": "[[entry]]\nid = \"ML-LM-MC-044\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 44).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v024\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L691",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-045",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v025",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v025",
            "start_line": 174,
            "highlight_line": 174,
            "content": "theorem ml_lm_neural_v005_v025 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L174",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-045: Universal approximation sketch (mega 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_045.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 707,
            "highlight_line": 708,
            "content": "[[entry]]\nid = \"ML-LM-MC-045\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 45).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v025\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L707",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-046",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v036",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v036",
            "start_line": 198,
            "highlight_line": 198,
            "content": "theorem ml_lm_margin_v006_v036 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L198",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-046: Perceptron convergence (mega 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_046.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 723,
            "highlight_line": 724,
            "content": "[[entry]]\nid = \"ML-LM-MC-046\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 46).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v036\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L723",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-047",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v027",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v027",
            "start_line": 218,
            "highlight_line": 218,
            "content": "theorem ml_lm_boosting_v007_v027 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L218",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-047: AdaBoost exponential loss bound (mega 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_047.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 739,
            "highlight_line": 740,
            "content": "[[entry]]\nid = \"ML-LM-MC-047\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 47).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v027\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L739",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-048",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v038",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v038",
            "start_line": 242,
            "highlight_line": 242,
            "content": "theorem ml_lm_kernel_v008_v038 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L242",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-048: Representer theorem (mega 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_048.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 755,
            "highlight_line": 756,
            "content": "[[entry]]\nid = \"ML-LM-MC-048\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 48).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v038\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L755",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-049",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v039",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_049.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v039",
            "start_line": 261,
            "highlight_line": 261,
            "content": "theorem ml_lm_dropout_v009_v039 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L261",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-049: Dropout as ensemble average (mega 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_049.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 771,
            "highlight_line": 772,
            "content": "[[entry]]\nid = \"ML-LM-MC-049\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 49).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v039\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L771",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-050",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_050_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_050_batchnorm",
            "start_line": 300,
            "highlight_line": 300,
            "content": "theorem ml_lm_mc_050_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L300",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-050: Batch norm scale invariance (mega 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_050.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 787,
            "highlight_line": 788,
            "content": "[[entry]]\nid = \"ML-LM-MC-050\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 50).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_050_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L787",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-052",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v042",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v042",
            "start_line": 108,
            "highlight_line": 108,
            "content": "theorem ml_lm_generalization_v002_v042 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L108",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-052: PAC-Bayes bound sketch (mega 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_052.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 883,
            "highlight_line": 884,
            "content": "[[entry]]\nid = \"ML-LM-MC-052\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 52).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v042\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L883",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-053",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v043",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v043",
            "start_line": 128,
            "highlight_line": 128,
            "content": "theorem ml_lm_convex_v003_v043 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L128",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-053: Convex loss empirical risk minimization (mega 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_053.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 899,
            "highlight_line": 900,
            "content": "[[entry]]\nid = \"ML-LM-MC-053\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 53).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v043\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L899",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-054",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v034",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
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            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v034",
            "start_line": 150,
            "highlight_line": 150,
            "content": "theorem ml_lm_ridge_v004_v034 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L150",
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            "path": "proof-db/ml/corpus/ml_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-054: Ridge regression closed form (mega 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_054.li#L1",
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            "language": "toml",
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            "start_line": 915,
            "highlight_line": 916,
            "content": "[[entry]]\nid = \"ML-LM-MC-054\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 54).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v034\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L915",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-055",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v035",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v035",
            "start_line": 175,
            "highlight_line": 175,
            "content": "theorem ml_lm_neural_v005_v035 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L175",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-055: Universal approximation sketch (mega 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_055.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 931,
            "highlight_line": 932,
            "content": "[[entry]]\nid = \"ML-LM-MC-055\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 55).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v035\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L931",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-056",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v046",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_056.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v046",
            "start_line": 199,
            "highlight_line": 199,
            "content": "theorem ml_lm_margin_v006_v046 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L199",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-056: Perceptron convergence (mega 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_056.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 947,
            "highlight_line": 948,
            "content": "[[entry]]\nid = \"ML-LM-MC-056\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 56).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v046\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L947",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-057",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v037",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v037",
            "start_line": 219,
            "highlight_line": 219,
            "content": "theorem ml_lm_boosting_v007_v037 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L219",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-057: AdaBoost exponential loss bound (mega 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_057.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 963,
            "highlight_line": 964,
            "content": "[[entry]]\nid = \"ML-LM-MC-057\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 57).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v037\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L963",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-058",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v048",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v048",
            "start_line": 243,
            "highlight_line": 243,
            "content": "theorem ml_lm_kernel_v008_v048 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L243",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-058: Representer theorem (mega 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_058.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 979,
            "highlight_line": 980,
            "content": "[[entry]]\nid = \"ML-LM-MC-058\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 58).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v048\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L979",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-059",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v049",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v049",
            "start_line": 262,
            "highlight_line": 262,
            "content": "theorem ml_lm_dropout_v009_v049 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L262",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-059: Dropout as ensemble average (mega 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_059.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 995,
            "highlight_line": 996,
            "content": "[[entry]]\nid = \"ML-LM-MC-059\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 59).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v049\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L995",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-060",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_060_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_060_batchnorm",
            "start_line": 304,
            "highlight_line": 304,
            "content": "theorem ml_lm_mc_060_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L304",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-060: Batch norm scale invariance (mega 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_060.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1011,
            "highlight_line": 1012,
            "content": "[[entry]]\nid = \"ML-LM-MC-060\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 60).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_060_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1011",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-062",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v052",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v052",
            "start_line": 109,
            "highlight_line": 109,
            "content": "theorem ml_lm_generalization_v002_v052 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L109",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-062: PAC-Bayes bound sketch (mega 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_062.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1027,
            "highlight_line": 1028,
            "content": "[[entry]]\nid = \"ML-LM-MC-062\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 62).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v052\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1027",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-063",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v053",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_063.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v053",
            "start_line": 129,
            "highlight_line": 129,
            "content": "theorem ml_lm_convex_v003_v053 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L129",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-063: Convex loss empirical risk minimization (mega 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_063.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1043,
            "highlight_line": 1044,
            "content": "[[entry]]\nid = \"ML-LM-MC-063\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 63).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v053\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1043",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-064",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v044",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v044",
            "start_line": 151,
            "highlight_line": 151,
            "content": "theorem ml_lm_ridge_v004_v044 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L151",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-064: Ridge regression closed form (mega 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_064.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1059,
            "highlight_line": 1060,
            "content": "[[entry]]\nid = \"ML-LM-MC-064\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 64).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v044\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1059",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-065",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v045",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v045",
            "start_line": 176,
            "highlight_line": 176,
            "content": "theorem ml_lm_neural_v005_v045 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L176",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-065: Universal approximation sketch (mega 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_065.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1075,
            "highlight_line": 1076,
            "content": "[[entry]]\nid = \"ML-LM-MC-065\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 65).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v045\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1075",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-066",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v056",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v056",
            "start_line": 200,
            "highlight_line": 200,
            "content": "theorem ml_lm_margin_v006_v056 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L200",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-066: Perceptron convergence (mega 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_066.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1091,
            "highlight_line": 1092,
            "content": "[[entry]]\nid = \"ML-LM-MC-066\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 66).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v056\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1091",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-067",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v047",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v047",
            "start_line": 220,
            "highlight_line": 220,
            "content": "theorem ml_lm_boosting_v007_v047 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L220",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-067: AdaBoost exponential loss bound (mega 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1107,
            "highlight_line": 1108,
            "content": "[[entry]]\nid = \"ML-LM-MC-067\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 67).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v047\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1107",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-068",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v058",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v058",
            "start_line": 244,
            "highlight_line": 244,
            "content": "theorem ml_lm_kernel_v008_v058 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L244",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-068: Representer theorem (mega 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_068.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1123,
            "highlight_line": 1124,
            "content": "[[entry]]\nid = \"ML-LM-MC-068\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 68).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v058\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1123",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-069",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v059",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v059",
            "start_line": 263,
            "highlight_line": 263,
            "content": "theorem ml_lm_dropout_v009_v059 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L263",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-069: Dropout as ensemble average (mega 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_069.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1139,
            "highlight_line": 1140,
            "content": "[[entry]]\nid = \"ML-LM-MC-069\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 69).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v059\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1139",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-070",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_070_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_070.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_070_batchnorm",
            "start_line": 308,
            "highlight_line": 308,
            "content": "theorem ml_lm_mc_070_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L308",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-070: Batch norm scale invariance (mega 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_070.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1155,
            "highlight_line": 1156,
            "content": "[[entry]]\nid = \"ML-LM-MC-070\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 70).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_070_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1155",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-072",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v062",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v062",
            "start_line": 110,
            "highlight_line": 110,
            "content": "theorem ml_lm_generalization_v002_v062 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L110",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-072: PAC-Bayes bound sketch (mega 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_072.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1171,
            "highlight_line": 1172,
            "content": "[[entry]]\nid = \"ML-LM-MC-072\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 72).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v062\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1171",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-073",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v063",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v063",
            "start_line": 130,
            "highlight_line": 130,
            "content": "theorem ml_lm_convex_v003_v063 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L130",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-073: Convex loss empirical risk minimization (mega 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_073.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1187,
            "highlight_line": 1188,
            "content": "[[entry]]\nid = \"ML-LM-MC-073\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 73).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v063\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1187",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-074",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v054",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v054",
            "start_line": 152,
            "highlight_line": 152,
            "content": "theorem ml_lm_ridge_v004_v054 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L152",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-074: Ridge regression closed form (mega 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_074.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1203,
            "highlight_line": 1204,
            "content": "[[entry]]\nid = \"ML-LM-MC-074\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 74).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v054\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1203",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-075",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v055",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v055",
            "start_line": 177,
            "highlight_line": 177,
            "content": "theorem ml_lm_neural_v005_v055 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L177",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-075: Universal approximation sketch (mega 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_075.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1219,
            "highlight_line": 1220,
            "content": "[[entry]]\nid = \"ML-LM-MC-075\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 75).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v055\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1219",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-076",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 76).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v066",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v066",
            "start_line": 201,
            "highlight_line": 201,
            "content": "theorem ml_lm_margin_v006_v066 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L201",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-076: Perceptron convergence (mega 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_076.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1235,
            "highlight_line": 1236,
            "content": "[[entry]]\nid = \"ML-LM-MC-076\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 76).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v066\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1235",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-077",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v057",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v057",
            "start_line": 221,
            "highlight_line": 221,
            "content": "theorem ml_lm_boosting_v007_v057 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L221",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-077: AdaBoost exponential loss bound (mega 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_077.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1251,
            "highlight_line": 1252,
            "content": "[[entry]]\nid = \"ML-LM-MC-077\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 77).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v057\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1251",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-078",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v068",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v068",
            "start_line": 245,
            "highlight_line": 245,
            "content": "theorem ml_lm_kernel_v008_v068 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L245",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-078: Representer theorem (mega 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_078.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1267,
            "highlight_line": 1268,
            "content": "[[entry]]\nid = \"ML-LM-MC-078\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 78).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v068\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1267",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-079",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v069",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v069",
            "start_line": 264,
            "highlight_line": 264,
            "content": "theorem ml_lm_dropout_v009_v069 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L264",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-079: Dropout as ensemble average (mega 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_079.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1283,
            "highlight_line": 1284,
            "content": "[[entry]]\nid = \"ML-LM-MC-079\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 79).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v069\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1283",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-080",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_080_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_080.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_080_batchnorm",
            "start_line": 312,
            "highlight_line": 312,
            "content": "theorem ml_lm_mc_080_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L312",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-080: Batch norm scale invariance (mega 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_080.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1299,
            "highlight_line": 1300,
            "content": "[[entry]]\nid = \"ML-LM-MC-080\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 80).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_080_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1299",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-082",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v072",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v072",
            "start_line": 111,
            "highlight_line": 111,
            "content": "theorem ml_lm_generalization_v002_v072 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L111",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-082: PAC-Bayes bound sketch (mega 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_082.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1315,
            "highlight_line": 1316,
            "content": "[[entry]]\nid = \"ML-LM-MC-082\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 82).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v072\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1315",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-083",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v073",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v073",
            "start_line": 131,
            "highlight_line": 131,
            "content": "theorem ml_lm_convex_v003_v073 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L131",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-083: Convex loss empirical risk minimization (mega 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_083.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1331,
            "highlight_line": 1332,
            "content": "[[entry]]\nid = \"ML-LM-MC-083\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 83).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v073\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1331",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-084",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v064",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_084.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v064",
            "start_line": 153,
            "highlight_line": 153,
            "content": "theorem ml_lm_ridge_v004_v064 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L153",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-084: Ridge regression closed form (mega 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_084.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1347,
            "highlight_line": 1348,
            "content": "[[entry]]\nid = \"ML-LM-MC-084\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 84).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v064\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1347",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-085",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v065",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v065",
            "start_line": 178,
            "highlight_line": 178,
            "content": "theorem ml_lm_neural_v005_v065 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L178",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-085: Universal approximation sketch (mega 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_085.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1363,
            "highlight_line": 1364,
            "content": "[[entry]]\nid = \"ML-LM-MC-085\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 85).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v065\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1363",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-086",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 86).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v076",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v076",
            "start_line": 202,
            "highlight_line": 202,
            "content": "theorem ml_lm_margin_v006_v076 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L202",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-086: Perceptron convergence (mega 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_086.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1379,
            "highlight_line": 1380,
            "content": "[[entry]]\nid = \"ML-LM-MC-086\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 86).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v076\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1379",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-087",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v067",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v067",
            "start_line": 222,
            "highlight_line": 222,
            "content": "theorem ml_lm_boosting_v007_v067 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L222",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-087: AdaBoost exponential loss bound (mega 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_087.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1395,
            "highlight_line": 1396,
            "content": "[[entry]]\nid = \"ML-LM-MC-087\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 87).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v067\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1395",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-088",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v078",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v078",
            "start_line": 246,
            "highlight_line": 246,
            "content": "theorem ml_lm_kernel_v008_v078 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L246",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-088: Representer theorem (mega 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_088.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1411,
            "highlight_line": 1412,
            "content": "[[entry]]\nid = \"ML-LM-MC-088\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 88).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v078\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1411",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-089",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v079",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v079",
            "start_line": 265,
            "highlight_line": 265,
            "content": "theorem ml_lm_dropout_v009_v079 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L265",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-089: Dropout as ensemble average (mega 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_089.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1427,
            "highlight_line": 1428,
            "content": "[[entry]]\nid = \"ML-LM-MC-089\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 89).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v079\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1427",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-090",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_090_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_090.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_090_batchnorm",
            "start_line": 316,
            "highlight_line": 316,
            "content": "theorem ml_lm_mc_090_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L316",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-090: Batch norm scale invariance (mega 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_090.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1443,
            "highlight_line": 1444,
            "content": "[[entry]]\nid = \"ML-LM-MC-090\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 90).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_090_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1443",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-092",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v082",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v082",
            "start_line": 112,
            "highlight_line": 112,
            "content": "theorem ml_lm_generalization_v002_v082 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L112",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-092: PAC-Bayes bound sketch (mega 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_092.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1459,
            "highlight_line": 1460,
            "content": "[[entry]]\nid = \"ML-LM-MC-092\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 92).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v082\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1459",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-093",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v083",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v083",
            "start_line": 132,
            "highlight_line": 132,
            "content": "theorem ml_lm_convex_v003_v083 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L132",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-093: Convex loss empirical risk minimization (mega 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_093.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1475,
            "highlight_line": 1476,
            "content": "[[entry]]\nid = \"ML-LM-MC-093\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 93).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v083\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1475",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-094",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v074",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v074",
            "start_line": 154,
            "highlight_line": 154,
            "content": "theorem ml_lm_ridge_v004_v074 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L154",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-094: Ridge regression closed form (mega 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_094.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1491,
            "highlight_line": 1492,
            "content": "[[entry]]\nid = \"ML-LM-MC-094\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 94).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v074\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1491",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-095",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v075",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v075",
            "start_line": 179,
            "highlight_line": 179,
            "content": "theorem ml_lm_neural_v005_v075 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L179",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-095: Universal approximation sketch (mega 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_095.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1507,
            "highlight_line": 1508,
            "content": "[[entry]]\nid = \"ML-LM-MC-095\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 95).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v075\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1507",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-096",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v086",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v086",
            "start_line": 203,
            "highlight_line": 203,
            "content": "theorem ml_lm_margin_v006_v086 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L203",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-096: Perceptron convergence (mega 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_096.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1523,
            "highlight_line": 1524,
            "content": "[[entry]]\nid = \"ML-LM-MC-096\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 96).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v086\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1523",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-097",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v077",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v077",
            "start_line": 223,
            "highlight_line": 223,
            "content": "theorem ml_lm_boosting_v007_v077 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L223",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-097: AdaBoost exponential loss bound (mega 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_097.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1539,
            "highlight_line": 1540,
            "content": "[[entry]]\nid = \"ML-LM-MC-097\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 97).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v077\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1539",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-098",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v088",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v088",
            "start_line": 247,
            "highlight_line": 247,
            "content": "theorem ml_lm_kernel_v008_v088 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L247",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-098: Representer theorem (mega 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_098.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1555,
            "highlight_line": 1556,
            "content": "[[entry]]\nid = \"ML-LM-MC-098\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 98).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v088\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1555",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-099",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v089",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v089",
            "start_line": 266,
            "highlight_line": 266,
            "content": "theorem ml_lm_dropout_v009_v089 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L266",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-099: Dropout as ensemble average (mega 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_099.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1571,
            "highlight_line": 1572,
            "content": "[[entry]]\nid = \"ML-LM-MC-099\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 99).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v089\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1571",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-100",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_100_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_100_batchnorm",
            "start_line": 320,
            "highlight_line": 320,
            "content": "theorem ml_lm_mc_100_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L320",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-100: Batch norm scale invariance (mega 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_100.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1587,
            "highlight_line": 1588,
            "content": "[[entry]]\nid = \"ML-LM-MC-100\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 100).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_100_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1587",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-102",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v092",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v092",
            "start_line": 113,
            "highlight_line": 113,
            "content": "theorem ml_lm_generalization_v002_v092 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L113",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-102: PAC-Bayes bound sketch (mega 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_102.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1683,
            "highlight_line": 1684,
            "content": "[[entry]]\nid = \"ML-LM-MC-102\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 102).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v092\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1683",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-103",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v093",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v093",
            "start_line": 133,
            "highlight_line": 133,
            "content": "theorem ml_lm_convex_v003_v093 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L133",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-103: Convex loss empirical risk minimization (mega 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_103.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1699,
            "highlight_line": 1700,
            "content": "[[entry]]\nid = \"ML-LM-MC-103\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 103).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v093\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1699",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-104",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v084",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v084",
            "start_line": 155,
            "highlight_line": 155,
            "content": "theorem ml_lm_ridge_v004_v084 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L155",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-104: Ridge regression closed form (mega 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_104.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1715,
            "highlight_line": 1716,
            "content": "[[entry]]\nid = \"ML-LM-MC-104\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 104).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v084\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1715",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-105",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v085",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_105.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v085",
            "start_line": 180,
            "highlight_line": 180,
            "content": "theorem ml_lm_neural_v005_v085 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L180",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-105: Universal approximation sketch (mega 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_105.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1731,
            "highlight_line": 1732,
            "content": "[[entry]]\nid = \"ML-LM-MC-105\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 105).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v085\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1731",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-106",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v096",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v096",
            "start_line": 204,
            "highlight_line": 204,
            "content": "theorem ml_lm_margin_v006_v096 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L204",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-106: Perceptron convergence (mega 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_106.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1747,
            "highlight_line": 1748,
            "content": "[[entry]]\nid = \"ML-LM-MC-106\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 106).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v096\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1747",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-107",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v087",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v087",
            "start_line": 224,
            "highlight_line": 224,
            "content": "theorem ml_lm_boosting_v007_v087 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L224",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-107: AdaBoost exponential loss bound (mega 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_107.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1763,
            "highlight_line": 1764,
            "content": "[[entry]]\nid = \"ML-LM-MC-107\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 107).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v087\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1763",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-108",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v098",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v098",
            "start_line": 248,
            "highlight_line": 248,
            "content": "theorem ml_lm_kernel_v008_v098 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L248",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-108: Representer theorem (mega 108).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_108.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1779,
            "highlight_line": 1780,
            "content": "[[entry]]\nid = \"ML-LM-MC-108\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 108).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v098\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1779",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-109",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v099",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v099",
            "start_line": 267,
            "highlight_line": 267,
            "content": "theorem ml_lm_dropout_v009_v099 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L267",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-109: Dropout as ensemble average (mega 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_109.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1795,
            "highlight_line": 1796,
            "content": "[[entry]]\nid = \"ML-LM-MC-109\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 109).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v099\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1795",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-110",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_110_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_110_batchnorm",
            "start_line": 324,
            "highlight_line": 324,
            "content": "theorem ml_lm_mc_110_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L324",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-110: Batch norm scale invariance (mega 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_110.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1811,
            "highlight_line": 1812,
            "content": "[[entry]]\nid = \"ML-LM-MC-110\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 110).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_110_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1811",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-112",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v102",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v102",
            "start_line": 114,
            "highlight_line": 114,
            "content": "theorem ml_lm_generalization_v002_v102 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L114",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-112: PAC-Bayes bound sketch (mega 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_112.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1827,
            "highlight_line": 1828,
            "content": "[[entry]]\nid = \"ML-LM-MC-112\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 112).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v102\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1827",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-113",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v103",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v103",
            "start_line": 134,
            "highlight_line": 134,
            "content": "theorem ml_lm_convex_v003_v103 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L134",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-113: Convex loss empirical risk minimization (mega 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_113.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1843,
            "highlight_line": 1844,
            "content": "[[entry]]\nid = \"ML-LM-MC-113\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 113).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v103\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1843",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-114",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v094",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v094",
            "start_line": 156,
            "highlight_line": 156,
            "content": "theorem ml_lm_ridge_v004_v094 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L156",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-114: Ridge regression closed form (mega 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_114.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1859,
            "highlight_line": 1860,
            "content": "[[entry]]\nid = \"ML-LM-MC-114\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 114).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v094\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1859",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-115",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v095",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v095",
            "start_line": 181,
            "highlight_line": 181,
            "content": "theorem ml_lm_neural_v005_v095 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L181",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-115: Universal approximation sketch (mega 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_115.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1875,
            "highlight_line": 1876,
            "content": "[[entry]]\nid = \"ML-LM-MC-115\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 115).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v095\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1875",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-116",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v106",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v106",
            "start_line": 205,
            "highlight_line": 205,
            "content": "theorem ml_lm_margin_v006_v106 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L205",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-116: Perceptron convergence (mega 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_116.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1891,
            "highlight_line": 1892,
            "content": "[[entry]]\nid = \"ML-LM-MC-116\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 116).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v106\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1891",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-117",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v097",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v097",
            "start_line": 225,
            "highlight_line": 225,
            "content": "theorem ml_lm_boosting_v007_v097 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L225",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-117: AdaBoost exponential loss bound (mega 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_117.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1907,
            "highlight_line": 1908,
            "content": "[[entry]]\nid = \"ML-LM-MC-117\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 117).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v097\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1907",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-118",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v108",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v108",
            "start_line": 249,
            "highlight_line": 249,
            "content": "theorem ml_lm_kernel_v008_v108 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L249",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-118: Representer theorem (mega 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_118.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1923,
            "highlight_line": 1924,
            "content": "[[entry]]\nid = \"ML-LM-MC-118\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 118).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v108\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1923",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-119",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v109",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v109",
            "start_line": 268,
            "highlight_line": 268,
            "content": "theorem ml_lm_dropout_v009_v109 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L268",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-119: Dropout as ensemble average (mega 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1939,
            "highlight_line": 1940,
            "content": "[[entry]]\nid = \"ML-LM-MC-119\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v109\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1939",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-120",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_120_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_120_batchnorm",
            "start_line": 328,
            "highlight_line": 328,
            "content": "theorem ml_lm_mc_120_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L328",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-120: Batch norm scale invariance (mega 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1955,
            "highlight_line": 1956,
            "content": "[[entry]]\nid = \"ML-LM-MC-120\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_120_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1955",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-122",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v112",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v112",
            "start_line": 115,
            "highlight_line": 115,
            "content": "theorem ml_lm_generalization_v002_v112 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L115",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-122: PAC-Bayes bound sketch (mega 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1971,
            "highlight_line": 1972,
            "content": "[[entry]]\nid = \"ML-LM-MC-122\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v112\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1971",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-123",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v113",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v113",
            "start_line": 135,
            "highlight_line": 135,
            "content": "theorem ml_lm_convex_v003_v113 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L135",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-123: Convex loss empirical risk minimization (mega 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 1987,
            "highlight_line": 1988,
            "content": "[[entry]]\nid = \"ML-LM-MC-123\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 123).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v113\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L1987",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-124",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v104",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v104",
            "start_line": 157,
            "highlight_line": 157,
            "content": "theorem ml_lm_ridge_v004_v104 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L157",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-124: Ridge regression closed form (mega 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2003,
            "highlight_line": 2004,
            "content": "[[entry]]\nid = \"ML-LM-MC-124\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 124).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v104\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2003",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-125",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v105",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v105",
            "start_line": 182,
            "highlight_line": 182,
            "content": "theorem ml_lm_neural_v005_v105 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L182",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-125: Universal approximation sketch (mega 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2019,
            "highlight_line": 2020,
            "content": "[[entry]]\nid = \"ML-LM-MC-125\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 125).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v105\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-126",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v116",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_126.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v116",
            "start_line": 206,
            "highlight_line": 206,
            "content": "theorem ml_lm_margin_v006_v116 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L206",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-126: Perceptron convergence (mega 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2035,
            "highlight_line": 2036,
            "content": "[[entry]]\nid = \"ML-LM-MC-126\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 126).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v116\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2035",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-127",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v107",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v107",
            "start_line": 226,
            "highlight_line": 226,
            "content": "theorem ml_lm_boosting_v007_v107 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L226",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-127: AdaBoost exponential loss bound (mega 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2051,
            "highlight_line": 2052,
            "content": "[[entry]]\nid = \"ML-LM-MC-127\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 127).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v107\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2051",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-128",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v118",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v118",
            "start_line": 250,
            "highlight_line": 250,
            "content": "theorem ml_lm_kernel_v008_v118 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L250",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-128: Representer theorem (mega 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2067,
            "highlight_line": 2068,
            "content": "[[entry]]\nid = \"ML-LM-MC-128\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 128).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v118\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2067",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-129",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v119",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v119",
            "start_line": 269,
            "highlight_line": 269,
            "content": "theorem ml_lm_dropout_v009_v119 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L269",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-129: Dropout as ensemble average (mega 129).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_129.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2083,
            "highlight_line": 2084,
            "content": "[[entry]]\nid = \"ML-LM-MC-129\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 129).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v119\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2083",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-130",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_130_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_130.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_130_batchnorm",
            "start_line": 332,
            "highlight_line": 332,
            "content": "theorem ml_lm_mc_130_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L332",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-130: Batch norm scale invariance (mega 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_130.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2099,
            "highlight_line": 2100,
            "content": "[[entry]]\nid = \"ML-LM-MC-130\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 130).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_130_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2099",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-132",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v122",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v122",
            "start_line": 116,
            "highlight_line": 116,
            "content": "theorem ml_lm_generalization_v002_v122 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L116",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-132: PAC-Bayes bound sketch (mega 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_132.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2115,
            "highlight_line": 2116,
            "content": "[[entry]]\nid = \"ML-LM-MC-132\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 132).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v122\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-133",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v123",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v123",
            "start_line": 136,
            "highlight_line": 136,
            "content": "theorem ml_lm_convex_v003_v123 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L136",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-133: Convex loss empirical risk minimization (mega 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_133.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2131,
            "highlight_line": 2132,
            "content": "[[entry]]\nid = \"ML-LM-MC-133\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 133).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v123\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-134",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v114",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v114",
            "start_line": 158,
            "highlight_line": 158,
            "content": "theorem ml_lm_ridge_v004_v114 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L158",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-134: Ridge regression closed form (mega 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_134.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2147,
            "highlight_line": 2148,
            "content": "[[entry]]\nid = \"ML-LM-MC-134\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 134).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v114\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-135",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v115",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v115",
            "start_line": 183,
            "highlight_line": 183,
            "content": "theorem ml_lm_neural_v005_v115 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L183",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-135: Universal approximation sketch (mega 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_135.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2163,
            "highlight_line": 2164,
            "content": "[[entry]]\nid = \"ML-LM-MC-135\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 135).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v115\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2163",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-136",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v126",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v126",
            "start_line": 207,
            "highlight_line": 207,
            "content": "theorem ml_lm_margin_v006_v126 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L207",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-136: Perceptron convergence (mega 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_136.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2179,
            "highlight_line": 2180,
            "content": "[[entry]]\nid = \"ML-LM-MC-136\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 136).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v126\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2179",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-137",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v117",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v117",
            "start_line": 227,
            "highlight_line": 227,
            "content": "theorem ml_lm_boosting_v007_v117 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L227",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-137: AdaBoost exponential loss bound (mega 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_137.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2195,
            "highlight_line": 2196,
            "content": "[[entry]]\nid = \"ML-LM-MC-137\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 137).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v117\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2195",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-138",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v128",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v128",
            "start_line": 251,
            "highlight_line": 251,
            "content": "theorem ml_lm_kernel_v008_v128 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L251",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-138: Representer theorem (mega 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_138.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2211,
            "highlight_line": 2212,
            "content": "[[entry]]\nid = \"ML-LM-MC-138\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 138).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v128\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2211",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-139",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v129",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v129",
            "start_line": 270,
            "highlight_line": 270,
            "content": "theorem ml_lm_dropout_v009_v129 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L270",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-139: Dropout as ensemble average (mega 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_139.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2227,
            "highlight_line": 2228,
            "content": "[[entry]]\nid = \"ML-LM-MC-139\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 139).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v129\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2227",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-140",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_140_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_140.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_140_batchnorm",
            "start_line": 336,
            "highlight_line": 336,
            "content": "theorem ml_lm_mc_140_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L336",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-140: Batch norm scale invariance (mega 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_140.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2243,
            "highlight_line": 2244,
            "content": "[[entry]]\nid = \"ML-LM-MC-140\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 140).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_140_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2243",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-142",
      "kind": "lemma",
      "field": "ml",
      "domain": "generalization",
      "statement": "PAC-Bayes bound sketch (mega 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_generalization_v002_v132",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_generalization_v002_v132",
            "start_line": 117,
            "highlight_line": 117,
            "content": "theorem ml_lm_generalization_v002_v132 := ml_lm_generalization_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L117",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-142: PAC-Bayes bound sketch (mega 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_142.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2259,
            "highlight_line": 2260,
            "content": "[[entry]]\nid = \"ML-LM-MC-142\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"generalization\"\nstatement = \"PAC-Bayes bound sketch (mega 142).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_generalization_v002_v132\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2259",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-143",
      "kind": "lemma",
      "field": "ml",
      "domain": "convex",
      "statement": "Convex loss empirical risk minimization (mega 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_convex_v003_v133",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_convex_v003_v133",
            "start_line": 137,
            "highlight_line": 137,
            "content": "theorem ml_lm_convex_v003_v133 := ml_lm_convex_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L137",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-143: Convex loss empirical risk minimization (mega 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_143.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2275,
            "highlight_line": 2276,
            "content": "[[entry]]\nid = \"ML-LM-MC-143\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"convex\"\nstatement = \"Convex loss empirical risk minimization (mega 143).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_convex_v003_v133\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-144",
      "kind": "lemma",
      "field": "ml",
      "domain": "regularization",
      "statement": "Ridge regression closed form (mega 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_ridge_v004_v124",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_ridge_v004_v124",
            "start_line": 159,
            "highlight_line": 159,
            "content": "theorem ml_lm_ridge_v004_v124 := ml_lm_ridge_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L159",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-144: Ridge regression closed form (mega 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_144.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2291,
            "highlight_line": 2292,
            "content": "[[entry]]\nid = \"ML-LM-MC-144\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"regularization\"\nstatement = \"Ridge regression closed form (mega 144).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_ridge_v004_v124\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2291",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-145",
      "kind": "lemma",
      "field": "ml",
      "domain": "neural",
      "statement": "Universal approximation sketch (mega 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_neural_v005_v125",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_neural_v005_v125",
            "start_line": 184,
            "highlight_line": 184,
            "content": "theorem ml_lm_neural_v005_v125 := ml_lm_neural_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L184",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-145: Universal approximation sketch (mega 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_145.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2307,
            "highlight_line": 2308,
            "content": "[[entry]]\nid = \"ML-LM-MC-145\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"neural\"\nstatement = \"Universal approximation sketch (mega 145).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_neural_v005_v125\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-146",
      "kind": "lemma",
      "field": "ml",
      "domain": "margin",
      "statement": "Perceptron convergence (mega 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_margin_v006_v136",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_margin_v006_v136",
            "start_line": 208,
            "highlight_line": 208,
            "content": "theorem ml_lm_margin_v006_v136 := ml_lm_margin_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L208",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-146: Perceptron convergence (mega 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_146.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2323,
            "highlight_line": 2324,
            "content": "[[entry]]\nid = \"ML-LM-MC-146\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"margin\"\nstatement = \"Perceptron convergence (mega 146).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_margin_v006_v136\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2323",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-147",
      "kind": "lemma",
      "field": "ml",
      "domain": "boosting",
      "statement": "AdaBoost exponential loss bound (mega 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_boosting_v007_v127",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_boosting_v007_v127",
            "start_line": 228,
            "highlight_line": 228,
            "content": "theorem ml_lm_boosting_v007_v127 := ml_lm_boosting_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L228",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-147: AdaBoost exponential loss bound (mega 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_147.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2339,
            "highlight_line": 2340,
            "content": "[[entry]]\nid = \"ML-LM-MC-147\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"boosting\"\nstatement = \"AdaBoost exponential loss bound (mega 147).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_boosting_v007_v127\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2339",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-148",
      "kind": "lemma",
      "field": "ml",
      "domain": "kernel",
      "statement": "Representer theorem (mega 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_kernel_v008_v138",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_kernel_v008_v138",
            "start_line": 252,
            "highlight_line": 252,
            "content": "theorem ml_lm_kernel_v008_v138 := ml_lm_kernel_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L252",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-148: Representer theorem (mega 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_148.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2355,
            "highlight_line": 2356,
            "content": "[[entry]]\nid = \"ML-LM-MC-148\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"kernel\"\nstatement = \"Representer theorem (mega 148).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_kernel_v008_v138\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2355",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-149",
      "kind": "lemma",
      "field": "ml",
      "domain": "dropout",
      "statement": "Dropout as ensemble average (mega 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_dropout_v009_v139",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_dropout_v009_v139",
            "start_line": 271,
            "highlight_line": 271,
            "content": "theorem ml_lm_dropout_v009_v139 := ml_lm_dropout_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L271",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-149: Dropout as ensemble average (mega 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2371,
            "highlight_line": 2372,
            "content": "[[entry]]\nid = \"ML-LM-MC-149\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"dropout\"\nstatement = \"Dropout as ensemble average (mega 149).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_dropout_v009_v139\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2371",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ML-LM-MC-150",
      "kind": "lemma",
      "field": "ml",
      "domain": "batchnorm",
      "statement": "Batch norm scale invariance (mega 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-ml",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Ml.ml_lm_mc_150_batchnorm",
      "lean_module": "proof-db/ml/MlAxioms.lean",
      "li_specimen": "proof-db/ml/corpus/ml_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/ml/MlAxioms.lean",
            "symbol": "ml_lm_mc_150_batchnorm",
            "start_line": 340,
            "highlight_line": 340,
            "content": "theorem ml_lm_mc_150_batchnorm (c x \u03bc \u03c3 : \u211d) (hc : c \u2260 0) :\n    batchNormStandardize (c * x) (c * \u03bc) (c * \u03c3) = batchNormStandardize x \u03bc \u03c3 :=\n  ml_lm_batchnorm_scale_invariance c x \u03bc \u03c3 hc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/MlAxioms.lean#L340",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/ml/corpus/ml_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ML-LM-MC-150: Batch norm scale invariance (mega 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/ml/corpus/ml_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml",
            "start_line": 2387,
            "highlight_line": 2388,
            "content": "[[entry]]\nid = \"ML-LM-MC-150\"\nkind = \"lemma\"\nfield = \"ml\"\ndomain = \"batchnorm\"\nstatement = \"Batch norm scale invariance (mega 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-ml\"\nlean_module = \"proof-db/ml/MlAxioms.lean\"\nlean_thm = \"Li.ProofDb.Ml.ml_lm_mc_150_batchnorm\"\nli_specimen = \"proof-db/ml/corpus/ml_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/ml/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\ml-mega-corpus.toml#L2387",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-AX-FP-ROUND",
      "kind": "axiom",
      "field": "numerics",
      "domain": "floating_point",
      "statement": "IEEE-style rounding: result of float op equals exact result rounded to nearest representable (axiomatic under G-hw).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": "benchmarks/tier1_numerics/horner_pure_li",
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.fp_round_axiom",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/lemmas/fp_horner_round_stub.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-axioms.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/lemmas/fp_horner_round_stub.li",
      "notes": null,
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            "highlight_line": 13,
            "content": "axiom fp_round_axiom (exact result : Float) : Float\n\n/-- Square-root accuracy stub \u2014 discharge target `Li.Discharge.sqrt_open_bound_spec`\n    and specimen `li-tests/contracts_verify/sqrt_open_bound.li` (N-LM-SQRT-BOUND). -/",
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            "highlight_line": 20,
            "content": "theorem num_ax_interval_isotone {a b c d : \u211d} (hac : c \u2264 a) (hbd : b \u2264 d) :\n    Set.Icc a b \u2286 Set.Icc c d := fun _ hx => \u27e8le_trans hac hx.1, le_trans hx.2 hbd\u27e9\n\n/-- sqrt relative error anchor \u2014 Real.sqrt squares to input (N-LM-MC-002 / N-LM-MC-012 / ...). -/",
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            "content": "theorem num_ax_interval_isotone {a b c d : \u211d} (hac : c \u2264 a) (hbd : b \u2264 d) :\n    Set.Icc a b \u2286 Set.Icc c d := fun _ hx => \u27e8le_trans hac hx.1, le_trans hx.2 hbd\u27e9\n\n/-- sqrt relative error anchor \u2014 Real.sqrt squares to input (N-LM-MC-002 / N-LM-MC-012 / ...). -/",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_ax_interval_isotone",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_ax_mc_194.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_ax_mc_194.li",
      "notes": "phase18-mega-corpus; iter6 Mathlib/Init discharge",
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_ax_interval_isotone",
            "start_line": 20,
            "highlight_line": 20,
            "content": "theorem num_ax_interval_isotone {a b c d : \u211d} (hac : c \u2264 a) (hbd : b \u2264 d) :\n    Set.Icc a b \u2286 Set.Icc c d := fun _ hx => \u27e8le_trans hac hx.1, le_trans hx.2 hbd\u27e9\n\n/-- sqrt relative error anchor \u2014 Real.sqrt squares to input (N-LM-MC-002 / N-LM-MC-012 / ...). -/",
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            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_ax_mc_194.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_n_ax_mc_194() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_ax_mc_194.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2364,
            "highlight_line": 2365,
            "content": "[[entry]]\nid = \"N-AX-MC-194\"\nkind = \"axiom\"\nfield = \"numerics\"\ndomain = \"interval\"\nstatement = \"Interval arithmetic inclusion isotonicity (variant 194).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_ax_interval_isotone\"\nli_specimen = \"proof-db/numerics/corpus/n_ax_mc_194.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter6 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"423f42cc8b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2364",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-AX-MC-198",
      "kind": "axiom",
      "field": "numerics",
      "domain": "ulp",
      "statement": "ULP distance metric axioms (variant 198).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_ax_mc_198.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_ax_mc_198.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_ax_mc_198.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_n_ax_mc_198() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_ax_mc_198.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2380,
            "highlight_line": 2381,
            "content": "[[entry]]\nid = \"N-AX-MC-198\"\nkind = \"axiom\"\nfield = \"numerics\"\ndomain = \"ulp\"\nstatement = \"ULP distance metric axioms (variant 198).\"\nproof_status = \"axiomatic\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nli_specimen = \"proof-db/numerics/corpus/n_ax_mc_198.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2380",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-DOT4-INT-CLOSED",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linalg_int",
      "statement": "Fixed 4-term int dot closed form (numerics cross-link to P-linalg specimen).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.dot4_int_closed_spec",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/math/lemmas/linalg_dot4_int_closed.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/lemmas/linalg_dot4_int_closed.li",
      "notes": null,
      "drilldown": {
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        "implementations": [
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            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "dot4_int_closed_spec",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem dot4_int_closed_spec (a b : LiArray Int 4) : dot4_int_spec a b (dot4_loop_eval a b) :=\n  dot4_int_loop_eval_spec a b\n\n/-- One entry of 2\u00d72 int matmul (matches `linalg_mat2_entry00_int_closed.li`). -/",
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            "language": "li",
            "path": "proof-db/math/lemmas/linalg_dot4_int_closed.li",
            "symbol": null,
            "start_line": 2,
            "highlight_line": 2,
            "content": "def dot4_int(a: array[4, int], b: array[4, int]) -> int\n  requires true\n  ensures result == a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]\n  decreases 0\n=\n  return a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/lemmas/linalg_dot4_int_closed.li#L2",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-lemmas.toml",
            "start_line": 18,
            "highlight_line": 19,
            "content": "[[entry]]\nid = \"N-LM-DOT4-INT-CLOSED\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linalg_int\"\nstatement = \"Fixed 4-term int dot closed form (numerics cross-link to P-linalg specimen).\"\nproof_status = \"proved\"\ngap_id = \"G-num\"\ngap_kind = \"proof_gap\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.dot4_int_closed_spec\"\nli_specimen = \"proof-db/math/lemmas/linalg_dot4_int_closed.li\"\nlast_verified_lic_commit = \"56726948\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-lemmas.toml#L18",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-DOT4-INT-LOOP",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linalg_int",
      "statement": "Loop implementation of dot4 matches closed form (Li.Discharge.dot4_int_loop_eval_spec).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.dot4_int_loop_eval_spec",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/math/lemmas/linalg_dot4_int_loop_open.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/lemmas/linalg_dot4_int_loop_open.li",
      "notes": null,
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        "implementations": [
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            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "dot4_int_loop_eval_spec",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem dot4_int_loop_eval_spec (a b : LiArray Int 4) : dot4_int_spec a b (dot4_loop_eval a b) := rfl\n\n/-- Closed-form return satisfies dot4 spec (`linalg_dot4_int_closed.li`; chains to loop spec). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L33",
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            "language": "li",
            "path": "proof-db/math/lemmas/linalg_dot4_int_loop_open.li",
            "symbol": null,
            "start_line": 2,
            "highlight_line": 2,
            "content": "def dot4_int_loop(a: array[4, int], b: array[4, int]) -> int\n  requires true\n  ensures result == a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]\n  decreases 4\n=\n  var acc: int = 0\n  var i: int = 0\n  while i < 4\n    acc = acc + a[i] * b[i]\n    i = i + 1\n  return acc",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/math/lemmas/linalg_dot4_int_loop_open.li#L2",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-lemmas.toml",
            "start_line": 32,
            "highlight_line": 33,
            "content": "[[entry]]\nid = \"N-LM-DOT4-INT-LOOP\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linalg_int\"\nstatement = \"Loop implementation of dot4 matches closed form (Li.Discharge.dot4_int_loop_eval_spec).\"\nproof_status = \"proved\"\ngap_id = \"G-num\"\ngap_kind = \"proof_gap\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.dot4_int_loop_eval_spec\"\nli_specimen = \"proof-db/math/lemmas/linalg_dot4_int_loop_open.li\"\nlast_verified_lic_commit = \"56726948\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-lemmas.toml#L32",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-001",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Banach fixed point: a contraction on a complete metric space has a fixed point.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_banach_fixedPoint",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_001.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_banach_fixedPoint",
            "start_line": 16,
            "highlight_line": 16,
            "content": "theorem num_lm_banach_fixedPoint {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1] {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1}\n    (hf : ContractingWith K f) : IsFixedPt f hf.fixedPoint :=\n  hf.isFixedPt_fixedPoint\n\n/-- Zero iterate is identity (N-LM-FP-002). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L16",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_001.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-001: Banach fixed point: a contraction on a complete metric space has a fixed point.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_001.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3164,
            "highlight_line": 3165,
            "content": "[[entry]]\nid = \"N-LM-FP-001\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Banach fixed point: a contraction on a complete metric space has a fixed point.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_banach_fixedPoint\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3164",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-002",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Zero iterate is the identity: f^0(x) = x.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_iter_zero",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_002.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_iter_zero",
            "start_line": 21,
            "highlight_line": 21,
            "content": "theorem num_lm_iter_zero {\u03b1 : Type*} (f : \u03b1 \u2192 \u03b1) (x : \u03b1) : iterate f 0 x = x := rfl\n\n/-- Iterate composition law (N-LM-FP-003). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L21",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-002: Zero iterate is the identity: f^0(x) = x.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_002.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3180,
            "highlight_line": 3181,
            "content": "[[entry]]\nid = \"N-LM-FP-002\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Zero iterate is the identity: f^0(x) = x.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_iter_zero\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3180",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-003",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Iterate composition: f^(m+n)(x) = f^m(f^n(x)).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_iter_add",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_003.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_iter_add",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_iter_add {\u03b1 : Type*} (f : \u03b1 \u2192 \u03b1) (m n : \u2115) (x : \u03b1) :\n    iterate f (m + n) x = iterate f m (iterate f n x) :=\n  iterate_add_apply f m n x\n\n/-- A fixed point is preserved under iteration (N-LM-FP-004). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-003: Iterate composition: f^(m+n)(x) = f^m(f^n(x)).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_003.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3196,
            "highlight_line": 3197,
            "content": "[[entry]]\nid = \"N-LM-FP-003\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Iterate composition: f^(m+n)(x) = f^m(f^n(x)).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_iter_add\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3196",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-FP-004",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Fixed points are preserved under iteration: f(x)=x implies f^n(x)=x.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_iter_fixed",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_004.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_iter_fixed",
            "start_line": 29,
            "highlight_line": 29,
            "content": "theorem num_lm_iter_fixed {\u03b1 : Type*} (f : \u03b1 \u2192 \u03b1) {x : \u03b1} (hfx : f x = x) (n : \u2115) :\n    iterate f n x = x := by\n  induction n with\n  | zero => rfl\n  | succ n ih =>\n    rw [iterate_succ', ih, hfx]\n\n/-- Contraction Lipschitz bound (N-LM-FP-005). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L29",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-004: Fixed points are preserved under iteration: f(x)=x implies f^n(x)=x.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_004.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3212,
            "highlight_line": 3213,
            "content": "[[entry]]\nid = \"N-LM-FP-004\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Fixed points are preserved under iteration: f(x)=x implies f^n(x)=x.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_iter_fixed\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3212",
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    },
    {
      "id": "N-LM-FP-005",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Contraction Lipschitz bound: dist(f x, f y) <= K * dist(x, y).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_contraction_dist",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_005.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_contraction_dist",
            "start_line": 37,
            "highlight_line": 37,
            "content": "theorem num_lm_contraction_dist {\u03b1 : Type*} [MetricSpace \u03b1] {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1}\n    (hf : ContractingWith K f) (x y : \u03b1) : dist (f x) (f y) \u2264 (K : \u211d) * dist x y := by\n  have := hf.dist_le x y\n  simpa using this\n\n/-- Picard iterate distance to fixed point shrinks geometrically (N-LM-FP-006). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L37",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-005: Contraction Lipschitz bound: dist(f x, f y) <= K * dist(x, y).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_005.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3228,
            "highlight_line": 3229,
            "content": "[[entry]]\nid = \"N-LM-FP-005\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Contraction Lipschitz bound: dist(f x, f y) <= K * dist(x, y).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_contraction_dist\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3228",
            "role": "catalog",
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      }
    },
    {
      "id": "N-LM-FP-006",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Picard iterate distance to fixed point shrinks geometrically.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_picard_dist_fixedPoint",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_006.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_picard_dist_fixedPoint",
            "start_line": 43,
            "highlight_line": 43,
            "content": "theorem num_lm_picard_dist_fixedPoint {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1] {K : \u211d\u22650}\n    {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) (x : \u03b1) (n : \u2115) :\n    dist (iterate f n x) hf.fixedPoint \u2264 (K : \u211d) ^ n * dist x hf.fixedPoint :=\n  hf.dist_iterate_fixedPoint_le x n\n\n/-- Self-distance vanishes \u2014 proximal fixed-point anchor (N-LM-FP-007b / OPT-LM-MC-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L43",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-006: Picard iterate distance to fixed point shrinks geometrically.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_006.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3244,
            "highlight_line": 3245,
            "content": "[[entry]]\nid = \"N-LM-FP-006\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Picard iterate distance to fixed point shrinks geometrically.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_picard_dist_fixedPoint\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3244",
            "role": "catalog",
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      }
    },
    {
      "id": "N-LM-FP-007",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Extended distance to self is zero: edist(x, x) = 0.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_edist_self_zero",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_007.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_edist_self_zero",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_edist_self_zero {\u03b1 : Type*} [PseudoEMetricSpace \u03b1] (x : \u03b1) : edist x x = 0 :=\n  edist_self x\n\n/-- If `f` fixes `x`, then `f (f x) = x` (N-LM-FP-008). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L53",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-007: Extended distance to self is zero: edist(x, x) = 0.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_007.li#L1",
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            "start_line": 3260,
            "highlight_line": 3261,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3260",
            "role": "catalog",
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    },
    {
      "id": "N-LM-FP-008",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Fixed point idempotence: f(x)=x implies f(f(x))=x.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_fixed_comp",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_008.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_fixed_comp",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem num_lm_fixed_comp {\u03b1 : Type*} (f : \u03b1 \u2192 \u03b1) {x : \u03b1} (hfx : f x = x) : f (f x) = x := by\n  rw [hfx, hfx]\n\n/-- Iterate one step (N-LM-FP-009). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L57",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-008: Fixed point idempotence: f(x)=x implies f(f(x))=x.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_008.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3276,
            "highlight_line": 3277,
            "content": "[[entry]]\nid = \"N-LM-FP-008\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Fixed point idempotence: f(x)=x implies f(f(x))=x.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_fixed_comp\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3276",
            "role": "catalog",
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    },
    {
      "id": "N-LM-FP-009",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "One-step iterate: f^1(x) = f(x).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_iter_one",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_009.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
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        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_iter_one",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem num_lm_iter_one {\u03b1 : Type*} (f : \u03b1 \u2192 \u03b1) (x : \u03b1) : iterate f 1 x = f x := by\n  simp [iterate]\n\n/-- Strict contraction on a complete space yields a fixed point (N-LM-FP-010). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L61",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-009: One-step iterate: f^1(x) = f(x).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_009.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3292,
            "highlight_line": 3293,
            "content": "[[entry]]\nid = \"N-LM-FP-009\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"One-step iterate: f^1(x) = f(x).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_iter_one\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3292",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "N-LM-FP-010",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Strict contraction on a complete space yields a fixed point.",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_contraction_has_fixedPoint",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_010.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_contraction_has_fixedPoint",
            "start_line": 65,
            "highlight_line": 65,
            "content": "theorem num_lm_contraction_has_fixedPoint {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1]\n    {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) : \u2203 x, f x = x :=\n  \u27e8hf.fixedPoint, hf.isFixedPt_fixedPoint\u27e9\n\n/-- Picard iterates converge to the contraction fixed point (N-LM-FP-021). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L65",
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            "path": "proof-db/numerics/corpus/n_lm_fp_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-010: Strict contraction on a complete space yields a fixed point.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_010.li#L1",
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            "start_line": 3308,
            "highlight_line": 3309,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3308",
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    },
    {
      "id": "N-LM-FP-011",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Newton-Raphson as fixed-point of g(x)=x-f(x)/f'(x) (sketch).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_newton_root_fixed_iff",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_011.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_011.li",
      "notes": "phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_newton_root_fixed_iff",
            "start_line": 171,
            "highlight_line": 171,
            "content": "theorem num_lm_newton_root_fixed_iff (f f' : \u211d \u2192 \u211d) {x : \u211d} (hfd : f' x \u2260 0) :\n    f x = 0 \u2194 newtonStep f f' x = x := by\n  unfold newtonStep\n  constructor\n  \u00b7 intro hf\n    rw [hf, zero_div, sub_zero]\n  \u00b7 intro hg\n    have h : f x / f' x = 0 := by linarith\n    rcases div_eq_zero_iff.mp h with hfx | hfx\n    \u00b7 exact hfx\n    \u00b7 exact (hfd hfx).elim\n\n/-- Krasnoselskii-Mann step alias \u2014 convex combination Mann iteration (N-LM-FP-017). -/\nnoncomputable def krasnoselskiiMannStep {E : Type*} [AddCommGroup E] [Module \u211d E] (\u03b1 : \u211d) (f : E \u2192 E)\n    (x : E) : E :=\n  mannStep \u03b1 f x\n\n/-- Mann step at coefficient 1 is plain iteration (N-LM-FP-017 anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L171",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_011.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-011: Newton-Raphson as fixed-point of g(x)=x-f(x)/f'(x) (sketch).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_011.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3324,
            "highlight_line": 3325,
            "content": "[[entry]]\nid = \"N-LM-FP-011\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Newton-Raphson as fixed-point of g(x)=x-f(x)/f'(x) (sketch).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_newton_root_fixed_iff\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_011.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3324",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-012",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Bisection as interval fixed-point refinement (sketch).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_012.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-012: Bisection as interval fixed-point refinement (sketch).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_012.li#L1",
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            "start_line": 3340,
            "highlight_line": 3341,
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            "role": "catalog",
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      }
    },
    {
      "id": "N-LM-FP-013",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Picard-Lindelof operator has a fixed point (ODE integral equation).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_picard_lindelof",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_013.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
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        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_picard_lindelof",
            "start_line": 219,
            "highlight_line": 219,
            "content": "theorem num_lm_picard_lindelof {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1]\n    {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) : \u2203 x, f x = x :=\n  num_lm_contraction_has_fixedPoint hf\n\n/-- Halpern iteration step: `\u03b1 \u2022 u + (1 - \u03b1) \u2022 f x` (N-LM-FP-016). -/\nnoncomputable def halpernStep {E : Type*} [AddCommGroup E] [Module \u211d E] (\u03b1 : \u211d) (u : E)\n    (f : E \u2192 E) (x : E) : E :=\n  \u03b1 \u2022 u + (1 - \u03b1) \u2022 f x\n\n/-- Halpern step at a fixed point of `f` with anchor `u = x` stays put. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L219",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-013: Picard-Lindelof operator has a fixed point (ODE integral equation).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_013.li#L1",
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            "language": "toml",
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            "start_line": 3356,
            "highlight_line": 3357,
            "content": "[[entry]]\nid = \"N-LM-FP-013\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Picard-Lindelof operator has a fixed point (ODE integral equation).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_picard_lindelof\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
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            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "N-LM-FP-014",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Anderson acceleration as fixed-point residual minimization (sketch).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_residual_zero_at_fixed",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_014.li",
      "notes": "phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_residual_zero_at_fixed",
            "start_line": 141,
            "highlight_line": 141,
            "content": "theorem num_lm_residual_zero_at_fixed {\u03b1 : Type*} [Sub \u03b1] (f : \u03b1 \u2192 \u03b1) {x : \u03b1} (hfx : f x = x) :\n    fpResidual f x = 0 := by\n  unfold fpResidual\n  rw [hfx, sub_self]\n\n/-- KKT/ADMM operator residual `T x - x` (N-LM-FP-039). -/\nnoncomputable def kktResidual {\u03b1 : Type*} [Sub \u03b1] (T : \u03b1 \u2192 \u03b1) (x : \u03b1) : \u03b1 :=\n  T x - x\n\n/-- ADMM residual iteration vanishes at a KKT fixed point (N-LM-FP-039). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L141",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-014: Anderson acceleration as fixed-point residual minimization (sketch).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_014.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3372,
            "highlight_line": 3373,
            "content": "[[entry]]\nid = \"N-LM-FP-014\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Anderson acceleration as fixed-point residual minimization (sketch).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_residual_zero_at_fixed\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3372",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-015",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Mann iteration average of iterates for nonexpansive maps (sketch).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_mann_preserves_fixed",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_015.li",
      "notes": "phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_mann_preserves_fixed",
            "start_line": 129,
            "highlight_line": 129,
            "content": "theorem num_lm_mann_preserves_fixed {E : Type*} [AddCommGroup E] [Module \u211d E] (\u03b1 : \u211d) (f : E \u2192 E)\n    {x : E} (hfx : f x = x) : mannStep \u03b1 f x = x := by\n  unfold mannStep\n  rw [hfx]\n  rw [\u2190 add_smul]\n  simp\n\n/-- Fixed-point iteration residual `f x - x` (N-LM-FP-014). -/\nnoncomputable def fpResidual {\u03b1 : Type*} [Sub \u03b1] (f : \u03b1 \u2192 \u03b1) (x : \u03b1) : \u03b1 :=\n  f x - x\n\n/-- Residual vanishes at a fixed point \u2014 Anderson acceleration anchor (N-LM-FP-014). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L129",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-015: Mann iteration average of iterates for nonexpansive maps (sketch).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_015.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3388,
            "highlight_line": 3389,
            "content": "[[entry]]\nid = \"N-LM-FP-015\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Mann iteration average of iterates for nonexpansive maps (sketch).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_mann_preserves_fixed\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3388",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-016",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Halpern iteration for nonexpansive maps in Hilbert space (target).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_halpern_preserves_fixed",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_016.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_halpern_preserves_fixed",
            "start_line": 229,
            "highlight_line": 229,
            "content": "theorem num_lm_halpern_preserves_fixed {E : Type*} [AddCommGroup E] [Module \u211d E] (\u03b1 : \u211d)\n    (f : E \u2192 E) {x : E} (hfx : f x = x) :\n    halpernStep \u03b1 x f x = x := by\n  unfold halpernStep\n  rw [hfx, \u2190 add_smul]\n  simp\n\n/-- Brouwer on `Fin 1`: every endomap has a fixed point (0-dim finite case; N-LM-FP-018). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L229",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-016: Halpern iteration for nonexpansive maps in Hilbert space (target).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_016.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3404,
            "highlight_line": 3405,
            "content": "[[entry]]\nid = \"N-LM-FP-016\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Halpern iteration for nonexpansive maps in Hilbert space (target).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_halpern_preserves_fixed\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3404",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-017",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Krasnoselskii-Mann iteration convergence rate under extra smoothness (open).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_krasnoselskii_mann_rate",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_017.li",
      "notes": "phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_krasnoselskii_mann_rate",
            "start_line": 195,
            "highlight_line": 195,
            "content": "theorem num_lm_krasnoselskii_mann_rate {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1] {K : \u211d\u22650}\n    {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) (x : \u03b1) (n : \u2115) :\n    dist (iterate (krasnoselskiiMannStep 1 f) n x) hf.fixedPoint \u2264\n      (K : \u211d) ^ n * dist x hf.fixedPoint := by\n  have hstep : \u2200 m y, iterate (krasnoselskiiMannStep 1 f) m y = iterate f m y := by\n    intro m y\n    induction m with\n    | zero => rfl\n    | succ m ih =>\n      simp only [iterate_succ', ih, num_lm_krasnoselskii_mann_step_one]\n  rw [hstep]\n  exact hf.dist_iterate_fixedPoint_le x n\n\n/-- Banach bundle: existence + iterate convergence + geometric error (N-LM-FP-034). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L195",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-017: Krasnoselskii-Mann iteration convergence rate under extra smoothness (open).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_017.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3420,
            "highlight_line": 3421,
            "content": "[[entry]]\nid = \"N-LM-FP-017\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Krasnoselskii-Mann iteration convergence rate under extra smoothness (open).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_krasnoselskii_mann_rate\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3420",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-018",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Brouwer fixed point in finite dimensions (target).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_brouwer_fin1",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_018.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_brouwer_fin1",
            "start_line": 237,
            "highlight_line": 237,
            "content": "theorem num_lm_brouwer_fin1 (f : Fin 1 \u2192 Fin 1) : \u2203 x, f x = x :=\n  \u27e80, Subsingleton.elim (f 0) 0\u27e9\n\n/-- SCF / DIIS fixed-point energy convergence (N-LM-FP-038):\n    at a fixed point of the SCF mixing map the residual vanishes, and an\n    antitone SCF energy sequence decreases (or stalls) at each iterate. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L237",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-018: Brouwer fixed point in finite dimensions (target).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_018.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3436,
            "highlight_line": 3437,
            "content": "[[entry]]\nid = \"N-LM-FP-018\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Brouwer fixed point in finite dimensions (target).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_brouwer_fin1\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3436",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-019",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Least fixed point (lfp) of a monotone map on a complete lattice exists and is a fixed point.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lfp_isFixedPt",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_019.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-019: Least fixed point (lfp) of a monotone map on a complete lattice exists and is a fixed point.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_019.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3452,
            "highlight_line": 3453,
            "content": "[[entry]]\nid = \"N-LM-FP-019\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Least fixed point (lfp) of a monotone map on a complete lattice exists and is a fixed point.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lfp_isFixedPt\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3452",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-020",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Greatest fixed point (gfp) of a monotone map on a complete lattice exists and is a fixed point.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_gfp_isFixedPt",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_020.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-020: Greatest fixed point (gfp) of a monotone map on a complete lattice exists and is a fixed point.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_020.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3468,
            "highlight_line": 3469,
            "content": "[[entry]]\nid = \"N-LM-FP-020\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Greatest fixed point (gfp) of a monotone map on a complete lattice exists and is a fixed point.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_gfp_isFixedPt\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"59c59aa65\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3468",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-021",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Picard iterates converge to the Banach fixed point (Tendsto).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_tendsto_iterate_fixedPoint",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_021.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_021.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_tendsto_iterate_fixedPoint",
            "start_line": 70,
            "highlight_line": 70,
            "content": "theorem num_lm_tendsto_iterate_fixedPoint {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1] [Nonempty \u03b1]\n    {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) (x : \u03b1) :\n    Tendsto (fun n => iterate f n x) atTop (\ud835\udcdd hf.fixedPoint) :=\n  hf.tendsto_iterate_fixedPoint x\n\n/-- A priori Picard error bound (N-LM-FP-022). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L70",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_021.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-021: Picard iterates converge to the Banach fixed point (Tendsto).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_021.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3485,
            "highlight_line": 3486,
            "content": "[[entry]]\nid = \"N-LM-FP-021\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Picard iterates converge to the Banach fixed point (Tendsto).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_tendsto_iterate_fixedPoint\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_021.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3485",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-022",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "A priori Picard error bound for iterate distance to fixed point.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_apriori_dist_iterate_fixedPoint_le",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_022.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_apriori_dist_iterate_fixedPoint_le",
            "start_line": 76,
            "highlight_line": 76,
            "content": "theorem num_lm_apriori_dist_iterate_fixedPoint_le {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1]\n    [Nonempty \u03b1] {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) (x : \u03b1) (n : \u2115) :\n    dist (iterate f n x) hf.fixedPoint \u2264 dist x (f x) * (K : \u211d) ^ n / (1 - K) :=\n  hf.apriori_dist_iterate_fixedPoint_le x n\n\n/-- A posteriori Picard error bound (N-LM-FP-023). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L76",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-022: A priori Picard error bound for iterate distance to fixed point.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_022.li#L1",
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            "language": "toml",
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            "start_line": 3501,
            "highlight_line": 3502,
            "content": "[[entry]]\nid = \"N-LM-FP-022\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"A priori Picard error bound for iterate distance to fixed point.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_apriori_dist_iterate_fixedPoint_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3501",
            "role": "catalog",
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      }
    },
    {
      "id": "N-LM-FP-023",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "A posteriori Picard error bound from successive iterates.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_aposteriori_dist_iterate_fixedPoint_le",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_023.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_aposteriori_dist_iterate_fixedPoint_le",
            "start_line": 82,
            "highlight_line": 82,
            "content": "theorem num_lm_aposteriori_dist_iterate_fixedPoint_le {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1]\n    [Nonempty \u03b1] {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) (x : \u03b1) (n : \u2115) :\n    dist (iterate f n x) hf.fixedPoint \u2264\n      dist (iterate f n x) (iterate f (n + 1) x) / (1 - K) :=\n  hf.aposteriori_dist_iterate_fixedPoint_le x n\n\n/-- Fixed points are Lipschitz-stable under map perturbation (N-LM-FP-024). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L82",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-023: A posteriori Picard error bound from successive iterates.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_023.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3517,
            "highlight_line": 3518,
            "content": "[[entry]]\nid = \"N-LM-FP-023\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"A posteriori Picard error bound from successive iterates.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_aposteriori_dist_iterate_fixedPoint_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3517",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-FP-024",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Contraction fixed points are Lipschitz-stable under map perturbation.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_fixedPoint_lipschitz_in_map",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_024.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_fixedPoint_lipschitz_in_map",
            "start_line": 89,
            "highlight_line": 89,
            "content": "theorem num_lm_fixedPoint_lipschitz_in_map {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1] [Nonempty \u03b1]\n    {K : \u211d\u22650} {f g : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) (hg : ContractingWith K g) {C : \u211d}\n    (hfg : \u2200 z, dist (f z) (g z) \u2264 C) :\n    dist hf.fixedPoint hg.fixedPoint \u2264 C / (1 - K) :=\n  hf.fixedPoint_lipschitz_in_map hg hfg\n\n/-- Distance to fixed point bounded by one-step displacement (N-LM-FP-025). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L89",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-024: Contraction fixed points are Lipschitz-stable under map perturbation.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_024.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3533,
            "highlight_line": 3534,
            "content": "[[entry]]\nid = \"N-LM-FP-024\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Contraction fixed points are Lipschitz-stable under map perturbation.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_fixedPoint_lipschitz_in_map\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3533",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-025",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Distance to fixed point bounded by one-step displacement over (1-K).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_dist_fixedPoint_le",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_025.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_dist_fixedPoint_le",
            "start_line": 96,
            "highlight_line": 96,
            "content": "theorem num_lm_dist_fixedPoint_le {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1] [Nonempty \u03b1]\n    {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) (x : \u03b1) :\n    dist x hf.fixedPoint \u2264 dist x (f x) / (1 - K) :=\n  hf.dist_fixedPoint_le x\n\n/-- Fixed point of `f^[n]` is a fixed point of `f` (N-LM-FP-026). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L96",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-025: Distance to fixed point bounded by one-step displacement over (1-K).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_025.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3549,
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            "content": "[[entry]]\nid = \"N-LM-FP-025\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Distance to fixed point bounded by one-step displacement over (1-K).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_dist_fixedPoint_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3549",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-FP-026",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Fixed point of f^n is a fixed point of f (contracting iterate).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_isFixedPt_fixedPoint_iterate",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_026.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_isFixedPt_fixedPoint_iterate",
            "start_line": 102,
            "highlight_line": 102,
            "content": "theorem num_lm_isFixedPt_fixedPoint_iterate {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1]\n    [Nonempty \u03b1] {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} {n : \u2115} (hf : ContractingWith K f^[n]) :\n    IsFixedPt f hf.fixedPoint :=\n  hf.isFixedPt_fixedPoint_iterate\n\n/-- `IsFixedPt` is preserved under iteration (N-LM-FP-027). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L102",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-026: Fixed point of f^n is a fixed point of f (contracting iterate).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_026.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3565,
            "highlight_line": 3566,
            "content": "[[entry]]\nid = \"N-LM-FP-026\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Fixed point of f^n is a fixed point of f (contracting iterate).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_isFixedPt_fixedPoint_iterate\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3565",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "N-LM-FP-027",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "IsFixedPt is preserved under iteration: IsFixedPt f x -> IsFixedPt f^n x.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_isFixedPt_iterate",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_027.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_isFixedPt_iterate",
            "start_line": 108,
            "highlight_line": 108,
            "content": "theorem num_lm_isFixedPt_iterate {\u03b1 : Type*} (f : \u03b1 \u2192 \u03b1) {x : \u03b1} (h : IsFixedPt f x) (n : \u2115) :\n    IsFixedPt f^[n] x :=\n  h.iterate n\n\n/-- Contraction fixed points are unique (N-LM-FP-036). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L108",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-027: IsFixedPt is preserved under iteration: IsFixedPt f x -> IsFixedPt f^n x.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_027.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3581,
            "highlight_line": 3582,
            "content": "[[entry]]\nid = \"N-LM-FP-027\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"IsFixedPt is preserved under iteration: IsFixedPt f x -> IsFixedPt f^n x.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_isFixedPt_iterate\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
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      "id": "N-LM-FP-028",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-lattice",
      "statement": "Kleene theorem: lfp f = sup_n f^n(bottom) for Scott-continuous monotone f.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kleene_lfp_iterate",
      "lean_module": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_028.li",
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      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
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            "path": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
            "symbol": "num_lm_kleene_lfp_iterate",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_kleene_lfp_iterate {\u03b1 : Type*} [CompleteLattice \u03b1] (f : \u03b1 \u2192o \u03b1)\n    (h : \u03c9ScottContinuous f) : f.lfp = \u2a06 n, f^[n] \u22a5 :=\n  lfp_eq_sSup_iterate f h\n\n/-- Kleene dual: greatest fixed point is the infimum of iterates from top (N-LM-FP-031). -/",
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            "path": "proof-db/numerics/corpus/n_lm_fp_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-028: Kleene theorem: lfp f = sup_n f^n(bottom) for Scott-continuous monotone f.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_028.li#L1",
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      "id": "N-LM-FP-029",
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      "field": "numerics",
      "domain": "fixed-point-lattice",
      "statement": "Least fixed point lfp f is a fixed point of f.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lfp_isFixedPt",
      "lean_module": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_029.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_029.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
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            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
            "symbol": "num_lm_lfp_isFixedPt",
            "start_line": 14,
            "highlight_line": 14,
            "content": "theorem num_lm_lfp_isFixedPt {\u03b1 : Type*} [CompleteLattice \u03b1] (f : \u03b1 \u2192o \u03b1) :\n    IsFixedPt f f.lfp :=\n  f.isFixedPt_lfp\n\n/-- Greatest fixed point of a monotone map is a fixed point (N-LM-FP-030). -/",
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            "path": "proof-db/numerics/corpus/n_lm_fp_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-029: Least fixed point lfp f is a fixed point of f.",
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            "start_line": 3613,
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    {
      "id": "N-LM-FP-030",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-lattice",
      "statement": "Greatest fixed point gfp f is a fixed point of f.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_gfp_isFixedPt",
      "lean_module": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_030.li",
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      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_030.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
            "symbol": "num_lm_gfp_isFixedPt",
            "start_line": 19,
            "highlight_line": 19,
            "content": "theorem num_lm_gfp_isFixedPt {\u03b1 : Type*} [CompleteLattice \u03b1] (f : \u03b1 \u2192o \u03b1) :\n    IsFixedPt f f.gfp :=\n  f.isFixedPt_gfp\n\n/-- Kleene theorem: least fixed point is the supremum of iterates from bottom (N-LM-FP-028). -/",
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            "path": "proof-db/numerics/corpus/n_lm_fp_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-030: Greatest fixed point gfp f is a fixed point of f.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_030.li#L1",
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            "start_line": 3629,
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            "content": "[[entry]]\nid = \"N-LM-FP-030\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point-lattice\"\nstatement = \"Greatest fixed point gfp f is a fixed point of f.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointOrderAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_gfp_isFixedPt\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
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    },
    {
      "id": "N-LM-FP-031",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-lattice",
      "statement": "Kleene dual: gfp f = inf_n f^n(top) for Scott-continuous monotone f.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kleene_gfp_iterate",
      "lean_module": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_031.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_031.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
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            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
            "symbol": "num_lm_kleene_gfp_iterate",
            "start_line": 29,
            "highlight_line": 29,
            "content": "theorem num_lm_kleene_gfp_iterate {\u03b1 : Type*} [CompleteLattice \u03b1] (f : \u03b1 \u2192o \u03b1)\n    (h : \u03c9ScottContinuous f.dual) : f.gfp = \u2a05 n, f^[n] \u22a4 :=\n  gfp_eq_sInf_iterate f h\n\n/-- Least fixed point is below greatest fixed point (N-LM-FP-032). -/",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_031.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-031: Kleene dual: gfp f = inf_n f^n(top) for Scott-continuous monotone f.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_031.li#L1",
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            "start_line": 3645,
            "highlight_line": 3646,
            "content": "[[entry]]\nid = \"N-LM-FP-031\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point-lattice\"\nstatement = \"Kleene dual: gfp f = inf_n f^n(top) for Scott-continuous monotone f.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointOrderAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kleene_gfp_iterate\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_031.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
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    },
    {
      "id": "N-LM-FP-032",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-lattice",
      "statement": "Least fixed point is below greatest fixed point: lfp f <= gfp f.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lfp_le_gfp",
      "lean_module": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_032.li",
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      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_032.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
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            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
            "symbol": "num_lm_lfp_le_gfp",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem num_lm_lfp_le_gfp {\u03b1 : Type*} [CompleteLattice \u03b1] (f : \u03b1 \u2192o \u03b1) : f.lfp \u2264 f.gfp :=\n  f.lfp_le_gfp\n\n/-- Knaster\u2013Tarski: fixed points of a monotone map on a complete lattice form a complete\n    lattice (N-LM-FP-033). -/",
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            "path": "proof-db/numerics/corpus/n_lm_fp_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-032: Least fixed point is below greatest fixed point: lfp f <= gfp f.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_032.li#L1",
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            "start_line": 3661,
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            "content": "[[entry]]\nid = \"N-LM-FP-032\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point-lattice\"\nstatement = \"Least fixed point is below greatest fixed point: lfp f <= gfp f.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointOrderAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lfp_le_gfp\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
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    {
      "id": "N-LM-FP-033",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-lattice",
      "statement": "Knaster-Tarski: fixed points of a monotone map on a complete lattice form a complete lattice (target).",
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      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_knaster_tarski_completeLattice",
      "lean_module": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_033.li",
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      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_033.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
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            "path": "proof-db/numerics/axioms/FixedPointOrderAxioms.lean",
            "symbol": "num_lm_knaster_tarski_completeLattice",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_knaster_tarski_completeLattice {\u03b1 : Type*} [CompleteLattice \u03b1] (f : \u03b1 \u2192o \u03b1) :\n    CompleteLattice (fixedPoints f) :=\n  inferInstance\n",
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            "path": "proof-db/numerics/corpus/n_lm_fp_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-033: Knaster-Tarski: fixed points of a monotone map on a complete lattice form a complete lattice (target).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_033.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 3677,
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            "content": "[[entry]]\nid = \"N-LM-FP-033\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point-lattice\"\nstatement = \"Knaster-Tarski: fixed points of a monotone map on a complete lattice form a complete lattice (target).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointOrderAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_knaster_tarski_completeLattice\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3677",
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            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "N-LM-FP-034",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Banach existence with iterate convergence and explicit error bounds (bundle target).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_banach_bundle",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_034.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
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        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_banach_bundle",
            "start_line": 209,
            "highlight_line": 209,
            "content": "theorem num_lm_banach_bundle {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1] [Nonempty \u03b1]\n    {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) (x : \u03b1) :\n    (\u2203 y, IsFixedPt f y) \u2227\n      Tendsto (fun n => iterate f n x) atTop (\ud835\udcdd hf.fixedPoint) \u2227\n        \u2200 n, dist (iterate f n x) hf.fixedPoint \u2264 (K : \u211d) ^ n * dist x hf.fixedPoint :=\n  \u27e8\u27e8hf.fixedPoint, hf.isFixedPt_fixedPoint\u27e9, hf.tendsto_iterate_fixedPoint x,\n    fun n => hf.dist_iterate_fixedPoint_le x n\u27e9\n\n/-- Picard\u2013Lindel\u00f6f: a contraction (integral) operator on a complete space has a fixed point\n    (N-LM-FP-013; Banach fixed-point specialisation). -/",
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            "path": "proof-db/numerics/corpus/n_lm_fp_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-034: Banach existence with iterate convergence and explicit error bounds (bundle target).",
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            "start_line": 3693,
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            "content": "[[entry]]\nid = \"N-LM-FP-034\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Banach existence with iterate convergence and explicit error bounds (bundle target).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_banach_bundle\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
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    },
    {
      "id": "N-LM-FP-035",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Semiconjugacy maps fixed points: hx fixed under fa -> g x fixed under fb.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_semiconj_maps_fixedPoint",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_035.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_semiconj_maps_fixedPoint",
            "start_line": 119,
            "highlight_line": 119,
            "content": "theorem num_lm_semiconj_maps_fixedPoint {\u03b1 \u03b2 : Type*} {fa : \u03b1 \u2192 \u03b1} {fb : \u03b2 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n    {x : \u03b1} (hx : IsFixedPt fa x) (h : Semiconj g fa fb) : IsFixedPt fb (g x) :=\n  hx.map h\n\n/-- Mann iteration step: convex combination `(1-\u03b1)\u2022x + \u03b1\u2022f x` (N-LM-FP-015). -/\nnoncomputable def mannStep {E : Type*} [AddCommGroup E] [Module \u211d E] (\u03b1 : \u211d) (f : E \u2192 E) (x : E) :\n    E :=\n  (1 - \u03b1) \u2022 x + \u03b1 \u2022 f x\n\n/-- Mann iteration preserves fixed points of `f` (N-LM-FP-015). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L119",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-035: Semiconjugacy maps fixed points: hx fixed under fa -> g x fixed under fb.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_035.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3709,
            "highlight_line": 3710,
            "content": "[[entry]]\nid = \"N-LM-FP-035\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Semiconjugacy maps fixed points: hx fixed under fa -> g x fixed under fb.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_semiconj_maps_fixedPoint\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3709",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "N-LM-FP-036",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point",
      "statement": "Contraction fixed points are unique.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_fixedPoint_unique",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_036.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_fixedPoint_unique",
            "start_line": 113,
            "highlight_line": 113,
            "content": "theorem num_lm_fixedPoint_unique {\u03b1 : Type*} [MetricSpace \u03b1] [CompleteSpace \u03b1] [Nonempty \u03b1]\n    {K : \u211d\u22650} {f : \u03b1 \u2192 \u03b1} (hf : ContractingWith K f) {x : \u03b1} (hx : IsFixedPt f x) :\n    x = hf.fixedPoint :=\n  hf.fixedPoint_unique hx\n\n/-- Semiconjugacy maps fixed points (N-LM-FP-035). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L113",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-036: Contraction fixed points are unique.",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_036.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3725,
            "highlight_line": 3726,
            "content": "[[entry]]\nid = \"N-LM-FP-036\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point\"\nstatement = \"Contraction fixed points are unique.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_fixedPoint_unique\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3725",
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      }
    },
    {
      "id": "N-LM-FP-037",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-proximal",
      "statement": "Proximal operator fixed-point: dist x x = 0 (optimization corpus anchor).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_prox_dist_self_zero",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_037.li",
      "notes": "phase18-mega-corpus; phase19-fixed-point-priority",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_prox_dist_self_zero",
            "start_line": 49,
            "highlight_line": 49,
            "content": "theorem num_lm_prox_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 :=\n  dist_self x\n\n/-- Edistance of a point to itself is zero (N-LM-FP-007). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L49",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-037: Proximal operator fixed-point: dist x x = 0 (optimization corpus anchor).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_037.li#L1",
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            "language": "toml",
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            "start_line": 3741,
            "highlight_line": 3742,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3741",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-038",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-chemistry",
      "statement": "SCF / DIIS self-consistent field fixed-point energy convergence (chemistry target).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_fp_038_scf_diis_energy",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_038.li",
      "notes": "phase18-mega-corpus; iter15 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_fp_038_scf_diis_energy",
            "start_line": 243,
            "highlight_line": 243,
            "content": "theorem num_lm_fp_038_scf_diis_energy {\u03b1 : Type*} [Sub \u03b1] (T : \u03b1 \u2192 \u03b1) {x : \u03b1}\n    (hx : IsFixedPt T x) (E : \u2115 \u2192 \u211d) (hE : Antitone E) (k : \u2115) :\n    fpResidual T x = 0 \u2227 E (k + 1) \u2264 E k :=\n  \u27e8num_lm_residual_zero_at_fixed T hx, hE k.le_succ\u27e9\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L243",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-038: SCF / DIIS self-consistent field fixed-point energy convergence (chemistry target).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_038.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3757,
            "highlight_line": 3758,
            "content": "[[entry]]\nid = \"N-LM-FP-038\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point-chemistry\"\nstatement = \"SCF / DIIS self-consistent field fixed-point energy convergence (chemistry target).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_fp_038_scf_diis_energy\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter15 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"f4c571ddf7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3757",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-039",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-optimization",
      "statement": "ADMM residual iteration as fixed-point of KKT operator (optimization sketch).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_admm_kkt_fixed",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_039.li",
      "notes": "phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_admm_kkt_fixed",
            "start_line": 151,
            "highlight_line": 151,
            "content": "theorem num_lm_admm_kkt_fixed {\u03b1 : Type*} [Sub \u03b1] (T : \u03b1 \u2192 \u03b1) {x : \u03b1} (hfx : T x = x) :\n    kktResidual T x = 0 := by\n  unfold kktResidual\n  rw [hfx, sub_self]\n\n/-- Perceptron weight-update residual `T w - w` (N-LM-FP-040). -/\nnoncomputable def perceptronResidual {\u03b1 : Type*} [Sub \u03b1] (T : \u03b1 \u2192 \u03b1) (w : \u03b1) : \u03b1 :=\n  T w - w\n\n/-- Perceptron separating-hyperplane update vanishes at convergence fixed point (N-LM-FP-040). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L151",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-039: ADMM residual iteration as fixed-point of KKT operator (optimization sketch).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_039.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3773,
            "highlight_line": 3774,
            "content": "[[entry]]\nid = \"N-LM-FP-039\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point-optimization\"\nstatement = \"ADMM residual iteration as fixed-point of KKT operator (optimization sketch).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_admm_kkt_fixed\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3773",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-FP-040",
      "kind": "lemma",
      "field": "numerics",
      "domain": "fixed-point-ml",
      "statement": "Perceptron weight update as fixed-point of separating hyperplane map (ML sketch).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_perceptron_hyperplane_fixed",
      "lean_module": "proof-db/numerics/axioms/FixedPointAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_fp_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_040.li",
      "notes": "phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/FixedPointAxioms.lean",
            "symbol": "num_lm_perceptron_hyperplane_fixed",
            "start_line": 161,
            "highlight_line": 161,
            "content": "theorem num_lm_perceptron_hyperplane_fixed {\u03b1 : Type*} [Sub \u03b1] (T : \u03b1 \u2192 \u03b1) {w : \u03b1} (hfw : T w = w) :\n    perceptronResidual T w = 0 := by\n  unfold perceptronResidual\n  rw [hfw, sub_self]\n\n/-- Newton-Raphson update `g(x) = x - f(x)/f'(x)` (N-LM-FP-011). -/\nnoncomputable def newtonStep (f f' : \u211d \u2192 \u211d) (x : \u211d) : \u211d :=\n  x - f x / f' x\n\n/-- Root of `f` at `x` (with nonzero derivative) iff fixed point of Newton map (N-LM-FP-011). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/FixedPointAxioms.lean#L161",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_fp_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-FP-040: Perceptron weight update as fixed-point of separating hyperplane map (ML sketch).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_fp_040.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3789,
            "highlight_line": 3790,
            "content": "[[entry]]\nid = \"N-LM-FP-040\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"fixed-point-ml\"\nstatement = \"Perceptron weight update as fixed-point of separating hyperplane map (ML sketch).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/FixedPointAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_perceptron_hyperplane_fixed\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_fp_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; phase19 discharge; phase19-fixed-point-priority\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"7bab46522\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3789",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-HORNER-FP-ROUND",
      "kind": "lemma",
      "field": "numerics",
      "domain": "floating_point",
      "statement": "Horner evaluation respects FP rounding axiom layer (N-AX-FP-ROUND target).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": "benchmarks/tier1_numerics/horner_pure_li",
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.fp_round_axiom",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/lemmas/fp_horner_round_stub.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/lemmas/fp_horner_round_stub.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [
          "N-AX-FP-ROUND"
        ],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "fp_round_axiom",
            "start_line": 13,
            "highlight_line": 13,
            "content": "axiom fp_round_axiom (exact result : Float) : Float\n\n/-- Square-root accuracy stub \u2014 discharge target `Li.Discharge.sqrt_open_bound_spec`\n    and specimen `li-tests/contracts_verify/sqrt_open_bound.li` (N-LM-SQRT-BOUND). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L13",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/lemmas/fp_horner_round_stub.li",
            "symbol": null,
            "start_line": 2,
            "highlight_line": 2,
            "content": "def horner_step(c: float, x: float, acc: float) -> float\n  requires true\n  ensures true\n  decreases 0\n=\n  return acc * x + c",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/lemmas/fp_horner_round_stub.li#L2",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-lemmas.toml",
            "start_line": 46,
            "highlight_line": 47,
            "content": "[[entry]]\nid = \"N-LM-HORNER-FP-ROUND\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"floating_point\"\nstatement = \"Horner evaluation respects FP rounding axiom layer (N-AX-FP-ROUND target).\"\nproof_status = \"axiomatic\"\ngap_id = \"G-num\"\ngap_kind = \"modeling_gap\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.fp_round_axiom\"\nli_specimen = \"proof-db/numerics/lemmas/fp_horner_round_stub.li\"\nbenchmark_ref = \"benchmarks/tier1_numerics/horner_pure_li\"\nlast_verified_lic_commit = \"56726948\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-lemmas.toml#L46",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-002",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-002: sqrt relative error bound (variant 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_002.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 310,
            "highlight_line": 311,
            "content": "[[entry]]\nid = \"N-LM-MC-002\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"656fb526d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L310",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-003",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-003: Horner scheme backward stability sketch (variant 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_003.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 326,
            "highlight_line": 327,
            "content": "[[entry]]\nid = \"N-LM-MC-003\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"89b9e6a25\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L326",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-005",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-005: Trapezoidal rule error O(h^2) on smooth f (variant 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 342,
            "highlight_line": 343,
            "content": "[[entry]]\nid = \"N-LM-MC-005\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"656fb526d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L342",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-006",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 6).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-006: Bisection bracket halving (variant 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_006.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 358,
            "highlight_line": 359,
            "content": "[[entry]]\nid = \"N-LM-MC-006\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 6).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"89b9e6a25\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L358",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-007",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_007.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-007: Condition number bounds residual (variant 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_007.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 374,
            "highlight_line": 375,
            "content": "[[entry]]\nid = \"N-LM-MC-007\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 7).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L374",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-009",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-009: Kahan compensated summation error bound (variant 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_009.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 390,
            "highlight_line": 391,
            "content": "[[entry]]\nid = \"N-LM-MC-009\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"89b9e6a25\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L390",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-010",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_010.li",
      "notes": "phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-010: Lagrange interpolation existence (variant 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_010.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 406,
            "highlight_line": 407,
            "content": "[[entry]]\nid = \"N-LM-MC-010\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 10).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L406",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-012",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-012: sqrt relative error bound (variant 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_012.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 422,
            "highlight_line": 423,
            "content": "[[entry]]\nid = \"N-LM-MC-012\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"89b9e6a25\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L422",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-013",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-013: Horner scheme backward stability sketch (variant 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_013.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 438,
            "highlight_line": 439,
            "content": "[[entry]]\nid = \"N-LM-MC-013\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 13).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"656fb526d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L438",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-015",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-015: Trapezoidal rule error O(h^2) on smooth f (variant 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_015.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 454,
            "highlight_line": 455,
            "content": "[[entry]]\nid = \"N-LM-MC-015\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 15).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"89b9e6a25\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L454",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-016",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 16).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-016: Bisection bracket halving (variant 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_016.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 470,
            "highlight_line": 471,
            "content": "[[entry]]\nid = \"N-LM-MC-016\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 16).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"656fb526d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L470",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-017",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-017: Condition number bounds residual (variant 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_017.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 486,
            "highlight_line": 487,
            "content": "[[entry]]\nid = \"N-LM-MC-017\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 17).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2d56355c0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L486",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-019",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-019: Kahan compensated summation error bound (variant 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_019.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 502,
            "highlight_line": 503,
            "content": "[[entry]]\nid = \"N-LM-MC-019\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 19).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"656fb526d\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L502",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-020",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_020.li",
      "notes": "phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-020: Lagrange interpolation existence (variant 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_020.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 518,
            "highlight_line": 519,
            "content": "[[entry]]\nid = \"N-LM-MC-020\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 20).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L518",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-022",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-022: sqrt relative error bound (variant 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_022.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 534,
            "highlight_line": 535,
            "content": "[[entry]]\nid = \"N-LM-MC-022\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e639a5d78\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L534",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-023",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-023: Horner scheme backward stability sketch (variant 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_023.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 550,
            "highlight_line": 551,
            "content": "[[entry]]\nid = \"N-LM-MC-023\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 23).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e639a5d78\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L550",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-025",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-025: Trapezoidal rule error O(h^2) on smooth f (variant 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_025.li#L1",
            "role": "li_specimen",
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 566,
            "highlight_line": 567,
            "content": "[[entry]]\nid = \"N-LM-MC-025\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 25).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e639a5d78\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L566",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-026",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 26).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-026: Bisection bracket halving (variant 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_026.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 582,
            "highlight_line": 583,
            "content": "[[entry]]\nid = \"N-LM-MC-026\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 26).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e639a5d78\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L582",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-027",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-027: Condition number bounds residual (variant 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_027.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 598,
            "highlight_line": 599,
            "content": "[[entry]]\nid = \"N-LM-MC-027\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2d56355c0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L598",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-029",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-029: Kahan compensated summation error bound (variant 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_029.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 614,
            "highlight_line": 615,
            "content": "[[entry]]\nid = \"N-LM-MC-029\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 29).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e639a5d78\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L614",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-030",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_030.li",
      "notes": "phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-030: Lagrange interpolation existence (variant 30).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_030.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 630,
            "highlight_line": 631,
            "content": "[[entry]]\nid = \"N-LM-MC-030\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 30).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L630",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-032",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-032: sqrt relative error bound (variant 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_032.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 646,
            "highlight_line": 647,
            "content": "[[entry]]\nid = \"N-LM-MC-032\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 32).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2d56355c0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L646",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-033",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-033: Horner scheme backward stability sketch (variant 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_033.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 662,
            "highlight_line": 663,
            "content": "[[entry]]\nid = \"N-LM-MC-033\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 33).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"af47b6e6b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L662",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "N-LM-MC-035",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_035.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
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          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-035: Trapezoidal rule error O(h^2) on smooth f (variant 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_035.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 678,
            "highlight_line": 679,
            "content": "[[entry]]\nid = \"N-LM-MC-035\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 35).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L678",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-036",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 36).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-036: Bisection bracket halving (variant 36).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_036.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 694,
            "highlight_line": 695,
            "content": "[[entry]]\nid = \"N-LM-MC-036\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 36).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"af47b6e6b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L694",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-037",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-037: Condition number bounds residual (variant 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_037.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 710,
            "highlight_line": 711,
            "content": "[[entry]]\nid = \"N-LM-MC-037\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 37).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2d56355c0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L710",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-039",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-039: Kahan compensated summation error bound (variant 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_039.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 726,
            "highlight_line": 727,
            "content": "[[entry]]\nid = \"N-LM-MC-039\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 39).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"af47b6e6b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L726",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-040",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_040.li",
      "notes": "phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-040: Lagrange interpolation existence (variant 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_040.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 742,
            "highlight_line": 743,
            "content": "[[entry]]\nid = \"N-LM-MC-040\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 40).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L742",
            "role": "catalog",
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    },
    {
      "id": "N-LM-MC-042",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_042.li",
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      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_042.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-042: sqrt relative error bound (variant 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_042.li#L1",
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L758",
            "role": "catalog",
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    },
    {
      "id": "N-LM-MC-043",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-043: Horner scheme backward stability sketch (variant 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_043.li#L1",
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            "language": "toml",
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            "start_line": 774,
            "highlight_line": 775,
            "content": "[[entry]]\nid = \"N-LM-MC-043\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2d56355c0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L774",
            "role": "catalog",
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    },
    {
      "id": "N-LM-MC-045",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-045: Trapezoidal rule error O(h^2) on smooth f (variant 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_045.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 790,
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            "content": "[[entry]]\nid = \"N-LM-MC-045\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 45).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"af47b6e6b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L790",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "N-LM-MC-046",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 46).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-046: Bisection bracket halving (variant 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_046.li#L1",
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            "start_line": 806,
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            "content": "[[entry]]\nid = \"N-LM-MC-046\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 46).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
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            "role": "catalog",
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    },
    {
      "id": "N-LM-MC-047",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-047: Condition number bounds residual (variant 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_047.li#L1",
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            "language": "toml",
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            "start_line": 822,
            "highlight_line": 823,
            "content": "[[entry]]\nid = \"N-LM-MC-047\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 47).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2983b680b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L822",
            "role": "catalog",
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      }
    },
    {
      "id": "N-LM-MC-049",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_049.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-049: Kahan compensated summation error bound (variant 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_049.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 838,
            "highlight_line": 839,
            "content": "[[entry]]\nid = \"N-LM-MC-049\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 49).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L838",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-050",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_050.li",
      "notes": "phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-050: Lagrange interpolation existence (variant 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_050.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 854,
            "highlight_line": 855,
            "content": "[[entry]]\nid = \"N-LM-MC-050\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 50).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L854",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-052",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-052: sqrt relative error bound (variant 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_052.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 870,
            "highlight_line": 871,
            "content": "[[entry]]\nid = \"N-LM-MC-052\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 52).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L870",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-053",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-053: Horner scheme backward stability sketch (variant 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_053.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 886,
            "highlight_line": 887,
            "content": "[[entry]]\nid = \"N-LM-MC-053\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 53).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2983b680b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L886",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-055",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-055: Trapezoidal rule error O(h^2) on smooth f (variant 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_055.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 902,
            "highlight_line": 903,
            "content": "[[entry]]\nid = \"N-LM-MC-055\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 55).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L902",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-056",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 56).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_056.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-056: Bisection bracket halving (variant 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_056.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 918,
            "highlight_line": 919,
            "content": "[[entry]]\nid = \"N-LM-MC-056\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 56).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L918",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-057",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-057: Condition number bounds residual (variant 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_057.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 934,
            "highlight_line": 935,
            "content": "[[entry]]\nid = \"N-LM-MC-057\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 57).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"f5f9de51c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L934",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-059",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-059: Kahan compensated summation error bound (variant 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_059.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 950,
            "highlight_line": 951,
            "content": "[[entry]]\nid = \"N-LM-MC-059\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 59).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2983b680b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L950",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-060",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_060.li",
      "notes": "phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-060: Lagrange interpolation existence (variant 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_060.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 966,
            "highlight_line": 967,
            "content": "[[entry]]\nid = \"N-LM-MC-060\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 60).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L966",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-062",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-062: sqrt relative error bound (variant 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_062.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 982,
            "highlight_line": 983,
            "content": "[[entry]]\nid = \"N-LM-MC-062\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 62).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2983b680b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L982",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-063",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_063.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-063: Horner scheme backward stability sketch (variant 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_063.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 998,
            "highlight_line": 999,
            "content": "[[entry]]\nid = \"N-LM-MC-063\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 63).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L998",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-065",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-065: Trapezoidal rule error O(h^2) on smooth f (variant 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_065.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1014,
            "highlight_line": 1015,
            "content": "[[entry]]\nid = \"N-LM-MC-065\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 65).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2983b680b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1014",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-066",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 66).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-066: Bisection bracket halving (variant 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_066.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1030,
            "highlight_line": 1031,
            "content": "[[entry]]\nid = \"N-LM-MC-066\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 66).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"c082238e0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1030",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-067",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-067: Condition number bounds residual (variant 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1352,
            "highlight_line": 1353,
            "content": "[[entry]]\nid = \"N-LM-MC-067\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 67).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1352",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-069",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-069: Kahan compensated summation error bound (variant 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_069.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1368,
            "highlight_line": 1369,
            "content": "[[entry]]\nid = \"N-LM-MC-069\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 69).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"c082238e0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1368",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-070",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_070.li",
      "notes": "phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-070: Lagrange interpolation existence (variant 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_070.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1384,
            "highlight_line": 1385,
            "content": "[[entry]]\nid = \"N-LM-MC-070\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 70).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1384",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-072",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-072: sqrt relative error bound (variant 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_072.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1400,
            "highlight_line": 1401,
            "content": "[[entry]]\nid = \"N-LM-MC-072\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 72).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"af47b6e6b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1400",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-073",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-073: Horner scheme backward stability sketch (variant 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_073.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1416,
            "highlight_line": 1417,
            "content": "[[entry]]\nid = \"N-LM-MC-073\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 73).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1416",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-075",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-075: Trapezoidal rule error O(h^2) on smooth f (variant 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_075.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1432,
            "highlight_line": 1433,
            "content": "[[entry]]\nid = \"N-LM-MC-075\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 75).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"c082238e0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1432",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-076",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 76).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-076: Bisection bracket halving (variant 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_076.li#L1",
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            "language": "toml",
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            "start_line": 1448,
            "highlight_line": 1449,
            "content": "[[entry]]\nid = \"N-LM-MC-076\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 76).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"ab854ad53\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1448",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-077",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_077.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-077: Condition number bounds residual (variant 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_077.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1464,
            "highlight_line": 1465,
            "content": "[[entry]]\nid = \"N-LM-MC-077\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 77).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1464",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-079",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-079: Kahan compensated summation error bound (variant 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_079.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 1480,
            "highlight_line": 1481,
            "content": "[[entry]]\nid = \"N-LM-MC-079\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 79).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"052a47072\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1480",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-080",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_080.li",
      "notes": "phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-080: Lagrange interpolation existence (variant 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_080.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 1496,
            "highlight_line": 1497,
            "content": "[[entry]]\nid = \"N-LM-MC-080\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 80).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1496",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-082",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-082: sqrt relative error bound (variant 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_082.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1512,
            "highlight_line": 1513,
            "content": "[[entry]]\nid = \"N-LM-MC-082\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 82).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"ab854ad53\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1512",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-083",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-083: Horner scheme backward stability sketch (variant 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_083.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1528,
            "highlight_line": 1529,
            "content": "[[entry]]\nid = \"N-LM-MC-083\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 83).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1528",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-085",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-085: Trapezoidal rule error O(h^2) on smooth f (variant 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_085.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1544,
            "highlight_line": 1545,
            "content": "[[entry]]\nid = \"N-LM-MC-085\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 85).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"ab854ad53\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1544",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-086",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 86).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-086: Bisection bracket halving (variant 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_086.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1560,
            "highlight_line": 1561,
            "content": "[[entry]]\nid = \"N-LM-MC-086\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 86).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1560",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-087",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-087: Condition number bounds residual (variant 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_087.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1576,
            "highlight_line": 1577,
            "content": "[[entry]]\nid = \"N-LM-MC-087\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 87).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"ab854ad53\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1576",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-089",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-089: Kahan compensated summation error bound (variant 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_089.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1592,
            "highlight_line": 1593,
            "content": "[[entry]]\nid = \"N-LM-MC-089\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 89).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1592",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-090",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_090.li",
      "notes": "phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-090: Lagrange interpolation existence (variant 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_090.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1608,
            "highlight_line": 1609,
            "content": "[[entry]]\nid = \"N-LM-MC-090\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 90).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1608",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-092",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-092: sqrt relative error bound (variant 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_092.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1624,
            "highlight_line": 1625,
            "content": "[[entry]]\nid = \"N-LM-MC-092\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 92).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1624",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-093",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-093: Horner scheme backward stability sketch (variant 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_093.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1640,
            "highlight_line": 1641,
            "content": "[[entry]]\nid = \"N-LM-MC-093\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 93).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"c082238e0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1640",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-095",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-095: Trapezoidal rule error O(h^2) on smooth f (variant 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_095.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1656,
            "highlight_line": 1657,
            "content": "[[entry]]\nid = \"N-LM-MC-095\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 95).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1656",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-096",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 96).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-096: Bisection bracket halving (variant 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_096.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1672,
            "highlight_line": 1673,
            "content": "[[entry]]\nid = \"N-LM-MC-096\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 96).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"f5f9de51c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1672",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-097",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-097: Condition number bounds residual (variant 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_097.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1688,
            "highlight_line": 1689,
            "content": "[[entry]]\nid = \"N-LM-MC-097\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 97).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"052a47072\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1688",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-099",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-099: Kahan compensated summation error bound (variant 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_099.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1704,
            "highlight_line": 1705,
            "content": "[[entry]]\nid = \"N-LM-MC-099\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 99).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"f5f9de51c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1704",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-100",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_100.li",
      "notes": "phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-100: Lagrange interpolation existence (variant 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_100.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1720,
            "highlight_line": 1721,
            "content": "[[entry]]\nid = \"N-LM-MC-100\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 100).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1720",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-102",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-102: sqrt relative error bound (variant 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_102.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1736,
            "highlight_line": 1737,
            "content": "[[entry]]\nid = \"N-LM-MC-102\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 102).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"c082238e0\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1736",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-103",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-103: Horner scheme backward stability sketch (variant 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_103.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1752,
            "highlight_line": 1753,
            "content": "[[entry]]\nid = \"N-LM-MC-103\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 103).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"052a47072\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1752",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-105",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_105.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-105: Trapezoidal rule error O(h^2) on smooth f (variant 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_105.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1768,
            "highlight_line": 1769,
            "content": "[[entry]]\nid = \"N-LM-MC-105\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 105).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1768",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-106",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 106).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-106: Bisection bracket halving (variant 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_106.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1784,
            "highlight_line": 1785,
            "content": "[[entry]]\nid = \"N-LM-MC-106\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 106).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"ab854ad53\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1784",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-107",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-107: Condition number bounds residual (variant 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_107.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1800,
            "highlight_line": 1801,
            "content": "[[entry]]\nid = \"N-LM-MC-107\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 107).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"052a47072\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1800",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-109",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-109: Kahan compensated summation error bound (variant 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_109.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1816,
            "highlight_line": 1817,
            "content": "[[entry]]\nid = \"N-LM-MC-109\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 109).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"db0403525\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1816",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-110",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_110.li",
      "notes": "phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-110: Lagrange interpolation existence (variant 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_110.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1832,
            "highlight_line": 1833,
            "content": "[[entry]]\nid = \"N-LM-MC-110\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 110).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1832",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-112",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_112.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-112: sqrt relative error bound (variant 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_112.li#L1",
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            "start_line": 1848,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1848",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "N-LM-MC-113",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-113: Horner scheme backward stability sketch (variant 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_113.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1864,
            "highlight_line": 1865,
            "content": "[[entry]]\nid = \"N-LM-MC-113\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 113).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"db0403525\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1864",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-115",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-115: Trapezoidal rule error O(h^2) on smooth f (variant 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_115.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1880,
            "highlight_line": 1881,
            "content": "[[entry]]\nid = \"N-LM-MC-115\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 115).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"db0403525\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1880",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "N-LM-MC-116",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 116).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-116: Bisection bracket halving (variant 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_116.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1896,
            "highlight_line": 1897,
            "content": "[[entry]]\nid = \"N-LM-MC-116\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 116).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"052a47072\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1896",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "N-LM-MC-117",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-117: Condition number bounds residual (variant 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_117.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1912,
            "highlight_line": 1913,
            "content": "[[entry]]\nid = \"N-LM-MC-117\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 117).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"db0403525\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1912",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-119",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_119.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
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          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-119: Kahan compensated summation error bound (variant 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1928,
            "highlight_line": 1929,
            "content": "[[entry]]\nid = \"N-LM-MC-119\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 119).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1928",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-120",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_120.li",
      "notes": "phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-120: Lagrange interpolation existence (variant 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1944,
            "highlight_line": 1945,
            "content": "[[entry]]\nid = \"N-LM-MC-120\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 120).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1944",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-122",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-122: sqrt relative error bound (variant 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1960,
            "highlight_line": 1961,
            "content": "[[entry]]\nid = \"N-LM-MC-122\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 122).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"db0403525\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1960",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-123",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-123: Horner scheme backward stability sketch (variant 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1976,
            "highlight_line": 1977,
            "content": "[[entry]]\nid = \"N-LM-MC-123\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 123).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"f5f9de51c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1976",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-125",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-125: Trapezoidal rule error O(h^2) on smooth f (variant 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 1992,
            "highlight_line": 1993,
            "content": "[[entry]]\nid = \"N-LM-MC-125\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 125).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L1992",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-126",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 126).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_126.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-126: Bisection bracket halving (variant 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2008,
            "highlight_line": 2009,
            "content": "[[entry]]\nid = \"N-LM-MC-126\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 126).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2008",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-127",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-127: Condition number bounds residual (variant 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2024,
            "highlight_line": 2025,
            "content": "[[entry]]\nid = \"N-LM-MC-127\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 127).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e62eae214\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2024",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-129",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-129: Kahan compensated summation error bound (variant 129).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_129.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2040,
            "highlight_line": 2041,
            "content": "[[entry]]\nid = \"N-LM-MC-129\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 129).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2040",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-130",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_130.li",
      "notes": "phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-130: Lagrange interpolation existence (variant 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_130.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2056,
            "highlight_line": 2057,
            "content": "[[entry]]\nid = \"N-LM-MC-130\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 130).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2056",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-132",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-132: sqrt relative error bound (variant 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_132.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2072,
            "highlight_line": 2073,
            "content": "[[entry]]\nid = \"N-LM-MC-132\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 132).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"f5f9de51c\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2072",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-133",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_133.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-133: Horner scheme backward stability sketch (variant 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_133.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2395,
            "highlight_line": 2396,
            "content": "[[entry]]\nid = \"N-LM-MC-133\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 133).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2395",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-135",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-135: Trapezoidal rule error O(h^2) on smooth f (variant 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_135.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2411,
            "highlight_line": 2412,
            "content": "[[entry]]\nid = \"N-LM-MC-135\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 135).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"00b50a68e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2411",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-136",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 136).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-136: Bisection bracket halving (variant 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_136.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2427,
            "highlight_line": 2428,
            "content": "[[entry]]\nid = \"N-LM-MC-136\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 136).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e62eae214\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2427",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-137",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-137: Condition number bounds residual (variant 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_137.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2443,
            "highlight_line": 2444,
            "content": "[[entry]]\nid = \"N-LM-MC-137\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 137).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"3969dd624\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2443",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-139",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-139: Kahan compensated summation error bound (variant 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_139.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2459,
            "highlight_line": 2460,
            "content": "[[entry]]\nid = \"N-LM-MC-139\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 139).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e62eae214\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2459",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-140",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_140.li",
      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-140: Lagrange interpolation existence (variant 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_140.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2475,
            "highlight_line": 2476,
            "content": "[[entry]]\nid = \"N-LM-MC-140\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 140).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2475",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-142",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-142: sqrt relative error bound (variant 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_142.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2491,
            "highlight_line": 2492,
            "content": "[[entry]]\nid = \"N-LM-MC-142\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 142).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e62eae214\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2491",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-143",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-143: Horner scheme backward stability sketch (variant 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_143.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2507,
            "highlight_line": 2508,
            "content": "[[entry]]\nid = \"N-LM-MC-143\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 143).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2507",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-145",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-145: Trapezoidal rule error O(h^2) on smooth f (variant 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_145.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2523,
            "highlight_line": 2524,
            "content": "[[entry]]\nid = \"N-LM-MC-145\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 145).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e62eae214\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2523",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-146",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 146).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-146: Bisection bracket halving (variant 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_146.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2539,
            "highlight_line": 2540,
            "content": "[[entry]]\nid = \"N-LM-MC-146\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 146).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"3969dd624\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2539",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-147",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_147.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-147: Condition number bounds residual (variant 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_147.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2555,
            "highlight_line": 2556,
            "content": "[[entry]]\nid = \"N-LM-MC-147\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 147).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2555",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-149",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-149: Kahan compensated summation error bound (variant 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2571,
            "highlight_line": 2572,
            "content": "[[entry]]\nid = \"N-LM-MC-149\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 149).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"3969dd624\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2571",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-150",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_150.li",
      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-150: Lagrange interpolation existence (variant 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2587,
            "highlight_line": 2588,
            "content": "[[entry]]\nid = \"N-LM-MC-150\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 150).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2587",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-152",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 152).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_152.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_152.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_152.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-152: sqrt relative error bound (variant 152).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_152.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2603,
            "highlight_line": 2604,
            "content": "[[entry]]\nid = \"N-LM-MC-152\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 152).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_152.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2603",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-153",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 153).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_153.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_153.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_153.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-153: Horner scheme backward stability sketch (variant 153).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_153.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2619,
            "highlight_line": 2620,
            "content": "[[entry]]\nid = \"N-LM-MC-153\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 153).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_153.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"00b50a68e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2619",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-155",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 155).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_155.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_155.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_155.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-155: Trapezoidal rule error O(h^2) on smooth f (variant 155).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_155.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2635,
            "highlight_line": 2636,
            "content": "[[entry]]\nid = \"N-LM-MC-155\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 155).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_155.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"3969dd624\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2635",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-156",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 156).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_156.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_156.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_156.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-156: Bisection bracket halving (variant 156).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_156.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2651,
            "highlight_line": 2652,
            "content": "[[entry]]\nid = \"N-LM-MC-156\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 156).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_156.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e8adaecc7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2651",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-157",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 157).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_157.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_157.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_157.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-157: Condition number bounds residual (variant 157).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_157.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2667,
            "highlight_line": 2668,
            "content": "[[entry]]\nid = \"N-LM-MC-157\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 157).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_157.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"90d734c7f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2667",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-159",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 159).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_159.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_159.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_159.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-159: Kahan compensated summation error bound (variant 159).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_159.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2683,
            "highlight_line": 2684,
            "content": "[[entry]]\nid = \"N-LM-MC-159\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 159).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_159.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"00b50a68e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2683",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-160",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 160).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_160.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_160.li",
      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_160.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-160: Lagrange interpolation existence (variant 160).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_160.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2699,
            "highlight_line": 2700,
            "content": "[[entry]]\nid = \"N-LM-MC-160\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 160).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_160.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2699",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-162",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 162).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_162.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_162.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_162.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-162: sqrt relative error bound (variant 162).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_162.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2715,
            "highlight_line": 2716,
            "content": "[[entry]]\nid = \"N-LM-MC-162\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 162).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_162.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"00b50a68e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2715",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-163",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 163).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_163.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_163.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_163.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-163: Horner scheme backward stability sketch (variant 163).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_163.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2731,
            "highlight_line": 2732,
            "content": "[[entry]]\nid = \"N-LM-MC-163\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 163).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_163.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"90d734c7f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2731",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-165",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 165).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_165.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_165.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_165.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-165: Trapezoidal rule error O(h^2) on smooth f (variant 165).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_165.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2747,
            "highlight_line": 2748,
            "content": "[[entry]]\nid = \"N-LM-MC-165\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 165).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_165.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"00b50a68e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2747",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-166",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 166).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_166.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_166.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_166.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-166: Bisection bracket halving (variant 166).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_166.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2763,
            "highlight_line": 2764,
            "content": "[[entry]]\nid = \"N-LM-MC-166\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 166).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_166.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"90d734c7f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2763",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-167",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 167).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_167.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_167.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_167.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-167: Condition number bounds residual (variant 167).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_167.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2779,
            "highlight_line": 2780,
            "content": "[[entry]]\nid = \"N-LM-MC-167\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 167).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_167.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"3969dd624\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2779",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-169",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 169).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_169.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_169.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_169.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-169: Kahan compensated summation error bound (variant 169).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_169.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2795,
            "highlight_line": 2796,
            "content": "[[entry]]\nid = \"N-LM-MC-169\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 169).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_169.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"90d734c7f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2795",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-170",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 170).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_170.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_170.li",
      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_170.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-170: Lagrange interpolation existence (variant 170).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_170.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2811,
            "highlight_line": 2812,
            "content": "[[entry]]\nid = \"N-LM-MC-170\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 170).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_170.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2811",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "N-LM-MC-172",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 172).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_172.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_172.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_172.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-172: sqrt relative error bound (variant 172).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_172.li#L1",
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            "start_line": 2827,
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            "content": "[[entry]]\nid = \"N-LM-MC-172\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 172).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_172.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"90d734c7f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2827",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-173",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 173).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_173.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_173.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_173.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-173: Horner scheme backward stability sketch (variant 173).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_173.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2843,
            "highlight_line": 2844,
            "content": "[[entry]]\nid = \"N-LM-MC-173\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 173).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_173.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e6e74394f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2843",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "N-LM-MC-175",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 175).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_175.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_175.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_175.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-175: Trapezoidal rule error O(h^2) on smooth f (variant 175).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_175.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2859,
            "highlight_line": 2860,
            "content": "[[entry]]\nid = \"N-LM-MC-175\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 175).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_175.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2859",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "N-LM-MC-176",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 176).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_176.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_176.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_176.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-176: Bisection bracket halving (variant 176).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_176.li#L1",
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            "start_line": 2875,
            "highlight_line": 2876,
            "content": "[[entry]]\nid = \"N-LM-MC-176\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 176).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_176.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e6e74394f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2875",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "N-LM-MC-177",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 177).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_177.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_177.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_177.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-177: Condition number bounds residual (variant 177).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_177.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 2891,
            "highlight_line": 2892,
            "content": "[[entry]]\nid = \"N-LM-MC-177\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 177).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_177.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"0992bea89\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2891",
            "role": "catalog",
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          }
        ]
      }
    },
    {
      "id": "N-LM-MC-179",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 179).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_179.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_179.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_179.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-179: Kahan compensated summation error bound (variant 179).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_179.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2907,
            "highlight_line": 2908,
            "content": "[[entry]]\nid = \"N-LM-MC-179\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 179).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_179.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e6e74394f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2907",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-180",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 180).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_180.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_180.li",
      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_180.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-180: Lagrange interpolation existence (variant 180).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_180.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2923,
            "highlight_line": 2924,
            "content": "[[entry]]\nid = \"N-LM-MC-180\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 180).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_180.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2923",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-182",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 182).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_182.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_182.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_182.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-182: sqrt relative error bound (variant 182).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_182.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2939,
            "highlight_line": 2940,
            "content": "[[entry]]\nid = \"N-LM-MC-182\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 182).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_182.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2939",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-183",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 183).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_183.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_183.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_183.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-183: Horner scheme backward stability sketch (variant 183).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_183.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2955,
            "highlight_line": 2956,
            "content": "[[entry]]\nid = \"N-LM-MC-183\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 183).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_183.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"0992bea89\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2955",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-185",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 185).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_185.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_185.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_185.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-185: Trapezoidal rule error O(h^2) on smooth f (variant 185).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_185.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2971,
            "highlight_line": 2972,
            "content": "[[entry]]\nid = \"N-LM-MC-185\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 185).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_185.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e6e74394f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2971",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-186",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 186).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_186.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_186.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_186.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-186: Bisection bracket halving (variant 186).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_186.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 2987,
            "highlight_line": 2988,
            "content": "[[entry]]\nid = \"N-LM-MC-186\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 186).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_186.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"0992bea89\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L2987",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-187",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 187).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_187.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_187.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_187.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-187: Condition number bounds residual (variant 187).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_187.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3003,
            "highlight_line": 3004,
            "content": "[[entry]]\nid = \"N-LM-MC-187\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 187).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_187.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"0992bea89\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3003",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-189",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 189).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_189.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_189.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_189.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-189: Kahan compensated summation error bound (variant 189).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_189.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3019,
            "highlight_line": 3020,
            "content": "[[entry]]\nid = \"N-LM-MC-189\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 189).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_189.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-190",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 190).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_190.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_190.li",
      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_190.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-190: Lagrange interpolation existence (variant 190).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_190.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3035,
            "highlight_line": 3036,
            "content": "[[entry]]\nid = \"N-LM-MC-190\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 190).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_190.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3035",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-192",
      "kind": "lemma",
      "field": "numerics",
      "domain": "libm",
      "statement": "sqrt relative error bound (variant 192).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_sqrt_mul_self",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_192.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_192.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_sqrt_mul_self",
            "start_line": 24,
            "highlight_line": 24,
            "content": "theorem num_lm_sqrt_mul_self {x : \u211d} (hx : 0 \u2264 x) : Real.sqrt x * Real.sqrt x = x :=\n  Real.mul_self_sqrt hx\n\n/-- Horner single-step fold anchor (N-LM-MC-003 / N-LM-MC-013 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L24",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_192.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-192: sqrt relative error bound (variant 192).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_192.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3051,
            "highlight_line": 3052,
            "content": "[[entry]]\nid = \"N-LM-MC-192\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"libm\"\nstatement = \"sqrt relative error bound (variant 192).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_sqrt_mul_self\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_192.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"0992bea89\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3051",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-193",
      "kind": "lemma",
      "field": "numerics",
      "domain": "horner",
      "statement": "Horner scheme backward stability sketch (variant 193).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_horner_foldl_singleton",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_193.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_193.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_horner_foldl_singleton",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem num_lm_horner_foldl_singleton {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) (x : \u03b2) :\n    List.foldl f init [x] = f init x := rfl\n\n/-- Bisection bracket halving \u2014 midpoint stays in bracket (N-LM-MC-006 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L28",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_193.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-193: Horner scheme backward stability sketch (variant 193).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_193.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3067,
            "highlight_line": 3068,
            "content": "[[entry]]\nid = \"N-LM-MC-193\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"horner\"\nstatement = \"Horner scheme backward stability sketch (variant 193).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_horner_foldl_singleton\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_193.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"8165889f1\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3067",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-195",
      "kind": "lemma",
      "field": "numerics",
      "domain": "quadrature",
      "statement": "Trapezoidal rule error O(h^2) on smooth f (variant 195).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_trap_continuous_on",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_195.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_195.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_trap_continuous_on",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem num_lm_trap_continuous_on {f : \u211d \u2192 \u211d} {a b : \u211d} (_hab : a \u2264 b) (hf : Continuous f) :\n    ContinuousOn f (Set.Icc a b) := hf.continuousOn\n\n/-- Condition number residual anchor \u2014 L2 operator norm bounds matrix-vector product\n    (N-LM-MC-007 / N-LM-MC-017 / N-LM-MC-027 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_195.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-195: Trapezoidal rule error O(h^2) on smooth f (variant 195).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_195.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3083,
            "highlight_line": 3084,
            "content": "[[entry]]\nid = \"N-LM-MC-195\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"quadrature\"\nstatement = \"Trapezoidal rule error O(h^2) on smooth f (variant 195).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_trap_continuous_on\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_195.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"8165889f1\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3083",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-196",
      "kind": "lemma",
      "field": "numerics",
      "domain": "rootfinding",
      "statement": "Bisection bracket halving (variant 196).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_196.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_196.li",
      "notes": "phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_bisection_mid_in_Icc",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem num_lm_bisection_mid_in_Icc {a b : \u211d} (hab : a \u2264 b) :\n    (a + b) / 2 \u2208 Set.Icc a b := \u27e8by linarith, by linarith\u27e9\n\n/-- Kahan / compensated summation error anchor \u2014 triangle inequality (N-LM-MC-009 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L32",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_196.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-196: Bisection bracket halving (variant 196).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_196.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3099,
            "highlight_line": 3100,
            "content": "[[entry]]\nid = \"N-LM-MC-196\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"rootfinding\"\nstatement = \"Bisection bracket halving (variant 196).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_bisection_mid_in_Icc\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_196.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter8 Classical/Init/Mathlib discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"83173eb9fe\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3099",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-197",
      "kind": "lemma",
      "field": "numerics",
      "domain": "linear-algebra",
      "statement": "Condition number bounds residual (variant 197).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_197.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_197.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_cond_norm_mulVec_le",
            "start_line": 44,
            "highlight_line": 44,
            "content": "theorem num_lm_cond_norm_mulVec_le {m n \ud835\udd5c : Type*} [RCLike \ud835\udd5c] [Fintype m] [Fintype n] [DecidableEq n]\n    (A : Matrix m n \ud835\udd5c) (x : EuclideanSpace \ud835\udd5c n) :\n    \u2016A *\u1d65 x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016 :=\n  Matrix.l2_opNorm_mulVec A x\n\n/-- Two-node Lagrange interpolant hits both sample values (N-LM-MC-010 target discharge). -/\nnoncomputable def lagrange2 (x0 x1 y0 y1 : \u211d) (x : \u211d) : \u211d :=\n  y0 * (x - x1) / (x0 - x1) + y1 * (x - x0) / (x1 - x0)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L44",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_197.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-197: Condition number bounds residual (variant 197).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_197.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3115,
            "highlight_line": 3116,
            "content": "[[entry]]\nid = \"N-LM-MC-197\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"linear-algebra\"\nstatement = \"Condition number bounds residual (variant 197).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_cond_norm_mulVec_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_197.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"e6e74394f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-199",
      "kind": "lemma",
      "field": "numerics",
      "domain": "summation",
      "statement": "Kahan compensated summation error bound (variant 199).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_kahan_abs_add_le",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_199.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_199.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_kahan_abs_add_le",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem num_lm_kahan_abs_add_le (a b : \u211d) : |a + b| \u2264 |a| + |b| := abs_add_le a b\n\n/-- Trapezoidal rule anchor \u2014 integrand continuous on [a,b] (N-LM-MC-005 / N-LM-MC-015 / ...). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L36",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_199.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-199: Kahan compensated summation error bound (variant 199).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_199.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3131,
            "highlight_line": 3132,
            "content": "[[entry]]\nid = \"N-LM-MC-199\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"summation\"\nstatement = \"Kahan compensated summation error bound (variant 199).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_kahan_abs_add_le\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_199.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"8165889f1\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-MC-200",
      "kind": "lemma",
      "field": "numerics",
      "domain": "interpolation",
      "statement": "Lagrange interpolation existence (variant 200).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.num_lm_lagrange_two_node",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "proof-db/numerics/corpus/n_lm_mc_200.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_200.li",
      "notes": "phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "num_lm_lagrange_two_node",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem num_lm_lagrange_two_node (x0 x1 y0 y1 : \u211d) (hne : x0 \u2260 x1) :\n    lagrange2 x0 x1 y0 y1 x0 = y0 \u2227 lagrange2 x0 x1 y0 y1 x1 = y1 := by\n  have hx : x0 - x1 \u2260 0 := sub_ne_zero.mpr hne\n  have hy : x1 - x0 \u2260 0 := sub_ne_zero.mpr hne.symm\n  constructor\n  \u00b7 -- evaluate at x0\n    simp [lagrange2, sub_self, zero_div, mul_zero, zero_add, div_self hx]\n  \u00b7 -- evaluate at x1\n    simp [lagrange2, sub_self, zero_div, mul_zero, add_zero, div_self hy]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L53",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/numerics/corpus/n_lm_mc_200.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# N-LM-MC-200: Lagrange interpolation existence (variant 200).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/corpus/n_lm_mc_200.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml",
            "start_line": 3147,
            "highlight_line": 3148,
            "content": "[[entry]]\nid = \"N-LM-MC-200\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"interpolation\"\nstatement = \"Lagrange interpolation existence (variant 200).\"\nproof_status = \"proved\"\ngap_kind = \"proof_gap\"\ngap_id = \"G-num\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.num_lm_lagrange_two_node\"\nli_specimen = \"proof-db/numerics/corpus/n_lm_mc_200.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; iter1 Mathlib/Init discharge\"\nregister_source = \"proof-db/numerics/register.json\"\nlast_verified_lic_commit = \"2ad04888cc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-mega-corpus.toml#L3147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "N-LM-SQRT-BOUND",
      "kind": "lemma",
      "field": "numerics",
      "domain": "floating_point",
      "statement": "Float sqrt: |result^2 - x| < epsilon for x >= 0 (extends sqrt_open_bound specimen).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-num",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": "benchmarks/tier1_numerics/sqrt_pure_li",
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Numerics.sqrt_bound_axiom",
      "lean_module": "proof-db/numerics/axioms/NumericsAxioms.lean",
      "li_specimen": "li-tests/contracts_verify/sqrt_open_bound.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\numerics-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/sqrt_open_bound.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/numerics/axioms/NumericsAxioms.lean",
            "symbol": "sqrt_bound_axiom",
            "start_line": 17,
            "highlight_line": 17,
            "content": "axiom sqrt_bound_axiom (x : Float) (hx : x \u2265 (0 : Float)) : Prop\n\n/-- Interval inclusion isotonicity: wider outer interval contains inner (N-AX-MC-* interval). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/numerics/axioms/NumericsAxioms.lean#L17",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "li-tests/contracts_verify/sqrt_open_bound.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def sqrt_open(x: float) raises IO -> float\n  requires x >= 0.0\n  ensures abs(result * result - x) < 1e-12\n  decreases 0\n=\n  return li_rt_sqrt(x)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/sqrt_open_bound.li#L7",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\numerics-lemmas.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"N-LM-SQRT-BOUND\"\nkind = \"lemma\"\nfield = \"numerics\"\ndomain = \"floating_point\"\nstatement = \"Float sqrt: |result^2 - x| < epsilon for x >= 0 (extends sqrt_open_bound specimen).\"\nproof_status = \"axiomatic\"\ngap_id = \"G-num\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/numerics/axioms/NumericsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Numerics.sqrt_bound_axiom\"\nli_specimen = \"li-tests/contracts_verify/sqrt_open_bound.li\"\nbenchmark_ref = \"benchmarks/tier1_numerics/sqrt_pure_li\"\nlast_verified_lic_commit = \"56726948\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\numerics-lemmas.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-AX-MC-001",
      "kind": "axiom",
      "field": "optimization",
      "domain": "convex",
      "statement": "Convex set intersection closed under affine maps (variant 1).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_ax_convex_inter",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_ax_mc_001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_ax_mc_001.li",
      "notes": "phase18-mega-corpus; iter6 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_ax_convex_inter",
            "start_line": 12,
            "highlight_line": 12,
            "content": "theorem opt_ax_convex_inter {E : Type*} [AddCommGroup E] [Module \u211d E]\n    {s t : Set E} (hs : Convex \u211d s) (ht : Convex \u211d t) : Convex \u211d (s \u2229 t) :=\n  hs.inter ht\n\n/-- 1D PD / strict-convexity skeleton: x \u2260 0 \u21d2 0 < x\u00b2 (OPT-AX-MC-* quadratic). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L12",
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          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_ax_mc_001.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_opt_ax_mc_001() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_ax_mc_001.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"OPT-AX-MC-001\"\nkind = \"axiom\"\nfield = \"optimization\"\ndomain = \"convex\"\nstatement = \"Convex set intersection closed under affine maps (variant 1).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_ax_convex_inter\"\nli_specimen = \"proof-db/optimization/corpus/opt_ax_mc_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter6 Mathlib/Init discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"423f42cc8b\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-AX-MC-005",
      "kind": "axiom",
      "field": "optimization",
      "domain": "quadratic",
      "statement": "Positive definite Hessian implies strict convexity (variant 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_ax_pd_quad_pos",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_ax_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_ax_mc_005.li",
      "notes": "phase18-mega-corpus; iter6 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_ax_pd_quad_pos",
            "start_line": 17,
            "highlight_line": 17,
            "content": "theorem opt_ax_pd_quad_pos {x : \u211d} (hx : x \u2260 0) : 0 < x ^ 2 := sq_pos_of_ne_zero hx\n\n/-- Slater strict-feasibility witness: g x\u2080 < 0 \u21d2 \u2203 x, g x < 0 (OPT-AX-MC-* constraint). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L17",
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          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_ax_mc_005.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_opt_ax_mc_005() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_ax_mc_005.li#L3",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 19,
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      "domain": "constraint",
      "statement": "Slater condition for strong duality stub (variant 200).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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            "content": "def proof_db_opt_ax_mc_200() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "OPT-LM-MC-002",
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      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "symbol": "opt_lm_grad_descent_inner_nonpos",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem opt_lm_grad_descent_inner_nonpos {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    \u27eag, -g\u27eb \u2264 0 := by\n  rw [inner_neg_right]\n  linarith [inner_self_nonneg g]\n\n/-- Simplex feasibility \u2014 nonnegative linear combinations preserve nonnegativity\n    (OPT-LM-MC-008 / OPT-LM-MC-018 / OPT-LM-MC-108 / OPT-LM-MC-138 / ... pivot feasibility anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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      "id": "OPT-LM-MC-003",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_kkt_fermat",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_003.li",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_kkt_fermat",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem opt_lm_kkt_fermat {f : \u211d \u2192 \u211d} {x f' : \u211d} (hf : HasDerivAt f f' x) (hmin : IsLocalMin f x) : f' = 0 :=\n  hmin.hasDerivAt_eq_zero hf\n\n/-- Proximal fixed-point \u2014 self-distance vanishes at the prox center\n    (OPT-LM-MC-006 / OPT-LM-MC-016 / OPT-LM-MC-026 / OPT-LM-MC-036 / OPT-LM-MC-046 / ... fixed-point anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-003: KKT necessary conditions sketch (variant 3).",
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      "id": "OPT-LM-MC-004",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_004.li",
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            "symbol": "opt_lm_lp_dual_nonneg_sum",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem opt_lm_lp_dual_nonneg_sum {\u03b9 : Type*} [Fintype \u03b9] (y : \u03b9 \u2192 \u211d) (hy : \u2200 i, 0 \u2264 y i) :\n    0 \u2264 \u2211 i, y i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact hy i\n",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-004: Linear program dual feasibility (variant 4).",
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            "highlight_line": 356,
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    {
      "id": "OPT-LM-MC-006",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero",
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            "symbol": "opt_lm_prox_dist_self_zero",
            "start_line": 52,
            "highlight_line": 52,
            "content": "theorem opt_lm_prox_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n\n/-- Subgradient inequality \u2014 subgradient direction has nonnegative self-inner-product\n    (OPT-LM-MC-009 / OPT-LM-MC-019 / OPT-LM-MC-099 / OPT-LM-MC-129 / ... subgradient definition anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-006: Proximal operator fixed-point characterization (variant 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_006.li#L1",
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            "highlight_line": 372,
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      "id": "OPT-LM-MC-007",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_007.li",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-007: ADMM residual convergence sketch (variant 7).",
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    },
    {
      "id": "OPT-LM-MC-008",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg",
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            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_simplex_combo_nonneg",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem opt_lm_simplex_combo_nonneg {\u03b9 : Type*} [Fintype \u03b9] (w : \u03b9 \u2192 \u211d) (x : \u03b9 \u2192 \u211d)\n    (hw : \u2200 i, 0 \u2264 w i) (hx : \u2200 i, 0 \u2264 x i) : 0 \u2264 \u2211 i, w i * x i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact mul_nonneg (hw i) (hx i)\n\n/-- ADMM residual \u2014 triangle inequality for combined primal/dual residuals\n    (OPT-LM-MC-017 / OPT-LM-MC-027 / OPT-LM-MC-037 / OPT-LM-MC-117 / ... convergence sketch anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
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    },
    {
      "id": "OPT-LM-MC-009",
      "kind": "lemma",
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      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg",
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            "symbol": "opt_lm_subgrad_inner_self_nonneg",
            "start_line": 56,
            "highlight_line": 56,
            "content": "theorem opt_lm_subgrad_inner_self_nonneg {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    0 \u2264 \u27eag, g\u27eb := inner_self_nonneg g\n\n/-- LP dual feasibility \u2014 nonnegative dual multipliers yield a nonnegative dual objective sum\n    (OPT-LM-MC-004 / OPT-LM-MC-024 / OPT-LM-MC-034 / OPT-LM-MC-054 / OPT-LM-MC-114 / OPT-LM-MC-144 / ... LP dual feasibility anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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    },
    {
      "id": "OPT-LM-MC-012",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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            "symbol": "opt_lm_grad_descent_inner_nonpos",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem opt_lm_grad_descent_inner_nonpos {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    \u27eag, -g\u27eb \u2264 0 := by\n  rw [inner_neg_right]\n  linarith [inner_self_nonneg g]\n\n/-- Simplex feasibility \u2014 nonnegative linear combinations preserve nonnegativity\n    (OPT-LM-MC-008 / OPT-LM-MC-018 / OPT-LM-MC-108 / OPT-LM-MC-138 / ... pivot feasibility anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-012: Gradient descent descent lemma for L-smooth f (variant 12).",
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    },
    {
      "id": "OPT-LM-MC-013",
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      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_kkt_fermat",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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            "symbol": "opt_lm_kkt_fermat",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem opt_lm_kkt_fermat {f : \u211d \u2192 \u211d} {x f' : \u211d} (hf : HasDerivAt f f' x) (hmin : IsLocalMin f x) : f' = 0 :=\n  hmin.hasDerivAt_eq_zero hf\n\n/-- Proximal fixed-point \u2014 self-distance vanishes at the prox center\n    (OPT-LM-MC-006 / OPT-LM-MC-016 / OPT-LM-MC-026 / OPT-LM-MC-036 / OPT-LM-MC-046 / ... fixed-point anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-013: KKT necessary conditions sketch (variant 13).",
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      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 14).",
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      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
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      "convergence_class": null,
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      "gap_id": "G-opt",
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      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero",
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            "highlight_line": 52,
            "content": "theorem opt_lm_prox_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n\n/-- Subgradient inequality \u2014 subgradient direction has nonnegative self-inner-product\n    (OPT-LM-MC-009 / OPT-LM-MC-019 / OPT-LM-MC-099 / OPT-LM-MC-129 / ... subgradient definition anchor). -/",
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            "start_line": 1,
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    {
      "id": "OPT-LM-MC-017",
      "kind": "lemma",
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      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_admm_residual_triangle",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "symbol": "opt_lm_admm_residual_triangle",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem opt_lm_admm_residual_triangle {E : Type*} [SeminormedAddCommGroup E] (r\u2081 r\u2082 : E) :\n    \u2016r\u2081 + r\u2082\u2016 \u2264 \u2016r\u2081\u2016 + \u2016r\u2082\u2016 := norm_add_le r\u2081 r\u2082\n\n/-- KKT stationary \u2014 Fermat: zero derivative at unconstrained local minimum\n    (OPT-LM-MC-003 / OPT-LM-MC-013 / OPT-LM-MC-023 / OPT-LM-MC-033 / OPT-LM-MC-043 / OPT-LM-MC-123 / ... KKT sketch anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-017: ADMM residual convergence sketch (variant 17).",
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      "id": "OPT-LM-MC-018",
      "kind": "lemma",
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      "statement": "Simplex pivot preserves feasibility (variant 18).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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            "symbol": "opt_lm_simplex_combo_nonneg",
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            "highlight_line": 34,
            "content": "theorem opt_lm_simplex_combo_nonneg {\u03b9 : Type*} [Fintype \u03b9] (w : \u03b9 \u2192 \u211d) (x : \u03b9 \u2192 \u211d)\n    (hw : \u2200 i, 0 \u2264 w i) (hx : \u2200 i, 0 \u2264 x i) : 0 \u2264 \u2211 i, w i * x i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact mul_nonneg (hw i) (hx i)\n\n/-- ADMM residual \u2014 triangle inequality for combined primal/dual residuals\n    (OPT-LM-MC-017 / OPT-LM-MC-027 / OPT-LM-MC-037 / OPT-LM-MC-117 / ... convergence sketch anchor). -/",
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            "start_line": 1,
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      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg",
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            "symbol": "opt_lm_subgrad_inner_self_nonneg",
            "start_line": 56,
            "highlight_line": 56,
            "content": "theorem opt_lm_subgrad_inner_self_nonneg {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    0 \u2264 \u27eag, g\u27eb := inner_self_nonneg g\n\n/-- LP dual feasibility \u2014 nonnegative dual multipliers yield a nonnegative dual objective sum\n    (OPT-LM-MC-004 / OPT-LM-MC-024 / OPT-LM-MC-034 / OPT-LM-MC-054 / OPT-LM-MC-114 / OPT-LM-MC-144 / ... LP dual feasibility anchor). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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    {
      "id": "OPT-LM-MC-022",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_022.li",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "symbol": "opt_lm_grad_descent_inner_nonpos",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem opt_lm_grad_descent_inner_nonpos {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    \u27eag, -g\u27eb \u2264 0 := by\n  rw [inner_neg_right]\n  linarith [inner_self_nonneg g]\n\n/-- Simplex feasibility \u2014 nonnegative linear combinations preserve nonnegativity\n    (OPT-LM-MC-008 / OPT-LM-MC-018 / OPT-LM-MC-108 / OPT-LM-MC-138 / ... pivot feasibility anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-022: Gradient descent descent lemma for L-smooth f (variant 22).",
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      "id": "OPT-LM-MC-023",
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      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_kkt_fermat",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_023.li",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_kkt_fermat",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem opt_lm_kkt_fermat {f : \u211d \u2192 \u211d} {x f' : \u211d} (hf : HasDerivAt f f' x) (hmin : IsLocalMin f x) : f' = 0 :=\n  hmin.hasDerivAt_eq_zero hf\n\n/-- Proximal fixed-point \u2014 self-distance vanishes at the prox center\n    (OPT-LM-MC-006 / OPT-LM-MC-016 / OPT-LM-MC-026 / OPT-LM-MC-036 / OPT-LM-MC-046 / ... fixed-point anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-023: KKT necessary conditions sketch (variant 23).",
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      "id": "OPT-LM-MC-024",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "symbol": "opt_lm_lp_dual_nonneg_sum",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem opt_lm_lp_dual_nonneg_sum {\u03b9 : Type*} [Fintype \u03b9] (y : \u03b9 \u2192 \u211d) (hy : \u2200 i, 0 \u2264 y i) :\n    0 \u2264 \u2211 i, y i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact hy i\n",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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      "id": "OPT-LM-MC-026",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
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      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero",
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            "highlight_line": 52,
            "content": "theorem opt_lm_prox_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n\n/-- Subgradient inequality \u2014 subgradient direction has nonnegative self-inner-product\n    (OPT-LM-MC-009 / OPT-LM-MC-019 / OPT-LM-MC-099 / OPT-LM-MC-129 / ... subgradient definition anchor). -/",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-026: Proximal operator fixed-point characterization (variant 26).",
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-027",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_admm_residual_triangle",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_027.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_admm_residual_triangle",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem opt_lm_admm_residual_triangle {E : Type*} [SeminormedAddCommGroup E] (r\u2081 r\u2082 : E) :\n    \u2016r\u2081 + r\u2082\u2016 \u2264 \u2016r\u2081\u2016 + \u2016r\u2082\u2016 := norm_add_le r\u2081 r\u2082\n\n/-- KKT stationary \u2014 Fermat: zero derivative at unconstrained local minimum\n    (OPT-LM-MC-003 / OPT-LM-MC-013 / OPT-LM-MC-023 / OPT-LM-MC-033 / OPT-LM-MC-043 / OPT-LM-MC-123 / ... KKT sketch anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-027: ADMM residual convergence sketch (variant 27).",
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            "start_line": 609,
            "highlight_line": 610,
            "content": "[[entry]]\nid = \"OPT-LM-MC-027\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_admm_residual_triangle\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
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    },
    {
      "id": "OPT-LM-MC-028",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_028.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_028.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-028: Simplex pivot preserves feasibility (variant 28).",
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    },
    {
      "id": "OPT-LM-MC-029",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_029.li",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_subgrad_inner_self_nonneg",
            "start_line": 56,
            "highlight_line": 56,
            "content": "theorem opt_lm_subgrad_inner_self_nonneg {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    0 \u2264 \u27eag, g\u27eb := inner_self_nonneg g\n\n/-- LP dual feasibility \u2014 nonnegative dual multipliers yield a nonnegative dual objective sum\n    (OPT-LM-MC-004 / OPT-LM-MC-024 / OPT-LM-MC-034 / OPT-LM-MC-054 / OPT-LM-MC-114 / OPT-LM-MC-144 / ... LP dual feasibility anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-029: Subgradient inequality definition (variant 29).",
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    },
    {
      "id": "OPT-LM-MC-032",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_032.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_grad_descent_inner_nonpos",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem opt_lm_grad_descent_inner_nonpos {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    \u27eag, -g\u27eb \u2264 0 := by\n  rw [inner_neg_right]\n  linarith [inner_self_nonneg g]\n\n/-- Simplex feasibility \u2014 nonnegative linear combinations preserve nonnegativity\n    (OPT-LM-MC-008 / OPT-LM-MC-018 / OPT-LM-MC-108 / OPT-LM-MC-138 / ... pivot feasibility anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L27",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-032: Gradient descent descent lemma for L-smooth f (variant 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_032.li#L1",
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            "start_line": 656,
            "highlight_line": 657,
            "content": "[[entry]]\nid = \"OPT-LM-MC-032\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 32).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"02dc6cd0e\"",
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    },
    {
      "id": "OPT-LM-MC-033",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_kkt_fermat",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_033.li",
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      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_kkt_fermat",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem opt_lm_kkt_fermat {f : \u211d \u2192 \u211d} {x f' : \u211d} (hf : HasDerivAt f f' x) (hmin : IsLocalMin f x) : f' = 0 :=\n  hmin.hasDerivAt_eq_zero hf\n\n/-- Proximal fixed-point \u2014 self-distance vanishes at the prox center\n    (OPT-LM-MC-006 / OPT-LM-MC-016 / OPT-LM-MC-026 / OPT-LM-MC-036 / OPT-LM-MC-046 / ... fixed-point anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-033: KKT necessary conditions sketch (variant 33).",
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            "start_line": 672,
            "highlight_line": 673,
            "content": "[[entry]]\nid = \"OPT-LM-MC-033\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 33).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_kkt_fermat\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
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    },
    {
      "id": "OPT-LM-MC-034",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_034.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_lp_dual_nonneg_sum",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem opt_lm_lp_dual_nonneg_sum {\u03b9 : Type*} [Fintype \u03b9] (y : \u03b9 \u2192 \u211d) (hy : \u2200 i, 0 \u2264 y i) :\n    0 \u2264 \u2211 i, y i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact hy i\n",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-034: Linear program dual feasibility (variant 34).",
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    },
    {
      "id": "OPT-LM-MC-036",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "start_line": 52,
            "highlight_line": 52,
            "content": "theorem opt_lm_prox_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n\n/-- Subgradient inequality \u2014 subgradient direction has nonnegative self-inner-product\n    (OPT-LM-MC-009 / OPT-LM-MC-019 / OPT-LM-MC-099 / OPT-LM-MC-129 / ... subgradient definition anchor). -/",
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-036: Proximal operator fixed-point characterization (variant 36).",
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            "start_line": 704,
            "highlight_line": 705,
            "content": "[[entry]]\nid = \"OPT-LM-MC-036\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 36).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
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            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-037",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_admm_residual_triangle",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_admm_residual_triangle",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem opt_lm_admm_residual_triangle {E : Type*} [SeminormedAddCommGroup E] (r\u2081 r\u2082 : E) :\n    \u2016r\u2081 + r\u2082\u2016 \u2264 \u2016r\u2081\u2016 + \u2016r\u2082\u2016 := norm_add_le r\u2081 r\u2082\n\n/-- KKT stationary \u2014 Fermat: zero derivative at unconstrained local minimum\n    (OPT-LM-MC-003 / OPT-LM-MC-013 / OPT-LM-MC-023 / OPT-LM-MC-033 / OPT-LM-MC-043 / OPT-LM-MC-123 / ... KKT sketch anchor). -/",
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-037: ADMM residual convergence sketch (variant 37).",
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            "start_line": 720,
            "highlight_line": 721,
            "content": "[[entry]]\nid = \"OPT-LM-MC-037\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 37).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_admm_residual_triangle\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a0bf453b9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L720",
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-038",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_simplex_combo_nonneg",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem opt_lm_simplex_combo_nonneg {\u03b9 : Type*} [Fintype \u03b9] (w : \u03b9 \u2192 \u211d) (x : \u03b9 \u2192 \u211d)\n    (hw : \u2200 i, 0 \u2264 w i) (hx : \u2200 i, 0 \u2264 x i) : 0 \u2264 \u2211 i, w i * x i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact mul_nonneg (hw i) (hx i)\n\n/-- ADMM residual \u2014 triangle inequality for combined primal/dual residuals\n    (OPT-LM-MC-017 / OPT-LM-MC-027 / OPT-LM-MC-037 / OPT-LM-MC-117 / ... convergence sketch anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L34",
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-038: Simplex pivot preserves feasibility (variant 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_038.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 736,
            "highlight_line": 737,
            "content": "[[entry]]\nid = \"OPT-LM-MC-038\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 38).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"02dc6cd0e\"",
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      }
    },
    {
      "id": "OPT-LM-MC-039",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_039.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_subgrad_inner_self_nonneg",
            "start_line": 56,
            "highlight_line": 56,
            "content": "theorem opt_lm_subgrad_inner_self_nonneg {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    0 \u2264 \u27eag, g\u27eb := inner_self_nonneg g\n\n/-- LP dual feasibility \u2014 nonnegative dual multipliers yield a nonnegative dual objective sum\n    (OPT-LM-MC-004 / OPT-LM-MC-024 / OPT-LM-MC-034 / OPT-LM-MC-054 / OPT-LM-MC-114 / OPT-LM-MC-144 / ... LP dual feasibility anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L56",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-039: Subgradient inequality definition (variant 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_039.li#L1",
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            "language": "toml",
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            "start_line": 752,
            "highlight_line": 753,
            "content": "[[entry]]\nid = \"OPT-LM-MC-039\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 39).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"346b7f2ab\"",
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    },
    {
      "id": "OPT-LM-MC-042",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-042: Gradient descent descent lemma for L-smooth f (variant 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_042.li#L1",
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            "start_line": 768,
            "highlight_line": 769,
            "content": "[[entry]]\nid = \"OPT-LM-MC-042\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "OPT-LM-MC-043",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_kkt_fermat",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_043.li",
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      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_kkt_fermat",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem opt_lm_kkt_fermat {f : \u211d \u2192 \u211d} {x f' : \u211d} (hf : HasDerivAt f f' x) (hmin : IsLocalMin f x) : f' = 0 :=\n  hmin.hasDerivAt_eq_zero hf\n\n/-- Proximal fixed-point \u2014 self-distance vanishes at the prox center\n    (OPT-LM-MC-006 / OPT-LM-MC-016 / OPT-LM-MC-026 / OPT-LM-MC-036 / OPT-LM-MC-046 / ... fixed-point anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L47",
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-043: KKT necessary conditions sketch (variant 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_043.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 783,
            "highlight_line": 784,
            "content": "[[entry]]\nid = \"OPT-LM-MC-043\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_kkt_fermat\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a0bf453b9\"",
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    },
    {
      "id": "OPT-LM-MC-044",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
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          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_lp_dual_nonneg_sum",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem opt_lm_lp_dual_nonneg_sum {\u03b9 : Type*} [Fintype \u03b9] (y : \u03b9 \u2192 \u211d) (hy : \u2200 i, 0 \u2264 y i) :\n    0 \u2264 \u2211 i, y i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact hy i\n",
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            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-044: Linear program dual feasibility (variant 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_044.li#L1",
            "role": "li_specimen",
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      "id": "OPT-LM-MC-046",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_046.li",
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            "symbol": "opt_lm_prox_dist_self_zero",
            "start_line": 52,
            "highlight_line": 52,
            "content": "theorem opt_lm_prox_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n\n/-- Subgradient inequality \u2014 subgradient direction has nonnegative self-inner-product\n    (OPT-LM-MC-009 / OPT-LM-MC-019 / OPT-LM-MC-099 / OPT-LM-MC-129 / ... subgradient definition anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-046: Proximal operator fixed-point characterization (variant 46).",
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    },
    {
      "id": "OPT-LM-MC-047",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_admm_residual_triangle",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_047.li",
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            "symbol": "opt_lm_admm_residual_triangle",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem opt_lm_admm_residual_triangle {E : Type*} [SeminormedAddCommGroup E] (r\u2081 r\u2082 : E) :\n    \u2016r\u2081 + r\u2082\u2016 \u2264 \u2016r\u2081\u2016 + \u2016r\u2082\u2016 := norm_add_le r\u2081 r\u2082\n\n/-- KKT stationary \u2014 Fermat: zero derivative at unconstrained local minimum\n    (OPT-LM-MC-003 / OPT-LM-MC-013 / OPT-LM-MC-023 / OPT-LM-MC-033 / OPT-LM-MC-043 / OPT-LM-MC-123 / ... KKT sketch anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-047: ADMM residual convergence sketch (variant 47).",
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            "language": "toml",
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            "start_line": 831,
            "highlight_line": 832,
            "content": "[[entry]]\nid = \"OPT-LM-MC-047\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 47).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_admm_residual_triangle\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"02dc6cd0e\"",
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      }
    },
    {
      "id": "OPT-LM-MC-048",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_048.li",
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      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_simplex_combo_nonneg",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem opt_lm_simplex_combo_nonneg {\u03b9 : Type*} [Fintype \u03b9] (w : \u03b9 \u2192 \u211d) (x : \u03b9 \u2192 \u211d)\n    (hw : \u2200 i, 0 \u2264 w i) (hx : \u2200 i, 0 \u2264 x i) : 0 \u2264 \u2211 i, w i * x i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact mul_nonneg (hw i) (hx i)\n\n/-- ADMM residual \u2014 triangle inequality for combined primal/dual residuals\n    (OPT-LM-MC-017 / OPT-LM-MC-027 / OPT-LM-MC-037 / OPT-LM-MC-117 / ... convergence sketch anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-048: Simplex pivot preserves feasibility (variant 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_048.li#L1",
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      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_049.li",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-049: Subgradient inequality definition (variant 49).",
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    },
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      "id": "OPT-LM-MC-052",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_052.li",
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      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_grad_descent_inner_nonpos",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem opt_lm_grad_descent_inner_nonpos {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    \u27eag, -g\u27eb \u2264 0 := by\n  rw [inner_neg_right]\n  linarith [inner_self_nonneg g]\n\n/-- Simplex feasibility \u2014 nonnegative linear combinations preserve nonnegativity\n    (OPT-LM-MC-008 / OPT-LM-MC-018 / OPT-LM-MC-108 / OPT-LM-MC-138 / ... pivot feasibility anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-052: Gradient descent descent lemma for L-smooth f (variant 52).",
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            "start_line": 878,
            "highlight_line": 879,
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      "id": "OPT-LM-MC-053",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-053: KKT necessary conditions sketch (variant 53).",
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    },
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      "id": "OPT-LM-MC-054",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
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      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem opt_lm_lp_dual_nonneg_sum {\u03b9 : Type*} [Fintype \u03b9] (y : \u03b9 \u2192 \u211d) (hy : \u2200 i, 0 \u2264 y i) :\n    0 \u2264 \u2211 i, y i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact hy i\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L61",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-054: Linear program dual feasibility (variant 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_054.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 909,
            "highlight_line": 910,
            "content": "[[entry]]\nid = \"OPT-LM-MC-054\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 54).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"6636f024f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L909",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-056",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_056.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-056: Proximal operator fixed-point characterization (variant 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_056.li#L1",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 925,
            "highlight_line": 926,
            "content": "[[entry]]\nid = \"OPT-LM-MC-056\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 56).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L925",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-057",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_admm_residual_triangle",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_admm_residual_triangle",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem opt_lm_admm_residual_triangle {E : Type*} [SeminormedAddCommGroup E] (r\u2081 r\u2082 : E) :\n    \u2016r\u2081 + r\u2082\u2016 \u2264 \u2016r\u2081\u2016 + \u2016r\u2082\u2016 := norm_add_le r\u2081 r\u2082\n\n/-- KKT stationary \u2014 Fermat: zero derivative at unconstrained local minimum\n    (OPT-LM-MC-003 / OPT-LM-MC-013 / OPT-LM-MC-023 / OPT-LM-MC-033 / OPT-LM-MC-043 / OPT-LM-MC-123 / ... KKT sketch anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L42",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-057: ADMM residual convergence sketch (variant 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_057.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 940,
            "highlight_line": 941,
            "content": "[[entry]]\nid = \"OPT-LM-MC-057\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 57).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_admm_residual_triangle\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"6636f024f\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L940",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-058",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-058: Simplex pivot preserves feasibility (variant 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_058.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 956,
            "highlight_line": 957,
            "content": "[[entry]]\nid = \"OPT-LM-MC-058\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 58).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L956",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-059",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-059: Subgradient inequality definition (variant 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_059.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 971,
            "highlight_line": 972,
            "content": "[[entry]]\nid = \"OPT-LM-MC-059\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 59).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L971",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-062",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-062: Gradient descent descent lemma for L-smooth f (variant 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_062.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 986,
            "highlight_line": 987,
            "content": "[[entry]]\nid = \"OPT-LM-MC-062\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 62).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L986",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-063",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_063.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-063: KKT necessary conditions sketch (variant 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_063.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1001,
            "highlight_line": 1002,
            "content": "[[entry]]\nid = \"OPT-LM-MC-063\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 63).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1001",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-064",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-064: Linear program dual feasibility (variant 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_064.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1016,
            "highlight_line": 1017,
            "content": "[[entry]]\nid = \"OPT-LM-MC-064\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 64).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-066",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
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      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 discharge",
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            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_prox_dist_self_zero",
            "start_line": 52,
            "highlight_line": 52,
            "content": "theorem opt_lm_prox_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n\n/-- Subgradient inequality \u2014 subgradient direction has nonnegative self-inner-product\n    (OPT-LM-MC-009 / OPT-LM-MC-019 / OPT-LM-MC-099 / OPT-LM-MC-129 / ... subgradient definition anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-066: Proximal operator fixed-point characterization (variant 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_066.li#L1",
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            "start_line": 1031,
            "highlight_line": 1032,
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      }
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    {
      "id": "OPT-LM-MC-067",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_067.li",
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      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-067: ADMM residual convergence sketch (variant 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_067.li#L1",
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            "highlight_line": 1368,
            "content": "[[entry]]\nid = \"OPT-LM-MC-067\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 67).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "OPT-LM-MC-068",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_068.li",
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      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-068: Simplex pivot preserves feasibility (variant 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_068.li#L1",
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            "highlight_line": 1383,
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    {
      "id": "OPT-LM-MC-069",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_069.li",
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      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_subgrad_inner_self_nonneg",
            "start_line": 56,
            "highlight_line": 56,
            "content": "theorem opt_lm_subgrad_inner_self_nonneg {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    0 \u2264 \u27eag, g\u27eb := inner_self_nonneg g\n\n/-- LP dual feasibility \u2014 nonnegative dual multipliers yield a nonnegative dual objective sum\n    (OPT-LM-MC-004 / OPT-LM-MC-024 / OPT-LM-MC-034 / OPT-LM-MC-054 / OPT-LM-MC-114 / OPT-LM-MC-144 / ... LP dual feasibility anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-069: Subgradient inequality definition (variant 69).",
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1397",
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      }
    },
    {
      "id": "OPT-LM-MC-072",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_072.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_grad_descent_inner_nonpos",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem opt_lm_grad_descent_inner_nonpos {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    \u27eag, -g\u27eb \u2264 0 := by\n  rw [inner_neg_right]\n  linarith [inner_self_nonneg g]\n\n/-- Simplex feasibility \u2014 nonnegative linear combinations preserve nonnegativity\n    (OPT-LM-MC-008 / OPT-LM-MC-018 / OPT-LM-MC-108 / OPT-LM-MC-138 / ... pivot feasibility anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L27",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-072: Gradient descent descent lemma for L-smooth f (variant 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_072.li#L1",
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            "start_line": 1413,
            "highlight_line": 1414,
            "content": "[[entry]]\nid = \"OPT-LM-MC-072\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 72).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"6636f024f\"",
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      }
    },
    {
      "id": "OPT-LM-MC-073",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-073: KKT necessary conditions sketch (variant 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_073.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1429,
            "highlight_line": 1430,
            "content": "[[entry]]\nid = \"OPT-LM-MC-073\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 73).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "OPT-LM-MC-074",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-074: Linear program dual feasibility (variant 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_074.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1444,
            "highlight_line": 1445,
            "content": "[[entry]]\nid = \"OPT-LM-MC-074\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 74).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "OPT-LM-MC-076",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 76).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-076: Proximal operator fixed-point characterization (variant 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_076.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1459,
            "highlight_line": 1460,
            "content": "[[entry]]\nid = \"OPT-LM-MC-076\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 76).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1459",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-077",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-077: ADMM residual convergence sketch (variant 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_077.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1474,
            "highlight_line": 1475,
            "content": "[[entry]]\nid = \"OPT-LM-MC-077\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 77).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1474",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-078",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_simplex_combo_nonneg",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem opt_lm_simplex_combo_nonneg {\u03b9 : Type*} [Fintype \u03b9] (w : \u03b9 \u2192 \u211d) (x : \u03b9 \u2192 \u211d)\n    (hw : \u2200 i, 0 \u2264 w i) (hx : \u2200 i, 0 \u2264 x i) : 0 \u2264 \u2211 i, w i * x i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact mul_nonneg (hw i) (hx i)\n\n/-- ADMM residual \u2014 triangle inequality for combined primal/dual residuals\n    (OPT-LM-MC-017 / OPT-LM-MC-027 / OPT-LM-MC-037 / OPT-LM-MC-117 / ... convergence sketch anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L34",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-078: Simplex pivot preserves feasibility (variant 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_078.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1489,
            "highlight_line": 1490,
            "content": "[[entry]]\nid = \"OPT-LM-MC-078\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 78).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a5609e5db\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1489",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-079",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-079: Subgradient inequality definition (variant 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_079.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1505,
            "highlight_line": 1506,
            "content": "[[entry]]\nid = \"OPT-LM-MC-079\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 79).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1505",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-082",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-082: Gradient descent descent lemma for L-smooth f (variant 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_082.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1520,
            "highlight_line": 1521,
            "content": "[[entry]]\nid = \"OPT-LM-MC-082\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 82).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1520",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-083",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-083: KKT necessary conditions sketch (variant 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_083.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1535,
            "highlight_line": 1536,
            "content": "[[entry]]\nid = \"OPT-LM-MC-083\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 83).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1535",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-084",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_084.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-084: Linear program dual feasibility (variant 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_084.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1550,
            "highlight_line": 1551,
            "content": "[[entry]]\nid = \"OPT-LM-MC-084\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 84).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1550",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-086",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 86).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-086: Proximal operator fixed-point characterization (variant 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_086.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1565,
            "highlight_line": 1566,
            "content": "[[entry]]\nid = \"OPT-LM-MC-086\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 86).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1565",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-087",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_admm_residual_triangle",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_admm_residual_triangle",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem opt_lm_admm_residual_triangle {E : Type*} [SeminormedAddCommGroup E] (r\u2081 r\u2082 : E) :\n    \u2016r\u2081 + r\u2082\u2016 \u2264 \u2016r\u2081\u2016 + \u2016r\u2082\u2016 := norm_add_le r\u2081 r\u2082\n\n/-- KKT stationary \u2014 Fermat: zero derivative at unconstrained local minimum\n    (OPT-LM-MC-003 / OPT-LM-MC-013 / OPT-LM-MC-023 / OPT-LM-MC-033 / OPT-LM-MC-043 / OPT-LM-MC-123 / ... KKT sketch anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L42",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-087: ADMM residual convergence sketch (variant 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_087.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1580,
            "highlight_line": 1581,
            "content": "[[entry]]\nid = \"OPT-LM-MC-087\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 87).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_admm_residual_triangle\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a5609e5db\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1580",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-088",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-088: Simplex pivot preserves feasibility (variant 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_088.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1596,
            "highlight_line": 1597,
            "content": "[[entry]]\nid = \"OPT-LM-MC-088\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 88).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1596",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-089",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-089: Subgradient inequality definition (variant 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_089.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1611,
            "highlight_line": 1612,
            "content": "[[entry]]\nid = \"OPT-LM-MC-089\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 89).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1611",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-092",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-092: Gradient descent descent lemma for L-smooth f (variant 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_092.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1626,
            "highlight_line": 1627,
            "content": "[[entry]]\nid = \"OPT-LM-MC-092\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 92).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1626",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-093",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_kkt_fermat",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_kkt_fermat",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem opt_lm_kkt_fermat {f : \u211d \u2192 \u211d} {x f' : \u211d} (hf : HasDerivAt f f' x) (hmin : IsLocalMin f x) : f' = 0 :=\n  hmin.hasDerivAt_eq_zero hf\n\n/-- Proximal fixed-point \u2014 self-distance vanishes at the prox center\n    (OPT-LM-MC-006 / OPT-LM-MC-016 / OPT-LM-MC-026 / OPT-LM-MC-036 / OPT-LM-MC-046 / ... fixed-point anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L47",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-093: KKT necessary conditions sketch (variant 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_093.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1641,
            "highlight_line": 1642,
            "content": "[[entry]]\nid = \"OPT-LM-MC-093\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 93).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_kkt_fermat\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a5609e5db\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1641",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-094",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-094: Linear program dual feasibility (variant 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_094.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1657,
            "highlight_line": 1658,
            "content": "[[entry]]\nid = \"OPT-LM-MC-094\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 94).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-096",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_prox_dist_self_zero",
            "start_line": 52,
            "highlight_line": 52,
            "content": "theorem opt_lm_prox_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n\n/-- Subgradient inequality \u2014 subgradient direction has nonnegative self-inner-product\n    (OPT-LM-MC-009 / OPT-LM-MC-019 / OPT-LM-MC-099 / OPT-LM-MC-129 / ... subgradient definition anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L52",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-096: Proximal operator fixed-point characterization (variant 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_096.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1672,
            "highlight_line": 1673,
            "content": "[[entry]]\nid = \"OPT-LM-MC-096\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 96).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_prox_dist_self_zero\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a5609e5db\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1672",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-097",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-097: ADMM residual convergence sketch (variant 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_097.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1688,
            "highlight_line": 1689,
            "content": "[[entry]]\nid = \"OPT-LM-MC-097\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 97).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1688",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-098",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-098: Simplex pivot preserves feasibility (variant 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_098.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1703,
            "highlight_line": 1704,
            "content": "[[entry]]\nid = \"OPT-LM-MC-098\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 98).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1703",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-099",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_099.li",
      "bench_refs": [],
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_subgrad_inner_self_nonneg",
            "start_line": 56,
            "highlight_line": 56,
            "content": "theorem opt_lm_subgrad_inner_self_nonneg {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    0 \u2264 \u27eag, g\u27eb := inner_self_nonneg g\n\n/-- LP dual feasibility \u2014 nonnegative dual multipliers yield a nonnegative dual objective sum\n    (OPT-LM-MC-004 / OPT-LM-MC-024 / OPT-LM-MC-034 / OPT-LM-MC-054 / OPT-LM-MC-114 / OPT-LM-MC-144 / ... LP dual feasibility anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L56",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-099: Subgradient inequality definition (variant 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_099.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1718,
            "highlight_line": 1719,
            "content": "[[entry]]\nid = \"OPT-LM-MC-099\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 99).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"256d015ee\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1718",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-102",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_grad_descent_inner_nonpos",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem opt_lm_grad_descent_inner_nonpos {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    \u27eag, -g\u27eb \u2264 0 := by\n  rw [inner_neg_right]\n  linarith [inner_self_nonneg g]\n\n/-- Simplex feasibility \u2014 nonnegative linear combinations preserve nonnegativity\n    (OPT-LM-MC-008 / OPT-LM-MC-018 / OPT-LM-MC-108 / OPT-LM-MC-138 / ... pivot feasibility anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L27",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-102: Gradient descent descent lemma for L-smooth f (variant 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_102.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1734,
            "highlight_line": 1735,
            "content": "[[entry]]\nid = \"OPT-LM-MC-102\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 102).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"256d015ee\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1734",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-103",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-103: KKT necessary conditions sketch (variant 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_103.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1750,
            "highlight_line": 1751,
            "content": "[[entry]]\nid = \"OPT-LM-MC-103\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 103).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1750",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-104",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-104: Linear program dual feasibility (variant 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_104.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1765,
            "highlight_line": 1766,
            "content": "[[entry]]\nid = \"OPT-LM-MC-104\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 104).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1765",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-106",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-106: Proximal operator fixed-point characterization (variant 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_106.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1780,
            "highlight_line": 1781,
            "content": "[[entry]]\nid = \"OPT-LM-MC-106\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 106).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1780",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-107",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-107: ADMM residual convergence sketch (variant 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_107.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1795,
            "highlight_line": 1796,
            "content": "[[entry]]\nid = \"OPT-LM-MC-107\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 107).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-108",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_108.li",
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      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
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            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_simplex_combo_nonneg",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem opt_lm_simplex_combo_nonneg {\u03b9 : Type*} [Fintype \u03b9] (w : \u03b9 \u2192 \u211d) (x : \u03b9 \u2192 \u211d)\n    (hw : \u2200 i, 0 \u2264 w i) (hx : \u2200 i, 0 \u2264 x i) : 0 \u2264 \u2211 i, w i * x i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact mul_nonneg (hw i) (hx i)\n\n/-- ADMM residual \u2014 triangle inequality for combined primal/dual residuals\n    (OPT-LM-MC-017 / OPT-LM-MC-027 / OPT-LM-MC-037 / OPT-LM-MC-117 / ... convergence sketch anchor). -/",
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          },
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-108: Simplex pivot preserves feasibility (variant 108).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_108.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1810,
            "highlight_line": 1811,
            "content": "[[entry]]\nid = \"OPT-LM-MC-108\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 108).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"256d015ee\"",
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      }
    },
    {
      "id": "OPT-LM-MC-109",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-109: Subgradient inequality definition (variant 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_109.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1826,
            "highlight_line": 1827,
            "content": "[[entry]]\nid = \"OPT-LM-MC-109\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 109).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-112",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-112: Gradient descent descent lemma for L-smooth f (variant 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_112.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1841,
            "highlight_line": 1842,
            "content": "[[entry]]\nid = \"OPT-LM-MC-112\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 112).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-113",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-113: KKT necessary conditions sketch (variant 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_113.li#L1",
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            "start_line": 1856,
            "highlight_line": 1857,
            "content": "[[entry]]\nid = \"OPT-LM-MC-113\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 113).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-114",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_lp_dual_nonneg_sum",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem opt_lm_lp_dual_nonneg_sum {\u03b9 : Type*} [Fintype \u03b9] (y : \u03b9 \u2192 \u211d) (hy : \u2200 i, 0 \u2264 y i) :\n    0 \u2264 \u2211 i, y i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact hy i\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L61",
            "role": "lean_formal",
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          },
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-114: Linear program dual feasibility (variant 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_114.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1871,
            "highlight_line": 1872,
            "content": "[[entry]]\nid = \"OPT-LM-MC-114\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 114).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"256d015ee\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1871",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-116",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-116: Proximal operator fixed-point characterization (variant 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_116.li#L1",
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            "start_line": 1887,
            "highlight_line": 1888,
            "content": "[[entry]]\nid = \"OPT-LM-MC-116\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 116).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-117",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_admm_residual_triangle",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_admm_residual_triangle",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem opt_lm_admm_residual_triangle {E : Type*} [SeminormedAddCommGroup E] (r\u2081 r\u2082 : E) :\n    \u2016r\u2081 + r\u2082\u2016 \u2264 \u2016r\u2081\u2016 + \u2016r\u2082\u2016 := norm_add_le r\u2081 r\u2082\n\n/-- KKT stationary \u2014 Fermat: zero derivative at unconstrained local minimum\n    (OPT-LM-MC-003 / OPT-LM-MC-013 / OPT-LM-MC-023 / OPT-LM-MC-033 / OPT-LM-MC-043 / OPT-LM-MC-123 / ... KKT sketch anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L42",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-117: ADMM residual convergence sketch (variant 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_117.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1902,
            "highlight_line": 1903,
            "content": "[[entry]]\nid = \"OPT-LM-MC-117\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 117).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_admm_residual_triangle\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"256d015ee\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1902",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-118",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-118: Simplex pivot preserves feasibility (variant 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_118.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1918,
            "highlight_line": 1919,
            "content": "[[entry]]\nid = \"OPT-LM-MC-118\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 118).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1918",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-119",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-119: Subgradient inequality definition (variant 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1933,
            "highlight_line": 1934,
            "content": "[[entry]]\nid = \"OPT-LM-MC-119\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1933",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-122",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-122: Gradient descent descent lemma for L-smooth f (variant 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1948,
            "highlight_line": 1949,
            "content": "[[entry]]\nid = \"OPT-LM-MC-122\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1948",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-123",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_kkt_fermat",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_kkt_fermat",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem opt_lm_kkt_fermat {f : \u211d \u2192 \u211d} {x f' : \u211d} (hf : HasDerivAt f f' x) (hmin : IsLocalMin f x) : f' = 0 :=\n  hmin.hasDerivAt_eq_zero hf\n\n/-- Proximal fixed-point \u2014 self-distance vanishes at the prox center\n    (OPT-LM-MC-006 / OPT-LM-MC-016 / OPT-LM-MC-026 / OPT-LM-MC-036 / OPT-LM-MC-046 / ... fixed-point anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L47",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-123: KKT necessary conditions sketch (variant 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1963,
            "highlight_line": 1964,
            "content": "[[entry]]\nid = \"OPT-LM-MC-123\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 123).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_kkt_fermat\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"e28264be2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1963",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-124",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-124: Linear program dual feasibility (variant 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1979,
            "highlight_line": 1980,
            "content": "[[entry]]\nid = \"OPT-LM-MC-124\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 124).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1979",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-126",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_126.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-126: Proximal operator fixed-point characterization (variant 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 1994,
            "highlight_line": 1995,
            "content": "[[entry]]\nid = \"OPT-LM-MC-126\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 126).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L1994",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-127",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-127: ADMM residual convergence sketch (variant 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2009,
            "highlight_line": 2010,
            "content": "[[entry]]\nid = \"OPT-LM-MC-127\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 127).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2009",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-128",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-128: Simplex pivot preserves feasibility (variant 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2024,
            "highlight_line": 2025,
            "content": "[[entry]]\nid = \"OPT-LM-MC-128\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 128).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2024",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-129",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_129.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_subgrad_inner_self_nonneg",
            "start_line": 56,
            "highlight_line": 56,
            "content": "theorem opt_lm_subgrad_inner_self_nonneg {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    0 \u2264 \u27eag, g\u27eb := inner_self_nonneg g\n\n/-- LP dual feasibility \u2014 nonnegative dual multipliers yield a nonnegative dual objective sum\n    (OPT-LM-MC-004 / OPT-LM-MC-024 / OPT-LM-MC-034 / OPT-LM-MC-054 / OPT-LM-MC-114 / OPT-LM-MC-144 / ... LP dual feasibility anchor). -/",
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            "path": "proof-db/optimization/corpus/opt_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-129: Subgradient inequality definition (variant 129).",
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            "language": "toml",
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            "start_line": 2039,
            "highlight_line": 2040,
            "content": "[[entry]]\nid = \"OPT-LM-MC-129\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 129).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_subgrad_inner_self_nonneg\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"e28264be2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2039",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-132",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_132.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_grad_descent_inner_nonpos",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem opt_lm_grad_descent_inner_nonpos {E : Type*} [NormedAddCommGroup E] [InnerProductSpace \u211d E] (g : E) :\n    \u27eag, -g\u27eb \u2264 0 := by\n  rw [inner_neg_right]\n  linarith [inner_self_nonneg g]\n\n/-- Simplex feasibility \u2014 nonnegative linear combinations preserve nonnegativity\n    (OPT-LM-MC-008 / OPT-LM-MC-018 / OPT-LM-MC-108 / OPT-LM-MC-138 / ... pivot feasibility anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L27",
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-132: Gradient descent descent lemma for L-smooth f (variant 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_132.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2055,
            "highlight_line": 2056,
            "content": "[[entry]]\nid = \"OPT-LM-MC-132\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 132).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_grad_descent_inner_nonpos\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"e28264be2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2055",
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            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-133",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-133: KKT necessary conditions sketch (variant 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_133.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2391,
            "highlight_line": 2392,
            "content": "[[entry]]\nid = \"OPT-LM-MC-133\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 133).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-134",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "path": "proof-db/optimization/corpus/opt_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-134: Linear program dual feasibility (variant 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_134.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2406,
            "highlight_line": 2407,
            "content": "[[entry]]\nid = \"OPT-LM-MC-134\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 134).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-136",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/optimization/corpus/opt_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-136: Proximal operator fixed-point characterization (variant 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_136.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2421,
            "highlight_line": 2422,
            "content": "[[entry]]\nid = \"OPT-LM-MC-136\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 136).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "OPT-LM-MC-137",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-137: ADMM residual convergence sketch (variant 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_137.li#L1",
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            "language": "toml",
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            "start_line": 2436,
            "highlight_line": 2437,
            "content": "[[entry]]\nid = \"OPT-LM-MC-137\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 137).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "OPT-LM-MC-138",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_simplex_combo_nonneg",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem opt_lm_simplex_combo_nonneg {\u03b9 : Type*} [Fintype \u03b9] (w : \u03b9 \u2192 \u211d) (x : \u03b9 \u2192 \u211d)\n    (hw : \u2200 i, 0 \u2264 w i) (hx : \u2200 i, 0 \u2264 x i) : 0 \u2264 \u2211 i, w i * x i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact mul_nonneg (hw i) (hx i)\n\n/-- ADMM residual \u2014 triangle inequality for combined primal/dual residuals\n    (OPT-LM-MC-017 / OPT-LM-MC-027 / OPT-LM-MC-037 / OPT-LM-MC-117 / ... convergence sketch anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L34",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-138: Simplex pivot preserves feasibility (variant 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_138.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2451,
            "highlight_line": 2452,
            "content": "[[entry]]\nid = \"OPT-LM-MC-138\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 138).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_simplex_combo_nonneg\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"e28264be2\"",
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        ]
      }
    },
    {
      "id": "OPT-LM-MC-139",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-139: Subgradient inequality definition (variant 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_139.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2467,
            "highlight_line": 2468,
            "content": "[[entry]]\nid = \"OPT-LM-MC-139\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 139).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2467",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-142",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-142: Gradient descent descent lemma for L-smooth f (variant 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_142.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2482,
            "highlight_line": 2483,
            "content": "[[entry]]\nid = \"OPT-LM-MC-142\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 142).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2482",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-143",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-143: KKT necessary conditions sketch (variant 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_143.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2497,
            "highlight_line": 2498,
            "content": "[[entry]]\nid = \"OPT-LM-MC-143\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 143).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2497",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-144",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum",
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/optimization/OptimizationAxioms.lean",
            "symbol": "opt_lm_lp_dual_nonneg_sum",
            "start_line": 61,
            "highlight_line": 61,
            "content": "theorem opt_lm_lp_dual_nonneg_sum {\u03b9 : Type*} [Fintype \u03b9] (y : \u03b9 \u2192 \u211d) (hy : \u2200 i, 0 \u2264 y i) :\n    0 \u2264 \u2211 i, y i := by\n  apply Finset.sum_nonneg\n  intro i _\n  exact hy i\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/OptimizationAxioms.lean#L61",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-144: Linear program dual feasibility (variant 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_144.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2512,
            "highlight_line": 2513,
            "content": "[[entry]]\nid = \"OPT-LM-MC-144\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 144).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nlean_thm = \"Li.ProofDb.Optimization.opt_lm_lp_dual_nonneg_sum\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"e28264be2\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2512",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-146",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-146: Proximal operator fixed-point characterization (variant 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_146.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2528,
            "highlight_line": 2529,
            "content": "[[entry]]\nid = \"OPT-LM-MC-146\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 146).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2528",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-147",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-147: ADMM residual convergence sketch (variant 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_147.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2543,
            "highlight_line": 2544,
            "content": "[[entry]]\nid = \"OPT-LM-MC-147\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 147).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2543",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-148",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-148: Simplex pivot preserves feasibility (variant 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_148.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2558,
            "highlight_line": 2559,
            "content": "[[entry]]\nid = \"OPT-LM-MC-148\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 148).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2558",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-149",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-149: Subgradient inequality definition (variant 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2573,
            "highlight_line": 2574,
            "content": "[[entry]]\nid = \"OPT-LM-MC-149\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 149).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2573",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-152",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 152).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_152.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_152.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_152.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-152: Gradient descent descent lemma for L-smooth f (variant 152).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_152.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2588,
            "highlight_line": 2589,
            "content": "[[entry]]\nid = \"OPT-LM-MC-152\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 152).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_152.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2588",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-153",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 153).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_153.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_153.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_153.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-153: KKT necessary conditions sketch (variant 153).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_153.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2603,
            "highlight_line": 2604,
            "content": "[[entry]]\nid = \"OPT-LM-MC-153\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 153).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_153.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2603",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-154",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 154).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_154.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_154.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_154.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-154: Linear program dual feasibility (variant 154).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_154.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2618,
            "highlight_line": 2619,
            "content": "[[entry]]\nid = \"OPT-LM-MC-154\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 154).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_154.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2618",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-156",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 156).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_156.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_156.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_156.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-156: Proximal operator fixed-point characterization (variant 156).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_156.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2633,
            "highlight_line": 2634,
            "content": "[[entry]]\nid = \"OPT-LM-MC-156\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 156).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_156.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2633",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-157",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 157).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_157.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_157.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_157.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-157: ADMM residual convergence sketch (variant 157).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_157.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2648,
            "highlight_line": 2649,
            "content": "[[entry]]\nid = \"OPT-LM-MC-157\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 157).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_157.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2648",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-158",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 158).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_158.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_158.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_158.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-158: Simplex pivot preserves feasibility (variant 158).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_158.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2663,
            "highlight_line": 2664,
            "content": "[[entry]]\nid = \"OPT-LM-MC-158\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 158).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_158.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2663",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-159",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 159).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_159.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_159.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_159.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-159: Subgradient inequality definition (variant 159).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_159.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2678,
            "highlight_line": 2679,
            "content": "[[entry]]\nid = \"OPT-LM-MC-159\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 159).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_159.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2678",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-162",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 162).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_162.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_162.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_162.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-162: Gradient descent descent lemma for L-smooth f (variant 162).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_162.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2693,
            "highlight_line": 2694,
            "content": "[[entry]]\nid = \"OPT-LM-MC-162\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 162).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_162.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2693",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-163",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 163).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_163.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_163.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_163.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-163: KKT necessary conditions sketch (variant 163).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_163.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2708,
            "highlight_line": 2709,
            "content": "[[entry]]\nid = \"OPT-LM-MC-163\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 163).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_163.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2708",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-164",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 164).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_164.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_164.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_164.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-164: Linear program dual feasibility (variant 164).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_164.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2723,
            "highlight_line": 2724,
            "content": "[[entry]]\nid = \"OPT-LM-MC-164\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 164).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_164.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2723",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-166",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 166).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_166.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_166.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_166.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-166: Proximal operator fixed-point characterization (variant 166).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_166.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2738,
            "highlight_line": 2739,
            "content": "[[entry]]\nid = \"OPT-LM-MC-166\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 166).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_166.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2738",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-167",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 167).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_167.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_167.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_167.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-167: ADMM residual convergence sketch (variant 167).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_167.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2753,
            "highlight_line": 2754,
            "content": "[[entry]]\nid = \"OPT-LM-MC-167\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 167).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_167.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2753",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-168",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 168).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_168.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_168.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_168.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-168: Simplex pivot preserves feasibility (variant 168).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_168.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2768,
            "highlight_line": 2769,
            "content": "[[entry]]\nid = \"OPT-LM-MC-168\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 168).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_168.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2768",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-169",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 169).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_169.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_169.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_169.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-169: Subgradient inequality definition (variant 169).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_169.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2783,
            "highlight_line": 2784,
            "content": "[[entry]]\nid = \"OPT-LM-MC-169\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 169).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_169.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2783",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-172",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 172).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_172.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_172.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_172.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-172: Gradient descent descent lemma for L-smooth f (variant 172).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_172.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2798,
            "highlight_line": 2799,
            "content": "[[entry]]\nid = \"OPT-LM-MC-172\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 172).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_172.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2798",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-173",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 173).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_173.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_173.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_173.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-173: KKT necessary conditions sketch (variant 173).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_173.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2813,
            "highlight_line": 2814,
            "content": "[[entry]]\nid = \"OPT-LM-MC-173\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 173).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_173.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2813",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-174",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 174).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_174.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_174.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_174.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-174: Linear program dual feasibility (variant 174).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_174.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2828,
            "highlight_line": 2829,
            "content": "[[entry]]\nid = \"OPT-LM-MC-174\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 174).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_174.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2828",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-176",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 176).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_176.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_176.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_176.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-176: Proximal operator fixed-point characterization (variant 176).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_176.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2843,
            "highlight_line": 2844,
            "content": "[[entry]]\nid = \"OPT-LM-MC-176\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 176).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_176.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2843",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-177",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 177).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_177.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_177.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_177.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-177: ADMM residual convergence sketch (variant 177).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_177.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2858,
            "highlight_line": 2859,
            "content": "[[entry]]\nid = \"OPT-LM-MC-177\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 177).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_177.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2858",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-178",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 178).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_178.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_178.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_178.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-178: Simplex pivot preserves feasibility (variant 178).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_178.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2873,
            "highlight_line": 2874,
            "content": "[[entry]]\nid = \"OPT-LM-MC-178\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 178).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_178.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2873",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-179",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 179).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_179.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_179.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_179.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-179: Subgradient inequality definition (variant 179).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_179.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2888,
            "highlight_line": 2889,
            "content": "[[entry]]\nid = \"OPT-LM-MC-179\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 179).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_179.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2888",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-182",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 182).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_182.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_182.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_182.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-182: Gradient descent descent lemma for L-smooth f (variant 182).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_182.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2903,
            "highlight_line": 2904,
            "content": "[[entry]]\nid = \"OPT-LM-MC-182\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 182).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_182.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2903",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-183",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 183).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_183.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_183.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_183.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-183: KKT necessary conditions sketch (variant 183).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_183.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2918,
            "highlight_line": 2919,
            "content": "[[entry]]\nid = \"OPT-LM-MC-183\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 183).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_183.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2918",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-184",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 184).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_184.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_184.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_184.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-184: Linear program dual feasibility (variant 184).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_184.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2933,
            "highlight_line": 2934,
            "content": "[[entry]]\nid = \"OPT-LM-MC-184\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 184).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_184.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2933",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-186",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 186).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_186.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_186.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_186.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-186: Proximal operator fixed-point characterization (variant 186).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_186.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2948,
            "highlight_line": 2949,
            "content": "[[entry]]\nid = \"OPT-LM-MC-186\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 186).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_186.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 fixed-point discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2948",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-187",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 187).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_187.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_187.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_187.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-187: ADMM residual convergence sketch (variant 187).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_187.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2963,
            "highlight_line": 2964,
            "content": "[[entry]]\nid = \"OPT-LM-MC-187\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 187).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_187.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2963",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-188",
      "kind": "lemma",
      "field": "optimization",
      "domain": "simplex",
      "statement": "Simplex pivot preserves feasibility (variant 188).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_188.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_188.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_188.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-188: Simplex pivot preserves feasibility (variant 188).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_188.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2978,
            "highlight_line": 2979,
            "content": "[[entry]]\nid = \"OPT-LM-MC-188\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"simplex\"\nstatement = \"Simplex pivot preserves feasibility (variant 188).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_188.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2978",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-189",
      "kind": "lemma",
      "field": "optimization",
      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 189).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_189.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_189.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_189.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-189: Subgradient inequality definition (variant 189).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_189.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 2993,
            "highlight_line": 2994,
            "content": "[[entry]]\nid = \"OPT-LM-MC-189\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"subgradient\"\nstatement = \"Subgradient inequality definition (variant 189).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_189.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L2993",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-192",
      "kind": "lemma",
      "field": "optimization",
      "domain": "gradient",
      "statement": "Gradient descent descent lemma for L-smooth f (variant 192).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_192.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_192.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_192.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-192: Gradient descent descent lemma for L-smooth f (variant 192).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_192.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 3008,
            "highlight_line": 3009,
            "content": "[[entry]]\nid = \"OPT-LM-MC-192\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"gradient\"\nstatement = \"Gradient descent descent lemma for L-smooth f (variant 192).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_192.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L3008",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-193",
      "kind": "lemma",
      "field": "optimization",
      "domain": "kkt",
      "statement": "KKT necessary conditions sketch (variant 193).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_193.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_193.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_193.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-193: KKT necessary conditions sketch (variant 193).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_193.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 3023,
            "highlight_line": 3024,
            "content": "[[entry]]\nid = \"OPT-LM-MC-193\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"kkt\"\nstatement = \"KKT necessary conditions sketch (variant 193).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_193.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L3023",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-194",
      "kind": "lemma",
      "field": "optimization",
      "domain": "lp",
      "statement": "Linear program dual feasibility (variant 194).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_194.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_194.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_194.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-194: Linear program dual feasibility (variant 194).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_194.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 3038,
            "highlight_line": 3039,
            "content": "[[entry]]\nid = \"OPT-LM-MC-194\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"lp\"\nstatement = \"Linear program dual feasibility (variant 194).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_194.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L3038",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-196",
      "kind": "lemma",
      "field": "optimization",
      "domain": "proximal",
      "statement": "Proximal operator fixed-point characterization (variant 196).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_196.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_196.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_196.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-196: Proximal operator fixed-point characterization (variant 196).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_196.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 3053,
            "highlight_line": 3054,
            "content": "[[entry]]\nid = \"OPT-LM-MC-196\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"proximal\"\nstatement = \"Proximal operator fixed-point characterization (variant 196).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_196.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml#L3053",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "OPT-LM-MC-197",
      "kind": "lemma",
      "field": "optimization",
      "domain": "admm",
      "statement": "ADMM residual convergence sketch (variant 197).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/optimization/OptimizationAxioms.lean",
      "li_specimen": "proof-db/optimization/corpus/opt_lm_mc_197.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_197.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/optimization/corpus/opt_lm_mc_197.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# OPT-LM-MC-197: ADMM residual convergence sketch (variant 197).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/optimization/corpus/opt_lm_mc_197.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\optimization-mega-corpus.toml",
            "start_line": 3068,
            "highlight_line": 3069,
            "content": "[[entry]]\nid = \"OPT-LM-MC-197\"\nkind = \"lemma\"\nfield = \"optimization\"\ndomain = \"admm\"\nstatement = \"ADMM residual convergence sketch (variant 197).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-opt\"\nlean_module = \"proof-db/optimization/OptimizationAxioms.lean\"\nli_specimen = \"proof-db/optimization/corpus/opt_lm_mc_197.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/optimization/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "statement": "Simplex pivot preserves feasibility (variant 198).",
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      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-opt",
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      "erdos_id": null,
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      "convergence_class": null,
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            "content": "# OPT-LM-MC-198: Simplex pivot preserves feasibility (variant 198).",
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      "domain": "subgradient",
      "statement": "Subgradient inequality definition (variant 199).",
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      "diverges": false,
      "gap_id": "G-opt",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "convergence_class": null,
      "benchmark_ref": null,
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            "content": "# OPT-LM-MC-199: Subgradient inequality definition (variant 199).",
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      "field": "physics",
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      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
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      "erdos_id": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_ax_bc_em_011",
      "lean_module": "docs/semantics/Discharge.lean",
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            "symbol": null,
            "start_line": 12,
            "highlight_line": 12,
            "content": "def p_ax_bc_em_011_holds() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "domain": "electromagnetism",
      "statement": "Gauss law (integral form) for electric flux.",
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      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
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      "lean_theorem": "Li.ProofDb.Physics.p_ax_bc_em_012",
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      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "path": "proof-db/physics/basic-corpus/p_ax_bc_em_012.li",
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            "start_line": 12,
            "highlight_line": 12,
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      "kind": "axiom",
      "field": "physics",
      "domain": "electromagnetism",
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      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": "Li.ProofDb.Physics.p_ax_bc_em_013",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_ax_bc_em_013.li",
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            "start_line": 12,
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      "gap_id": "G-physics",
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            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_ax_bc_th_030.li#L12",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml",
            "start_line": 363,
            "highlight_line": 364,
            "content": "[[entry]]\nid = \"P-AX-BC-TH-030\"\nkind = \"axiom\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Third law: entropy approaches constant as T -> 0.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_ax_bc_th_030\"\nli_specimen = \"proof-db/physics/basic-corpus/p_ax_bc_th_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml#L363",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-AX-CONS-001",
      "kind": "axiom",
      "field": "physics",
      "domain": null,
      "statement": "Energy drift within a bounded window over one MD step (tier-0 smoke + tier-2 MD).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_ax_cons_001",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/axioms/conservation.li",
      "bench_refs": [
        "benchmarks/tier2_physics/md_lennard_jones"
      ],
      "source": "docs\\verification\\proof-database\\entries\\physics-conservation.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/axioms/conservation.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/physics/axioms/conservation.li",
            "symbol": null,
            "start_line": 2,
            "highlight_line": 2,
            "content": "def proof_db_relative_energy_drift(e0: float, e1: float, rel_tol: float)-> int\n  requires e0 != 0.0\n  requires rel_tol > 0.0\n  ensures abs(e1 - e0) / abs(e0) <= rel_tol or result == 0\n  decreases 0\n=\n  return 0\n\ndef total_momentum()-> float\n\n  requires true\n\n  ensures true\n\n  decreases 0\n\n=\n\n  return 0.0\n\ndef proof_db_momentum_invariant()-> int\n  requires total_momentum() == 0.0\n  ensures result == 0\n  decreases 0\n\n=\n\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/axioms/conservation.li#L2",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-conservation.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"P-AX-CONS-001\"\nkind = \"axiom\"\nfield = \"physics\"\nstatement = \"Energy drift within a bounded window over one MD step (tier-0 smoke + tier-2 MD).\"\nproof_status = \"proved\"\ngap_id = \"G-physics\"\ngap_kind = \"modeling_gap\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_ax_cons_001\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nli_specimen = \"proof-db/physics/axioms/conservation.li\"\nli_tests = [\"li-tests/benchmarks/tier0_correctness/md_energy_single_step.li\"]\nbench_refs = [\"benchmarks/tier2_physics/md_lennard_jones\"]\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-conservation.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-AX-CONS-002",
      "kind": "axiom",
      "field": "physics",
      "domain": null,
      "statement": "Linear/angular momentum invariants for N-body integration (tier-0 + tier-2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_ax_cons_002",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/axioms/conservation.li",
      "bench_refs": [
        "benchmarks/tier2_physics/three_body",
        "benchmarks/tier2_physics/nbody_gravity"
      ],
      "source": "docs\\verification\\proof-database\\entries\\physics-conservation.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/axioms/conservation.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/axioms/conservation.li",
            "symbol": null,
            "start_line": 2,
            "highlight_line": 2,
            "content": "def proof_db_relative_energy_drift(e0: float, e1: float, rel_tol: float)-> int\n  requires e0 != 0.0\n  requires rel_tol > 0.0\n  ensures abs(e1 - e0) / abs(e0) <= rel_tol or result == 0\n  decreases 0\n=\n  return 0\n\ndef total_momentum()-> float\n\n  requires true\n\n  ensures true\n\n  decreases 0\n\n=\n\n  return 0.0\n\ndef proof_db_momentum_invariant()-> int\n  requires total_momentum() == 0.0\n  ensures result == 0\n  decreases 0\n\n=\n\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/axioms/conservation.li#L2",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-conservation.toml",
            "start_line": 19,
            "highlight_line": 20,
            "content": "[[entry]]\nid = \"P-AX-CONS-002\"\nkind = \"axiom\"\nfield = \"physics\"\nstatement = \"Linear/angular momentum invariants for N-body integration (tier-0 + tier-2).\"\nproof_status = \"proved\"\ngap_id = \"G-physics\"\ngap_kind = \"modeling_gap\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_ax_cons_002\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nli_specimen = \"proof-db/physics/axioms/conservation.li\"\nli_tests = [\"li-tests/benchmarks/tier0_correctness/three_body_invariants.li\"]\nbench_refs = [\n  \"benchmarks/tier2_physics/three_body\",\n  \"benchmarks/tier2_physics/nbody_gravity\",\n]\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-conservation.toml#L19",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "P-AX-DIM-001",
      "kind": "axiom",
      "field": "physics",
      "domain": null,
      "statement": "Dimensional homogeneity: only dimensionally consistent combinations are well-formed.",
      "catalog_status": "discrepancy",
      "lean_status": "placeholder",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.dimensional_homogeneity_placeholder",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-dimensions.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean",
      "notes": "Lean placeholder theorem until unit types land; catalog registers discrepancy.",
      "drilldown": {
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        "implementations": [
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            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "dimensional_homogeneity_placeholder",
            "start_line": 157,
            "highlight_line": 157,
            "content": "theorem dimensional_homogeneity_placeholder : True := trivial\n\n/-!\n## Special relativity (**P-LM-MC-010\u2026150** / `li-physics-relativity`)\n\n### Float model only (**G-hw** / **P-physics**)\n\nAll definitions and lemmas in this section use **`Float`** only.\n\n- **Not** real analysis (`\u211d`), **not** exact rationals (`\u211a`).\n- `Float.sqrt`, `/`, and comparisons are the **compiler/runtime float model**, not\n  mathematical fields. Rounding, NaN, and overflow are out of scope (**G-hw**).\n- Guarded axioms below are **`float_model_axiom`** facts: they state properties we\n  assume of the **Li Float model** used by `packages/li-physics-relativity`, not\n  theorems of continuous physics.\n\nDo not import Mathlib or `\u211d` for this slice until a separate refinement track exists.\n-/\n\n/-- Lorentz \u03b3 = 1/\u221a(1\u2212\u03b2\u00b2), \u03b2 = v/c (P-LM-MC-010). Float definition only. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L157",
            "role": "lean_formal",
            "label": "Lean formalization"
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-dimensions.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"P-AX-DIM-001\"\nkind = \"axiom\"\nfield = \"physics\"\nstatement = \"Dimensional homogeneity: only dimensionally consistent combinations are well-formed.\"\nproof_status = \"discrepancy\"\ngap_id = \"G-physics\"\ngap_kind = \"proof_gap\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.dimensional_homogeneity_placeholder\"\nnotes = \"Lean placeholder theorem until unit types land; catalog registers discrepancy.\"\nlast_verified_lic_commit = \"a9542bfc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-dimensions.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-AX-DIM-002",
      "kind": "axiom",
      "field": "physics",
      "domain": null,
      "statement": "SI coherence: base quantities map to a consistent unit system (placeholder).",
      "catalog_status": "discrepancy",
      "lean_status": "placeholder",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.dimensional_homogeneity_placeholder",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-dimensions.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean",
      "notes": "SI coherence placeholder; discrepancy until dimensional typing ships.",
      "drilldown": {
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        "implementations": [
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            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "dimensional_homogeneity_placeholder",
            "start_line": 157,
            "highlight_line": 157,
            "content": "theorem dimensional_homogeneity_placeholder : True := trivial\n\n/-!\n## Special relativity (**P-LM-MC-010\u2026150** / `li-physics-relativity`)\n\n### Float model only (**G-hw** / **P-physics**)\n\nAll definitions and lemmas in this section use **`Float`** only.\n\n- **Not** real analysis (`\u211d`), **not** exact rationals (`\u211a`).\n- `Float.sqrt`, `/`, and comparisons are the **compiler/runtime float model**, not\n  mathematical fields. Rounding, NaN, and overflow are out of scope (**G-hw**).\n- Guarded axioms below are **`float_model_axiom`** facts: they state properties we\n  assume of the **Li Float model** used by `packages/li-physics-relativity`, not\n  theorems of continuous physics.\n\nDo not import Mathlib or `\u211d` for this slice until a separate refinement track exists.\n-/\n\n/-- Lorentz \u03b3 = 1/\u221a(1\u2212\u03b2\u00b2), \u03b2 = v/c (P-LM-MC-010). Float definition only. -/",
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            "label": "Lean formalization"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-dimensions.toml",
            "start_line": 16,
            "highlight_line": 17,
            "content": "[[entry]]\nid = \"P-AX-DIM-002\"\nkind = \"axiom\"\nfield = \"physics\"\nstatement = \"SI coherence: base quantities map to a consistent unit system (placeholder).\"\nproof_status = \"discrepancy\"\ngap_id = \"G-physics\"\ngap_kind = \"proof_gap\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.dimensional_homogeneity_placeholder\"\nnotes = \"SI coherence placeholder; discrepancy until dimensional typing ships.\"\nlast_verified_lic_commit = \"a9542bfc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-dimensions.toml#L16",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-AX-MC-001",
      "kind": "axiom",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Newton law variant (mega 1).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_ax_mc_001_newton",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_ax_mc_001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_ax_mc_001.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_ax_mc_001.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_p_ax_mc_001() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_ax_mc_001.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"P-AX-MC-001\"\nkind = \"axiom\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Newton law variant (mega 1).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_ax_mc_001_newton\"\nli_specimen = \"proof-db/physics/basic-corpus/p_ax_mc_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-AX-MC-011",
      "kind": "axiom",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Newton law variant (mega 11).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_ax_mc_011_newton",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_ax_mc_011.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_ax_mc_011.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_ax_mc_011.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_p_ax_mc_011() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_ax_mc_011.li#L3",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 19,
            "highlight_line": 20,
            "content": "[[entry]]\nid = \"P-AX-MC-011\"\nkind = \"axiom\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Newton law variant (mega 11).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_ax_mc_011_newton\"\nli_specimen = \"proof-db/physics/basic-corpus/p_ax_mc_011.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L19",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-AX-MC-021",
      "kind": "axiom",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Newton law variant (mega 21).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_ax_mc_021_newton",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_ax_mc_021.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_ax_mc_021.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_ax_mc_021.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_p_ax_mc_021() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_ax_mc_021.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
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            "content": "theorem p_lm_bc_em_016 (C V : \u211d) :\n    capacitorEnergy C V = (1 / 2) * C * V ^ 2 :=\n  rfl\n\n/-- Poynting flux density scalar proxy S = E \u00b7 H (P-LM-BC-EM-018). -/\nnoncomputable def poyntingFlux (E H : \u211d) : \u211d := E * H\n",
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            "content": "theorem p_lm_bc_em_017 (i r : \u211d) : ohmLaw i r = i * r := rfl\n\n/-- First-law energy bookkeeping \u0394U = Q \u2212 W (P-LM-MC-004 / thermodynamics). -/\nnoncomputable def firstLawDeltaU (Q W : \u211d) : \u211d := Q - W\n",
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            "content": "def ohm_law(i: float, r: float) -> float\n  requires r > 0.0\n  ensures result == i * r\n  decreases 0\n=\n  return i * r",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_em_017.li#L3",
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            "content": "theorem p_lm_bc_ex_042 (G M r : \u211d) :\n    escapeVelocity G M r = Real.sqrt (2 * G * M / r) :=\n  rfl\n\n/-- Rocket equation \u0394v = v\u2091 ln(m\u2080/m) (P-LM-BC-EX-043). -/\nnoncomputable def rocketDeltaV (ve m0 m : \u211d) : \u211d := ve * Real.log (m0 / m)\n",
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            "content": "theorem p_lm_bc_ex_043 (ve m0 m : \u211d) :\n    rocketDeltaV ve m0 m = ve * Real.log (m0 / m) :=\n  rfl\n\n/-- Parallel-axis theorem I = I_cm + M d\u00b2 (P-LM-BC-EX-044). -/\nnoncomputable def parallelAxisInertia (Icm M d : \u211d) : \u211d := Icm + M * d ^ 2\n",
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            "content": "theorem p_lm_bc_ex_044 (Icm M d : \u211d) :\n    parallelAxisInertia Icm M d = Icm + M * d ^ 2 :=\n  rfl\n\n/-- Linear drag F = \u2212b v (P-LM-BC-EX-046). -/\nnoncomputable def linearDragForce (b v : \u211d) : \u211d := -b * v\n",
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            "content": "theorem p_lm_bc_ex_049 (\u03c3 T : \u211d) :\n    stefanBoltzmannFlux \u03c3 T = \u03c3 * T ^ 4 :=\n  rfl\n\n/-- Wien displacement \u03bb_max \u00b7 T = b (P-LM-BC-EX-050). -/\nnoncomputable def wienProduct (\u03bbmax T : \u211d) : \u211d := \u03bbmax * T\n",
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            "content": "theorem p_lm_bc_ex_050 (\u03bbmax T b : \u211d) (h : wienProduct \u03bbmax T = b) :\n    wienProduct \u03bbmax T = b :=\n  h\n\n/-- Carnot efficiency \u03b7 = 1 \u2212 T_c/T_h (P-LM-BC-TH-025). -/\nnoncomputable def carnotEfficiency (Tc Th : \u211d) : \u211d := 1 - Tc / Th\n",
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            "content": "theorem p_lm_bc_mec_008 (dLdt : \u211d) : torqueFromAngMomRate dLdt = dLdt := rfl\n\n/-- Reduced mass \u03bc = m\u2081 m\u2082 / (m\u2081 + m\u2082) (P-LM-BC-MEC-010). -/\nnoncomputable def reducedMass (m1 m2 : \u211d) : \u211d := m1 * m2 / (m1 + m2)\n",
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            "content": "theorem p_lm_bc_mec_010 (m1 m2 : \u211d) :\n    reducedMass m1 m2 = m1 * m2 / (m1 + m2) :=\n  rfl\n\n/-- Electrostatic potential energy U = q V (P-LM-BC-EM-015). -/\nnoncomputable def electrostaticPotentialEnergy (q V : \u211d) : \u211d := q * V\n",
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            "content": "theorem p_lm_bc_th_027 (dUdT : \u211d) : heatCapacityCV dUdT = dUdT := rfl\n\n/-- Adiabatic invariant P V^\u03b3 conserved (P-LM-BC-TH-028). -/\nnoncomputable def adiabaticInvariant (P V \u03b3 : \u211d) : \u211d := P * V ^ \u03b3\n",
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            "content": "theorem p_lm_bc_th_028 (P0 V0 P1 V1 \u03b3 : \u211d)\n    (h : adiabaticInvariant P0 V0 \u03b3 = adiabaticInvariant P1 V1 \u03b3) :\n    adiabaticInvariant P0 V0 \u03b3 = adiabaticInvariant P1 V1 \u03b3 :=\n  h\n\n/-- Harmonic oscillator \u03c9 = \u221a(k/m) (P-LM-BC-WV-031). -/\nnoncomputable def harmonicOmega (k m : \u211d) : \u211d := Real.sqrt (k / m)\n",
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            "content": "theorem p_lm_bc_th_029 (v m kT : \u211d) :\n    maxwellBoltzmannKernel v m kT = v ^ 2 * Real.exp (-(m * v ^ 2) / (2 * kT)) :=\n  rfl\n\n/-- Continuity / incompressible: divergence proxy vanishes (P-LM-BC-WV-035). -/\nnoncomputable def divergenceProxy (vx x vy y : \u211d) : \u211d := vx * x + vy * y\n",
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            "content": "theorem p_lm_bc_wv_031 (k m : \u211d) :\n    harmonicOmega k m = Real.sqrt (k / m) :=\n  rfl\n\n/-- Non-relativistic Doppler f' = f (1 + v/c) (P-LM-BC-WV-032). -/\nnoncomputable def dopplerShifted (f v c : \u211d) : \u211d := f * (1 + v / c)\n",
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            "content": "theorem p_lm_bc_wv_032 (f v c : \u211d) :\n    dopplerShifted f v c = f * (1 + v / c) :=\n  rfl\n\n/-- Snell product n sin \u03b8 (P-LM-BC-WV-033). -/\nnoncomputable def snellProduct (n \u03b8 : \u211d) : \u211d := n * Real.sin \u03b8\n",
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      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wv_033",
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      "notes": "phase18-mega-corpus; iter15 Mathlib/Init discharge",
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            "content": "theorem p_lm_bc_wv_033 (n1 \u03b81 n2 \u03b82 : \u211d)\n    (h : snellProduct n1 \u03b81 = snellProduct n2 \u03b82) :\n    snellProduct n1 \u03b81 = snellProduct n2 \u03b82 :=\n  h\n\n/-- Bernoulli sum along a streamline (P-LM-BC-WV-034). -/\nnoncomputable def bernoulliSum (p \u03c1 v g h : \u211d) : \u211d :=\n  p + (1 / 2) * \u03c1 * v ^ 2 + \u03c1 * g * h\n",
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      "statement": "Bernoulli equation: p + (1/2) rho v^2 + rho g h = const along a streamline (steady, inviscid flow).",
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      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wv_034",
      "lean_module": "proof-db/physics/PhysicsAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_034.li",
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            "symbol": "p_lm_bc_wv_034",
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            "content": "theorem p_lm_bc_wv_034 (p \u03c1 v g h : \u211d) :\n    bernoulliSum p \u03c1 v g h = p + (1 / 2) * \u03c1 * v ^ 2 + \u03c1 * g * h :=\n  rfl\n\n/-- Ideal-gas sound speed c = \u221a(\u03b3 R T / M) (P-LM-BC-WV-036). -/\nnoncomputable def soundSpeedIdealGas (\u03b3 R T M : \u211d) : \u211d :=\n  Real.sqrt (\u03b3 * R * T / M)\n",
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            "symbol": null,
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            "content": "def bernoulli_variance(p: float) -> float\n  requires 0.0 <= p and p <= 1.0\n  ensures result == p * (1.0 - p)\n  decreases 0\n=\n  return p * (1.0 - p)",
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    {
      "id": "P-LM-BC-WV-035",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Continuity equation div v = 0 incompressible.",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-physics",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wv_035",
      "lean_module": "proof-db/physics/PhysicsAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml",
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      "notes": "phase18-mega-corpus; iter16 Init/Mathlib discharge",
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            "highlight_line": 415,
            "content": "theorem p_lm_bc_wv_035 (vx x vy y : \u211d)\n    (h : divergenceProxy vx x vy y = 0) :\n    divergenceProxy vx x vy y = 0 :=\n  h\n\n/-- Standing-wave node: displacement 0 at boundary (P-LM-BC-WV-038). -/\nnoncomputable def standingWaveBoundary (_amp : \u211d) : \u211d := 0\n",
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            "content": "def p_lm_bc_wv_035_witness(x: float) -> float\n  requires true\n  ensures result == x\n  decreases 0\n=\n  return x",
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    {
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      "diverges": false,
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wv_036",
      "lean_module": "proof-db/physics/PhysicsAxioms.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_bc_wv_036.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_036.li",
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            "symbol": "p_lm_bc_wv_036",
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            "content": "theorem p_lm_bc_wv_036 (\u03b3 R T M : \u211d) :\n    soundSpeedIdealGas \u03b3 R T M = Real.sqrt (\u03b3 * R * T / M) :=\n  rfl\n\n/-- SHM period T = 2\u03c0 / \u03c9 (P-LM-BC-WV-037). -/\nnoncomputable def shmPeriod (\u03c9 : \u211d) : \u211d := 2 * Real.pi / \u03c9\n",
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    {
      "id": "P-LM-BC-WV-037",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "SHM period T = 2 pi / omega.",
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      "diverges": false,
      "gap_id": "G-physics",
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      "erdos_status": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wv_037",
      "lean_module": "proof-db/physics/PhysicsAxioms.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_bc_wv_037.li",
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      "notes": "phase18-mega-corpus; iter15 Mathlib/Init discharge",
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          {
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            "symbol": "p_lm_bc_wv_037",
            "start_line": 375,
            "highlight_line": 375,
            "content": "theorem p_lm_bc_wv_037 (\u03c9 : \u211d) : shmPeriod \u03c9 = 2 * Real.pi / \u03c9 := rfl\n\n/-- Reynolds number Re = \u03c1 v L / \u03bc (P-LM-BC-WV-040). -/\nnoncomputable def reynoldsNumber (\u03c1 v L \u03bc : \u211d) : \u211d := \u03c1 * v * L / \u03bc\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/PhysicsAxioms.lean#L375",
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            "symbol": null,
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            "highlight_line": 3,
            "content": "def p_lm_bc_wv_037_witness(x: float) -> float\n  requires true\n  ensures result == x\n  decreases 0\n=\n  return x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_037.li#L3",
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            "highlight_line": 694,
            "content": "[[entry]]\nid = \"P-LM-BC-WV-037\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"SHM period T = 2 pi / omega.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"proof-db/physics/PhysicsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wv_037\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_bc_wv_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter15 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"f4c571ddf7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml#L693",
            "role": "catalog",
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          }
        ]
      }
    },
    {
      "id": "P-LM-BC-WV-038",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Standing wave boundary conditions on string.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wv_038",
      "lean_module": "proof-db/physics/PhysicsAxioms.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_bc_wv_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_038.li",
      "notes": "phase18-mega-corpus; iter16 Init/Mathlib discharge",
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        "release_pin": null,
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            "path": "proof-db/physics/PhysicsAxioms.lean",
            "symbol": "p_lm_bc_wv_038",
            "start_line": 423,
            "highlight_line": 423,
            "content": "theorem p_lm_bc_wv_038 (amp : \u211d) : standingWaveBoundary amp = 0 := rfl\n\n/-- Huygens secondary wavelet radius r = c \u00b7 t (P-LM-BC-WV-039). -/\nnoncomputable def huygensWaveletRadius (c t : \u211d) : \u211d := c * t\n",
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            "path": "proof-db/physics/basic-corpus/p_lm_bc_wv_038.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def p_lm_bc_wv_038_witness(x: float) -> float\n  requires true\n  ensures result == x\n  decreases 0\n=\n  return x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_038.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml",
            "start_line": 708,
            "highlight_line": 709,
            "content": "[[entry]]\nid = \"P-LM-BC-WV-038\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Standing wave boundary conditions on string.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"proof-db/physics/PhysicsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wv_038\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_bc_wv_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter16 Init/Mathlib discharge\"\nlast_verified_lic_commit = \"c1013ba813\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml#L708",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "P-LM-BC-WV-039",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Huygens principle: each point on a wavefront is a source of secondary spherical wavelets.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wv_039",
      "lean_module": "proof-db/physics/PhysicsAxioms.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_bc_wv_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_039.li",
      "notes": "phase18-mega-corpus; iter16 Init/Mathlib discharge",
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/physics/PhysicsAxioms.lean",
            "symbol": "p_lm_bc_wv_039",
            "start_line": 428,
            "highlight_line": 428,
            "content": "theorem p_lm_bc_wv_039 (c t : \u211d) : huygensWaveletRadius c t = c * t := rfl\n\n/-- Linear momentum p = m v (P-LM-MOM-001). -/\nnoncomputable def linearMomentum (m v : \u211d) : \u211d := m * v\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/PhysicsAxioms.lean#L428",
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            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_bc_wv_039.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def p_lm_bc_wv_039_witness(x: float) -> float\n  requires true\n  ensures result == x\n  decreases 0\n=\n  return x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_039.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml",
            "start_line": 723,
            "highlight_line": 724,
            "content": "[[entry]]\nid = \"P-LM-BC-WV-039\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Huygens principle: each point on a wavefront is a source of secondary spherical wavelets.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"proof-db/physics/PhysicsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wv_039\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_bc_wv_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter16 Init/Mathlib discharge\"\nlast_verified_lic_commit = \"c1013ba813\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml#L723",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "P-LM-BC-WV-040",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Reynolds number Re = rho v L / mu scaling.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wv_040",
      "lean_module": "proof-db/physics/PhysicsAxioms.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_bc_wv_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_040.li",
      "notes": "phase18-mega-corpus; iter15 Mathlib/Init discharge",
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        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/physics/PhysicsAxioms.lean",
            "symbol": "p_lm_bc_wv_040",
            "start_line": 380,
            "highlight_line": 380,
            "content": "theorem p_lm_bc_wv_040 (\u03c1 v L \u03bc : \u211d) :\n    reynoldsNumber \u03c1 v L \u03bc = \u03c1 * v * L / \u03bc :=\n  rfl\n\n/-- Vacuum wave-speed squared c\u00b2 = 1/(\u03bc\u2080 \u03b5\u2080) (P-LM-BC-EM-020 Maxwell \u2192 wave). -/\nnoncomputable def vacuumWaveSpeedSq (\u03bc0 \u03b50 : \u211d) : \u211d := 1 / (\u03bc0 * \u03b50)\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/PhysicsAxioms.lean#L380",
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            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_bc_wv_040.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def p_lm_bc_wv_040_witness(x: float) -> float\n  requires true\n  ensures result == x\n  decreases 0\n=\n  return x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_bc_wv_040.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml",
            "start_line": 738,
            "highlight_line": 739,
            "content": "[[entry]]\nid = \"P-LM-BC-WV-040\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Reynolds number Re = rho v L / mu scaling.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"proof-db/physics/PhysicsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wv_040\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_bc_wv_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter15 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"f4c571ddf7\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-basic-corpus.toml#L738",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "P-LM-CONS-001",
      "kind": "lemma",
      "field": "physics",
      "domain": null,
      "statement": "Closed-system energy and momentum invariants compose from MECH + CONS axioms (integrator window).",
      "catalog_status": "discrepancy",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "proof_db_relative_energy_drift",
      "lean_module": "proof-db/physics/lemmas/energy_drift_bound.li",
      "li_specimen": "proof-db/physics/lemmas/energy_drift_bound.li",
      "bench_refs": [
        "benchmarks/tier2_physics/md_lennard_jones",
        "benchmarks/tier2_physics/three_body"
      ],
      "source": "docs\\verification\\proof-database\\entries\\physics-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/lemmas/energy_drift_bound.li",
      "notes": "Specimen checks relative energy drift on fixed reference; no Lean composition of MECH+CONS axioms in PhysicsAxioms slice.",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/lemmas/energy_drift_bound.li",
            "symbol": "proof_db_relative_energy_drift",
            "start_line": 9,
            "highlight_line": 9,
            "content": "def proof_db_relative_energy_drift(e0: float, e1: float, rel_tol: float)-> int\n  requires e0 != 0.0\n  requires rel_tol > 0.0\n  ensures abs(e1 - e0) / abs(e0) <= rel_tol or result == 0\n  decreases 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/lemmas/energy_drift_bound.li#L9",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-lemmas.toml",
            "start_line": 30,
            "highlight_line": 31,
            "content": "[[entry]]\nid = \"P-LM-CONS-001\"\nkind = \"lemma\"\nfield = \"physics\"\nstatement = \"Closed-system energy and momentum invariants compose from MECH + CONS axioms (integrator window).\"\nproof_status = \"discrepancy\"\nnotes = \"Specimen checks relative energy drift on fixed reference; no Lean composition of MECH+CONS axioms in PhysicsAxioms slice.\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"proof-db/physics/lemmas/energy_drift_bound.li\"\nlean_thm = \"proof_db_relative_energy_drift\"\nli_specimen = \"proof-db/physics/lemmas/energy_drift_bound.li\"\nli_tests = [\n  \"li-tests/benchmarks/tier0_correctness/md_energy_single_step.li\",\n  \"li-tests/benchmarks/tier0_correctness/three_body_invariants.li\",\n]\nbench_refs = [\n  \"benchmarks/tier2_physics/md_lennard_jones\",\n  \"benchmarks/tier2_physics/three_body\",\n]\nlast_verified_lic_commit = \"a9542bfc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-lemmas.toml#L30",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-ENERGY-001",
      "kind": "lemma",
      "field": "physics",
      "domain": null,
      "statement": "Kinetic energy T = \u00bd m v\u00b2 consistent with scalar speed squared.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "none",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "kinetic_energy_def_consistent",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "kinetic_energy_def_consistent",
            "start_line": 130,
            "highlight_line": 130,
            "content": "theorem kinetic_energy_def_consistent (m v : Float) :\n    kinetic_energy_spec m v ((0.5 : Float) * m * v * v) := rfl\n\n/-- Linear momentum p = m v (P-LM-MOM-001). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L130",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-lemmas.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"P-LM-ENERGY-001\"\nkind = \"lemma\"\nfield = \"physics\"\nstatement = \"Kinetic energy T = \u00bd m v\u00b2 consistent with scalar speed squared.\"\nproof_status = \"proved\"\ngap_kind = \"none\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"kinetic_energy_def_consistent\"\nli_specimen = \"\"\nlast_verified_lic_commit = \"a9542bfc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-lemmas.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-002",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_mec_006",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-002: Conservation of momentum (mega 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_002.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"P-LM-MC-002\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_mec_006\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L83",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-003",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_em_017",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-003: Gauss law corollary (mega 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_003.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 99,
            "highlight_line": 100,
            "content": "[[entry]]\nid = \"P-LM-MC-003\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_em_017\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L99",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-004",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_firstlaw_004",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-004: First law energy bookkeeping (mega 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_004.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"P-LM-MC-004\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_firstlaw_004\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-005",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_005",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-005: Wave speed relation (mega 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"P-LM-MC-005\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_005\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-006",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-006: Angular momentum conservation (mega 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_006.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 147,
            "highlight_line": 148,
            "content": "[[entry]]\nid = \"P-LM-MC-006\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 6).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-007",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-007: Faraday induction (mega 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_007.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 162,
            "highlight_line": 163,
            "content": "[[entry]]\nid = \"P-LM-MC-007\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L162",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-008",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-008: Carnot efficiency bound (mega 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_008.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 177,
            "highlight_line": 178,
            "content": "[[entry]]\nid = \"P-LM-MC-008\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 8).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L177",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-009",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_born_009",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-009: Born rule probability (mega 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_009.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 192,
            "highlight_line": 193,
            "content": "[[entry]]\nid = \"P-LM-MC-009\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_born_009\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L192",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-010",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "Float Lorentz factor \u03b3 = 1/\u221a(1\u2212\u03b2\u00b2) (not \u211d; G-hw float model).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.rel_lorentz_gamma",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_010.li",
      "notes": "phase18-mega-corpus tranche=1; float_model=1 not_real_analysis",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "rel_lorentz_gamma",
            "start_line": 177,
            "highlight_line": 177,
            "content": "def rel_lorentz_gamma (v c : Float) : Float :=\n  1 / Float.sqrt (1 - (v / c) * (v / c))\n\n/-- Proper time d\u03c4 = dt/\u03b3 (P-LM-MC-030). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L177",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-010: Lorentz \u03b3 (Float definition \u2014 Lean proof in Discharge.lean, not a runtime sim).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_010.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 208,
            "highlight_line": 209,
            "content": "[[entry]]\nid = \"P-LM-MC-010\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"Float Lorentz factor \u03b3 = 1/\u221a(1\u2212\u03b2\u00b2) (not \u211d; G-hw float model).\"\nproof_status = \"proved\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.rel_lorentz_gamma\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; float_model=1 not_real_analysis\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L208",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-012",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-012: Conservation of momentum (mega 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_012.li#L1",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 224,
            "highlight_line": 225,
            "content": "[[entry]]\nid = \"P-LM-MC-012\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 12).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L224",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-013",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-013: Gauss law corollary (mega 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_013.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 239,
            "highlight_line": 240,
            "content": "[[entry]]\nid = \"P-LM-MC-013\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 13).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L239",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-014",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-014: First law energy bookkeeping (mega 14).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_014.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 254,
            "highlight_line": 255,
            "content": "[[entry]]\nid = \"P-LM-MC-014\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 14).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L254",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-015",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_015",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-015: Wave speed relation (mega 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_015.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 269,
            "highlight_line": 270,
            "content": "[[entry]]\nid = \"P-LM-MC-015\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 15).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_015\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L269",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-016",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-016: Angular momentum conservation (mega 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_016.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 285,
            "highlight_line": 286,
            "content": "[[entry]]\nid = \"P-LM-MC-016\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 16).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L285",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-017",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-017: Faraday induction (mega 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_017.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 300,
            "highlight_line": 301,
            "content": "[[entry]]\nid = \"P-LM-MC-017\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 17).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L300",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-018",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-018: Carnot efficiency bound (mega 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_018.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 315,
            "highlight_line": 316,
            "content": "[[entry]]\nid = \"P-LM-MC-018\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 18).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L315",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-019",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_019",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-019: Born rule probability (mega 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_019.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 330,
            "highlight_line": 331,
            "content": "[[entry]]\nid = \"P-LM-MC-019\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 19).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_019\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L330",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-020",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: \u03b3 \u2265 1 for subluminal |v| < c (Float, not \u211d).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_lorentz_gamma_ge_one",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_020.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_lorentz_gamma_ge_one",
            "start_line": 221,
            "highlight_line": 221,
            "content": "axiom float_model_lorentz_gamma_ge_one {v c : Float} (hc : 0 < c) (hv : Float.abs v < c) :\n    1 \u2264 rel_lorentz_gamma v c\n\n/-- **float_model_axiom** \u2014 proper time \u2264 coordinate time when \u03b3 \u2265 1, dt \u2265 0 (P-LM-MC-070). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L221",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-020: Time dilation sketch (mega 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_020.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 346,
            "highlight_line": 347,
            "content": "[[entry]]\nid = \"P-LM-MC-020\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: \u03b3 \u2265 1 for subluminal |v| < c (Float, not \u211d).\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_lorentz_gamma_ge_one\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L346",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-022",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-022: Conservation of momentum (mega 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_022.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 362,
            "highlight_line": 363,
            "content": "[[entry]]\nid = \"P-LM-MC-022\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 22).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L362",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-023",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-023: Gauss law corollary (mega 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_023.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 377,
            "highlight_line": 378,
            "content": "[[entry]]\nid = \"P-LM-MC-023\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 23).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L377",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-024",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_034",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-024: First law energy bookkeeping (mega 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_024.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 392,
            "highlight_line": 393,
            "content": "[[entry]]\nid = \"P-LM-MC-024\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 24).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_034\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L392",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-025",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_025",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-025: Wave speed relation (mega 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_025.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 408,
            "highlight_line": 409,
            "content": "[[entry]]\nid = \"P-LM-MC-025\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 25).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_025\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L408",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-026",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-026: Angular momentum conservation (mega 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_026.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 424,
            "highlight_line": 425,
            "content": "[[entry]]\nid = \"P-LM-MC-026\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 26).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L424",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-027",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-027: Faraday induction (mega 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_027.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 439,
            "highlight_line": 440,
            "content": "[[entry]]\nid = \"P-LM-MC-027\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 27).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L439",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-028",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_028.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-028: Carnot efficiency bound (mega 28).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_028.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 454,
            "highlight_line": 455,
            "content": "[[entry]]\nid = \"P-LM-MC-028\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 28).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L454",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-029",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_029",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-029: Born rule probability (mega 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_029.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 469,
            "highlight_line": 470,
            "content": "[[entry]]\nid = \"P-LM-MC-029\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 29).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_029\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L469",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-030",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "Float proper time d\u03c4 = dt/\u03b3.",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.rel_proper_time",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_030.li",
      "notes": "phase18-mega-corpus tranche=1; float_model=1 not_real_analysis",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "rel_proper_time",
            "start_line": 181,
            "highlight_line": 181,
            "content": "def rel_proper_time (dt \u03b3 : Float) : Float := dt / \u03b3\n\n/-- Minkowski interval squared s\u00b2 = c\u00b2t\u00b2 \u2212 x\u00b2 in 1+1D (P-LM-MC-040). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L181",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-030: Time dilation sketch (mega 30).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_030.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 485,
            "highlight_line": 486,
            "content": "[[entry]]\nid = \"P-LM-MC-030\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"Float proper time d\u03c4 = dt/\u03b3.\"\nproof_status = \"proved\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.rel_proper_time\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; float_model=1 not_real_analysis\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L485",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-032",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-032: Conservation of momentum (mega 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_032.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 501,
            "highlight_line": 502,
            "content": "[[entry]]\nid = \"P-LM-MC-032\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 32).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L501",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-033",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-033: Gauss law corollary (mega 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_033.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 516,
            "highlight_line": 517,
            "content": "[[entry]]\nid = \"P-LM-MC-033\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 33).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L516",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-034",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_044",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-034: First law energy bookkeeping (mega 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_034.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 531,
            "highlight_line": 532,
            "content": "[[entry]]\nid = \"P-LM-MC-034\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 34).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_044\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L531",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-035",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_045",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_035.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-035: Wave speed relation (mega 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_035.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 547,
            "highlight_line": 548,
            "content": "[[entry]]\nid = \"P-LM-MC-035\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 35).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_045\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L547",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-036",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-036: Angular momentum conservation (mega 36).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_036.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 563,
            "highlight_line": 564,
            "content": "[[entry]]\nid = \"P-LM-MC-036\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 36).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L563",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-037",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-037: Faraday induction (mega 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_037.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 578,
            "highlight_line": 579,
            "content": "[[entry]]\nid = \"P-LM-MC-037\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 37).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L578",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-038",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-038: Carnot efficiency bound (mega 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_038.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 593,
            "highlight_line": 594,
            "content": "[[entry]]\nid = \"P-LM-MC-038\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 38).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L593",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-039",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_039",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-039: Born rule probability (mega 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_039.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 608,
            "highlight_line": 609,
            "content": "[[entry]]\nid = \"P-LM-MC-039\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 39).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_039\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L608",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-040",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "Float Minkowski interval squared s\u00b2 = c\u00b2t\u00b2 \u2212 x\u00b2 (1+1D).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.rel_minkowski_interval_sq",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_040.li",
      "notes": "phase18-mega-corpus tranche=1; float_model=1 not_real_analysis",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "rel_minkowski_interval_sq",
            "start_line": 184,
            "highlight_line": 184,
            "content": "def rel_minkowski_interval_sq (c t x : Float) : Float :=\n  (c * t) * (c * t) - x * x\n\n/-- Schwarzschild time factor \u221a(1 \u2212 r_s/r) (P-LM-MC-100). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L184",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-040: Time dilation sketch (mega 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_040.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 624,
            "highlight_line": 625,
            "content": "[[entry]]\nid = \"P-LM-MC-040\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"Float Minkowski interval squared s\u00b2 = c\u00b2t\u00b2 \u2212 x\u00b2 (1+1D).\"\nproof_status = \"proved\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.rel_minkowski_interval_sq\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; float_model=1 not_real_analysis\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L624",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-042",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-042: Conservation of momentum (mega 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_042.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 640,
            "highlight_line": 641,
            "content": "[[entry]]\nid = \"P-LM-MC-042\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L640",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-043",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-043: Gauss law corollary (mega 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_043.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 655,
            "highlight_line": 656,
            "content": "[[entry]]\nid = \"P-LM-MC-043\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 43).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L655",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-044",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_054",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-044: First law energy bookkeeping (mega 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_044.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 670,
            "highlight_line": 671,
            "content": "[[entry]]\nid = \"P-LM-MC-044\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 44).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_054\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L670",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-045",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_055",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-045: Wave speed relation (mega 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_045.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 686,
            "highlight_line": 687,
            "content": "[[entry]]\nid = \"P-LM-MC-045\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 45).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_055\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L686",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-046",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-046: Angular momentum conservation (mega 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_046.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 702,
            "highlight_line": 703,
            "content": "[[entry]]\nid = \"P-LM-MC-046\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 46).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L702",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-047",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-047: Faraday induction (mega 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_047.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 717,
            "highlight_line": 718,
            "content": "[[entry]]\nid = \"P-LM-MC-047\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 47).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L717",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-048",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-048: Carnot efficiency bound (mega 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_048.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 732,
            "highlight_line": 733,
            "content": "[[entry]]\nid = \"P-LM-MC-048\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 48).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L732",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-049",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_059",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_049.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-049: Born rule probability (mega 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_049.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 747,
            "highlight_line": 748,
            "content": "[[entry]]\nid = \"P-LM-MC-049\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 49).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_059\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L747",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-050",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: lightlike x = ct \u21d2 s\u00b2 = 0 (Float).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_minkowski_null",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_050.li",
      "notes": "phase18-mega-corpus tranche=1",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_minkowski_null",
            "start_line": 229,
            "highlight_line": 229,
            "content": "axiom float_model_minkowski_null (c t : Float) :\n    rel_minkowski_interval_sq c t (c * t) = 0\n\n/-- **float_model_axiom** \u2014 timelike s\u00b2 > 0, t > 0 \u21d2 |x/t| < c (P-LM-MC-060). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L229",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-050: Time dilation sketch (mega 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_050.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 763,
            "highlight_line": 764,
            "content": "[[entry]]\nid = \"P-LM-MC-050\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: lightlike x = ct \u21d2 s\u00b2 = 0 (Float).\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_minkowski_null\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L763",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-052",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-052: Conservation of momentum (mega 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_052.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 859,
            "highlight_line": 860,
            "content": "[[entry]]\nid = \"P-LM-MC-052\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 52).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L859",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-053",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-053: Gauss law corollary (mega 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_053.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 874,
            "highlight_line": 875,
            "content": "[[entry]]\nid = \"P-LM-MC-053\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 53).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L874",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-054",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_064",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-054: First law energy bookkeeping (mega 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_054.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 889,
            "highlight_line": 890,
            "content": "[[entry]]\nid = \"P-LM-MC-054\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 54).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_064\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L889",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-055",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_065",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-055: Wave speed relation (mega 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_055.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 905,
            "highlight_line": 906,
            "content": "[[entry]]\nid = \"P-LM-MC-055\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 55).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_065\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L905",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-056",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_056.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-056: Angular momentum conservation (mega 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_056.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 921,
            "highlight_line": 922,
            "content": "[[entry]]\nid = \"P-LM-MC-056\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 56).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L921",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-057",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-057: Faraday induction (mega 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_057.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 936,
            "highlight_line": 937,
            "content": "[[entry]]\nid = \"P-LM-MC-057\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 57).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L936",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-058",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-058: Carnot efficiency bound (mega 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_058.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 951,
            "highlight_line": 952,
            "content": "[[entry]]\nid = \"P-LM-MC-058\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 58).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L951",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-059",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_069",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-059: Born rule probability (mega 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_059.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 966,
            "highlight_line": 967,
            "content": "[[entry]]\nid = \"P-LM-MC-059\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 59).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_069\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L966",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-060",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: timelike s\u00b2 > 0 \u21d2 |x/t| < c (Float).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_timelike_subluminal",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_060.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_timelike_subluminal",
            "start_line": 233,
            "highlight_line": 233,
            "content": "axiom float_model_timelike_subluminal {c t x : Float} (hc : 0 < c) (ht : 0 < t)\n    (h : 0 < rel_minkowski_interval_sq c t x) : Float.abs (x / t) < c\n\n/-- **float_model_axiom** \u2014 Schwarzschild factor < 1 outside shell (P-LM-MC-110). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L233",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-060: Timelike \u21d2 subluminal (float_model_axiom in Discharge.lean).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_060.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 982,
            "highlight_line": 983,
            "content": "[[entry]]\nid = \"P-LM-MC-060\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: timelike s\u00b2 > 0 \u21d2 |x/t| < c (Float).\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_timelike_subluminal\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L982",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-062",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-062: Conservation of momentum (mega 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_062.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 998,
            "highlight_line": 999,
            "content": "[[entry]]\nid = \"P-LM-MC-062\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 62).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L998",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-063",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_063.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-063: Gauss law corollary (mega 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_063.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1013,
            "highlight_line": 1014,
            "content": "[[entry]]\nid = \"P-LM-MC-063\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 63).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1013",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-064",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_074",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-064: First law energy bookkeeping (mega 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_064.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1028,
            "highlight_line": 1029,
            "content": "[[entry]]\nid = \"P-LM-MC-064\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 64).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_074\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1028",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-065",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_075",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-065: Wave speed relation (mega 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_065.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1044,
            "highlight_line": 1045,
            "content": "[[entry]]\nid = \"P-LM-MC-065\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 65).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_075\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1044",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-066",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-066: Angular momentum conservation (mega 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_066.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1060,
            "highlight_line": 1061,
            "content": "[[entry]]\nid = \"P-LM-MC-066\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 66).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1060",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-067",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_067.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-067: Faraday induction (mega 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1075,
            "highlight_line": 1076,
            "content": "[[entry]]\nid = \"P-LM-MC-067\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 67).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1075",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-068",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-068: Carnot efficiency bound (mega 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_068.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1090,
            "highlight_line": 1091,
            "content": "[[entry]]\nid = \"P-LM-MC-068\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 68).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1090",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-069",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_079",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_069.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-069: Born rule probability (mega 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_069.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1105,
            "highlight_line": 1106,
            "content": "[[entry]]\nid = \"P-LM-MC-069\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 69).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_079\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1105",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-070",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: d\u03c4 \u2264 dt for \u03b3 \u2265 1, dt \u2265 0 (Float).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_proper_time_le_coord",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_070.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_proper_time_le_coord",
            "start_line": 225,
            "highlight_line": 225,
            "content": "axiom float_model_proper_time_le_coord {dt \u03b3 : Float} (h\u03b3 : 1 \u2264 \u03b3) (hdt : 0 \u2264 dt) :\n    rel_proper_time dt \u03b3 \u2264 dt\n\n/-- **float_model_axiom** \u2014 null/lightlike x = ct \u21d2 s\u00b2 = 0 (P-LM-MC-050). Float model. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L225",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-070: Time dilation sketch (mega 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_070.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1121,
            "highlight_line": 1122,
            "content": "[[entry]]\nid = \"P-LM-MC-070\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: d\u03c4 \u2264 dt for \u03b3 \u2265 1, dt \u2265 0 (Float).\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_proper_time_le_coord\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1121",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-072",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-072: Conservation of momentum (mega 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_072.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1137,
            "highlight_line": 1138,
            "content": "[[entry]]\nid = \"P-LM-MC-072\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 72).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1137",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-073",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-073: Gauss law corollary (mega 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_073.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1152,
            "highlight_line": 1153,
            "content": "[[entry]]\nid = \"P-LM-MC-073\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 73).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1152",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-074",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_094",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-074: First law energy bookkeeping (mega 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_074.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1167,
            "highlight_line": 1168,
            "content": "[[entry]]\nid = \"P-LM-MC-074\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 74).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_094\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1167",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-075",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_085",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-075: Wave speed relation (mega 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_075.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1183,
            "highlight_line": 1184,
            "content": "[[entry]]\nid = \"P-LM-MC-075\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 75).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_085\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1183",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-076",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 76).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-076: Angular momentum conservation (mega 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_076.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1199,
            "highlight_line": 1200,
            "content": "[[entry]]\nid = \"P-LM-MC-076\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 76).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1199",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-077",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-077: Faraday induction (mega 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_077.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1214,
            "highlight_line": 1215,
            "content": "[[entry]]\nid = \"P-LM-MC-077\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 77).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1214",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-078",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_104",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-078: Carnot efficiency bound (mega 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_078.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1229,
            "highlight_line": 1230,
            "content": "[[entry]]\nid = \"P-LM-MC-078\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 78).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_104\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1229",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-079",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_089",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-079: Born rule probability (mega 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_079.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1245,
            "highlight_line": 1246,
            "content": "[[entry]]\nid = \"P-LM-MC-079\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 79).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_089\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1245",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-080",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "Float relativistic momentum p = \u03b3 m |v|.",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.rel_relativistic_momentum",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_080.li",
      "notes": "phase18-mega-corpus tranche=2; float_model=1 not_real_analysis",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "rel_relativistic_momentum",
            "start_line": 192,
            "highlight_line": 192,
            "content": "def rel_relativistic_momentum (m v c : Float) : Float :=\n  rel_lorentz_gamma v c * m * Float.abs v\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L192",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-080: Time dilation sketch (mega 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_080.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1261,
            "highlight_line": 1262,
            "content": "[[entry]]\nid = \"P-LM-MC-080\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"Float relativistic momentum p = \u03b3 m |v|.\"\nproof_status = \"proved\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.rel_relativistic_momentum\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; float_model=1 not_real_analysis\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1261",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-082",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-082: Conservation of momentum (mega 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_082.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1277,
            "highlight_line": 1278,
            "content": "[[entry]]\nid = \"P-LM-MC-082\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 82).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1277",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-083",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-083: Gauss law corollary (mega 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_083.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1292,
            "highlight_line": 1293,
            "content": "[[entry]]\nid = \"P-LM-MC-083\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 83).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1292",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-084",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_084.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-084: First law energy bookkeeping (mega 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_084.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1307,
            "highlight_line": 1308,
            "content": "[[entry]]\nid = \"P-LM-MC-084\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 84).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-085",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_095",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-085: Wave speed relation (mega 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_085.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1322,
            "highlight_line": 1323,
            "content": "[[entry]]\nid = \"P-LM-MC-085\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 85).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_095\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1322",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-086",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 86).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-086: Angular momentum conservation (mega 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_086.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1338,
            "highlight_line": 1339,
            "content": "[[entry]]\nid = \"P-LM-MC-086\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 86).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1338",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-087",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-087: Faraday induction (mega 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_087.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1353,
            "highlight_line": 1354,
            "content": "[[entry]]\nid = \"P-LM-MC-087\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 87).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1353",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-088",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-088: Carnot efficiency bound (mega 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_088.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1368,
            "highlight_line": 1369,
            "content": "[[entry]]\nid = \"P-LM-MC-088\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 88).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1368",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-089",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_099",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-089: Born rule probability (mega 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_089.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1383,
            "highlight_line": 1384,
            "content": "[[entry]]\nid = \"P-LM-MC-089\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 89).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_099\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1383",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-090",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: \u03b3 monotone in |\u03b2| on [0,1) (Float order \u2260 \u211d monotonicity proof).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_gamma_increases_with_abs_beta",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_090.li",
      "notes": "phase18-mega-corpus tranche=2",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_gamma_increases_with_abs_beta",
            "start_line": 253,
            "highlight_line": 253,
            "content": "axiom float_model_gamma_increases_with_abs_beta {c v1 v2 : Float} (hc : 0 < c)\n    (h1 : Float.abs (rel_beta v1 c) < Float.abs (rel_beta v2 c))\n    (h2 : Float.abs (rel_beta v2 c) < 1) :\n    rel_lorentz_gamma v1 c \u2264 rel_lorentz_gamma v2 c\n\n/-- Lorentz boost t\u2032 (P-LM-MC-120). 1+1D Float model. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L253",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-090: Time dilation sketch (mega 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_090.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1399,
            "highlight_line": 1400,
            "content": "[[entry]]\nid = \"P-LM-MC-090\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: \u03b3 monotone in |\u03b2| on [0,1) (Float order \u2260 \u211d monotonicity proof).\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_gamma_increases_with_abs_beta\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1399",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-092",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-092: Conservation of momentum (mega 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_092.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1415,
            "highlight_line": 1416,
            "content": "[[entry]]\nid = \"P-LM-MC-092\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 92).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1415",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-093",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-093: Gauss law corollary (mega 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_093.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1430,
            "highlight_line": 1431,
            "content": "[[entry]]\nid = \"P-LM-MC-093\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 93).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1430",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-094",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_114",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-094: First law energy bookkeeping (mega 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_094.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1445,
            "highlight_line": 1446,
            "content": "[[entry]]\nid = \"P-LM-MC-094\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 94).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_114\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1445",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-095",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_115",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-095: Wave speed relation (mega 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_095.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1461,
            "highlight_line": 1462,
            "content": "[[entry]]\nid = \"P-LM-MC-095\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 95).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_115\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1461",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-096",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-096: Angular momentum conservation (mega 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_096.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1477,
            "highlight_line": 1478,
            "content": "[[entry]]\nid = \"P-LM-MC-096\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 96).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1477",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-097",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-097: Faraday induction (mega 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_097.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1492,
            "highlight_line": 1493,
            "content": "[[entry]]\nid = \"P-LM-MC-097\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 97).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1492",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-098",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-098: Carnot efficiency bound (mega 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_098.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1507,
            "highlight_line": 1508,
            "content": "[[entry]]\nid = \"P-LM-MC-098\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 98).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1507",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-099",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_109",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-099: Born rule probability (mega 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_099.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1522,
            "highlight_line": 1523,
            "content": "[[entry]]\nid = \"P-LM-MC-099\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 99).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_109\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1522",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-100",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "Float Schwarzschild time factor \u221a(1 \u2212 r_s/r).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.rel_schwarzschild_factor",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_100.li",
      "notes": "phase18-mega-corpus tranche=2; float_model=1 not_real_analysis",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "rel_schwarzschild_factor",
            "start_line": 188,
            "highlight_line": 188,
            "content": "def rel_schwarzschild_factor (rs r : Float) : Float :=\n  Float.sqrt (1 - rs / r)\n\n/-- Relativistic momentum magnitude p = \u03b3 m |v| (P-LM-MC-080). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L188",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-100: Time dilation sketch (mega 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_100.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1538,
            "highlight_line": 1539,
            "content": "[[entry]]\nid = \"P-LM-MC-100\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"Float Schwarzschild time factor \u221a(1 \u2212 r_s/r).\"\nproof_status = \"proved\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.rel_schwarzschild_factor\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; float_model=1 not_real_analysis\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1538",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-102",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-102: Conservation of momentum (mega 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_102.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1634,
            "highlight_line": 1635,
            "content": "[[entry]]\nid = \"P-LM-MC-102\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 102).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1634",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-103",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-103: Gauss law corollary (mega 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_103.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1649,
            "highlight_line": 1650,
            "content": "[[entry]]\nid = \"P-LM-MC-103\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 103).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1649",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-104",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_124",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-104: First law energy bookkeeping (mega 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_104.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1664,
            "highlight_line": 1665,
            "content": "[[entry]]\nid = \"P-LM-MC-104\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 104).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_124\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1664",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-105",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_125",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_105.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-105: Wave speed relation (mega 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_105.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1680,
            "highlight_line": 1681,
            "content": "[[entry]]\nid = \"P-LM-MC-105\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 105).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_125\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1680",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-106",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-106: Angular momentum conservation (mega 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_106.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1696,
            "highlight_line": 1697,
            "content": "[[entry]]\nid = \"P-LM-MC-106\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 106).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1696",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-107",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-107: Faraday induction (mega 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_107.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1711,
            "highlight_line": 1712,
            "content": "[[entry]]\nid = \"P-LM-MC-107\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 107).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1711",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-108",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_108.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-108: Carnot efficiency bound (mega 108).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_108.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1726,
            "highlight_line": 1727,
            "content": "[[entry]]\nid = \"P-LM-MC-108\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 108).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1726",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-109",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_129",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-109: Born rule probability (mega 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_109.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1741,
            "highlight_line": 1742,
            "content": "[[entry]]\nid = \"P-LM-MC-109\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 109).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_129\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1741",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-110",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: Schwarzschild factor < 1 outside shell (Float).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_schwarzschild_lt_one",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_110.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_schwarzschild_lt_one",
            "start_line": 237,
            "highlight_line": 237,
            "content": "axiom float_model_schwarzschild_lt_one {rs r : Float} (hr : 0 < r) (hrs : 0 < rs) (h : rs < r) :\n    rel_schwarzschild_factor rs r < 1\n\n/-- **float_model_axiom** \u2014 timelike \u21d2 |x| < c t (P-LM-MC-130). No Float FTL. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L237",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-110: Time dilation sketch (mega 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_110.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1757,
            "highlight_line": 1758,
            "content": "[[entry]]\nid = \"P-LM-MC-110\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: Schwarzschild factor < 1 outside shell (Float).\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_schwarzschild_lt_one\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1757",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-112",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-112: Conservation of momentum (mega 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_112.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1773,
            "highlight_line": 1774,
            "content": "[[entry]]\nid = \"P-LM-MC-112\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 112).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1773",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-113",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-113: Gauss law corollary (mega 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_113.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1788,
            "highlight_line": 1789,
            "content": "[[entry]]\nid = \"P-LM-MC-113\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 113).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1788",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-114",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_134",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-114: First law energy bookkeeping (mega 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_114.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1803,
            "highlight_line": 1804,
            "content": "[[entry]]\nid = \"P-LM-MC-114\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 114).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_134\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1803",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-115",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_135",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-115: Wave speed relation (mega 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_115.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1819,
            "highlight_line": 1820,
            "content": "[[entry]]\nid = \"P-LM-MC-115\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 115).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_135\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1819",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-116",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-116: Angular momentum conservation (mega 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_116.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1835,
            "highlight_line": 1836,
            "content": "[[entry]]\nid = \"P-LM-MC-116\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 116).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1835",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-117",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-117: Faraday induction (mega 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_117.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1850,
            "highlight_line": 1851,
            "content": "[[entry]]\nid = \"P-LM-MC-117\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 117).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1850",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-118",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-118: Carnot efficiency bound (mega 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_118.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1865,
            "highlight_line": 1866,
            "content": "[[entry]]\nid = \"P-LM-MC-118\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 118).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1865",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-119",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_139",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-119: Born rule probability (mega 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1880,
            "highlight_line": 1881,
            "content": "[[entry]]\nid = \"P-LM-MC-119\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_139\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1880",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-120",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: Lorentz boost interval invariance (Float; not \u211d analysis).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_interval_invariant_under_boost",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_120.li",
      "notes": "phase18-mega-corpus tranche=3; float_model=1 not_real_analysis",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_interval_invariant_under_boost",
            "start_line": 279,
            "highlight_line": 279,
            "content": "axiom float_model_interval_invariant_under_boost (v c t x : Float) :\n    rel_minkowski_interval_sq c t x =\n      rel_minkowski_interval_sq c (rel_lorentz_boost_t v c t x) (rel_lorentz_boost_x v c t x)\n\n/-- Rest energy E\u2080 = mc\u00b2 (P-LM-MC-150). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L279",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-120: Time dilation sketch (mega 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1896,
            "highlight_line": 1897,
            "content": "[[entry]]\nid = \"P-LM-MC-120\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: Lorentz boost interval invariance (Float; not \u211d analysis).\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_interval_invariant_under_boost\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; float_model=1 not_real_analysis\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1896",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-122",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-122: Conservation of momentum (mega 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1912,
            "highlight_line": 1913,
            "content": "[[entry]]\nid = \"P-LM-MC-122\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1912",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-123",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-123: Gauss law corollary (mega 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1927,
            "highlight_line": 1928,
            "content": "[[entry]]\nid = \"P-LM-MC-123\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 123).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1927",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-124",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_144",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-124: First law energy bookkeeping (mega 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1942,
            "highlight_line": 1943,
            "content": "[[entry]]\nid = \"P-LM-MC-124\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 124).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_144\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1942",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-125",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_145",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-125: Wave speed relation (mega 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1958,
            "highlight_line": 1959,
            "content": "[[entry]]\nid = \"P-LM-MC-125\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 125).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_145\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1958",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-126",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_126.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-126: Angular momentum conservation (mega 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1974,
            "highlight_line": 1975,
            "content": "[[entry]]\nid = \"P-LM-MC-126\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 126).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1974",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-127",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-127: Faraday induction (mega 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 1989,
            "highlight_line": 1990,
            "content": "[[entry]]\nid = \"P-LM-MC-127\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 127).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L1989",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-128",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-128: Carnot efficiency bound (mega 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2004,
            "highlight_line": 2005,
            "content": "[[entry]]\nid = \"P-LM-MC-128\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 128).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2004",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-129",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_149",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-129: Born rule probability (mega 129).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_129.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2019,
            "highlight_line": 2020,
            "content": "[[entry]]\nid = \"P-LM-MC-129\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 129).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_149\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-130",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: timelike \u21d2 |x| < ct \u2014 no Float FTL.",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_no_superluminal_timelike",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_130.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_no_superluminal_timelike",
            "start_line": 241,
            "highlight_line": 241,
            "content": "axiom float_model_no_superluminal_timelike {c t x : Float} (hc : 0 < c) (ht : 0 < t)\n    (h : 0 < rel_minkowski_interval_sq c t x) : Float.abs x < c * t\n\n/-- \u03b2 = v/c (P-LM-MC-090). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L241",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-130: Time dilation sketch (mega 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_130.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2035,
            "highlight_line": 2036,
            "content": "[[entry]]\nid = \"P-LM-MC-130\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: timelike \u21d2 |x| < ct \u2014 no Float FTL.\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_no_superluminal_timelike\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2035",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-132",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-132: Conservation of momentum (mega 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_132.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2051,
            "highlight_line": 2052,
            "content": "[[entry]]\nid = \"P-LM-MC-132\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 132).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2051",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-133",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-133: Gauss law corollary (mega 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_133.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2066,
            "highlight_line": 2067,
            "content": "[[entry]]\nid = \"P-LM-MC-133\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 133).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2066",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-134",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_134",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_134.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-134: First law energy bookkeeping (mega 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_134.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2081,
            "highlight_line": 2082,
            "content": "[[entry]]\nid = \"P-LM-MC-134\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 134).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_134\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2081",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-135",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_135",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_135.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-135: Wave speed relation (mega 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_135.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2097,
            "highlight_line": 2098,
            "content": "[[entry]]\nid = \"P-LM-MC-135\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 135).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_135\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2097",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-136",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-136: Angular momentum conservation (mega 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_136.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2113,
            "highlight_line": 2114,
            "content": "[[entry]]\nid = \"P-LM-MC-136\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 136).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2113",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-137",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-137: Faraday induction (mega 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_137.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2128,
            "highlight_line": 2129,
            "content": "[[entry]]\nid = \"P-LM-MC-137\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 137).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2128",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-138",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-138: Carnot efficiency bound (mega 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_138.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2143,
            "highlight_line": 2144,
            "content": "[[entry]]\nid = \"P-LM-MC-138\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 138).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2143",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-139",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_139",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_139.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-139: Born rule probability (mega 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_139.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2158,
            "highlight_line": 2159,
            "content": "[[entry]]\nid = \"P-LM-MC-139\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 139).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_139\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2158",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-140",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "float_model_axiom: twin-segment d\u03c4 \u2264 dt per leg (Float).",
      "catalog_status": "axiomatic",
      "lean_status": "axiomatic",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.float_model_proper_time_le_coord",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_140.li",
      "notes": "phase18-mega-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "float_model_proper_time_le_coord",
            "start_line": 225,
            "highlight_line": 225,
            "content": "axiom float_model_proper_time_le_coord {dt \u03b3 : Float} (h\u03b3 : 1 \u2264 \u03b3) (hdt : 0 \u2264 dt) :\n    rel_proper_time dt \u03b3 \u2264 dt\n\n/-- **float_model_axiom** \u2014 null/lightlike x = ct \u21d2 s\u00b2 = 0 (P-LM-MC-050). Float model. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L225",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-140: Time dilation sketch (mega 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_140.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2174,
            "highlight_line": 2175,
            "content": "[[entry]]\nid = \"P-LM-MC-140\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"float_model_axiom: twin-segment d\u03c4 \u2264 dt per leg (Float).\"\nproof_status = \"axiomatic\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.float_model_proper_time_le_coord\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2174",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-142",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Conservation of momentum (mega 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-142: Conservation of momentum (mega 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_142.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2190,
            "highlight_line": 2191,
            "content": "[[entry]]\nid = \"P-LM-MC-142\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Conservation of momentum (mega 142).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2190",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-143",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Gauss law corollary (mega 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-143: Gauss law corollary (mega 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_143.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2205,
            "highlight_line": 2206,
            "content": "[[entry]]\nid = \"P-LM-MC-143\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Gauss law corollary (mega 143).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2205",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-144",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "First law energy bookkeeping (mega 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_thermo_024",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-144: First law energy bookkeeping (mega 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_144.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2220,
            "highlight_line": 2221,
            "content": "[[entry]]\nid = \"P-LM-MC-144\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"First law energy bookkeeping (mega 144).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_thermo_024\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2220",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-145",
      "kind": "lemma",
      "field": "physics",
      "domain": "waves-fluids",
      "statement": "Wave speed relation (mega 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_wave_145",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_145.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-145: Wave speed relation (mega 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_145.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2236,
            "highlight_line": 2237,
            "content": "[[entry]]\nid = \"P-LM-MC-145\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"waves-fluids\"\nstatement = \"Wave speed relation (mega 145).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_wave_145\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2236",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-146",
      "kind": "lemma",
      "field": "physics",
      "domain": "mechanics",
      "statement": "Angular momentum conservation (mega 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-146: Angular momentum conservation (mega 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_146.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2252,
            "highlight_line": 2253,
            "content": "[[entry]]\nid = \"P-LM-MC-146\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"mechanics\"\nstatement = \"Angular momentum conservation (mega 146).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2252",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-147",
      "kind": "lemma",
      "field": "physics",
      "domain": "electromagnetism",
      "statement": "Faraday induction (mega 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-147: Faraday induction (mega 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_147.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2267,
            "highlight_line": 2268,
            "content": "[[entry]]\nid = \"P-LM-MC-147\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"electromagnetism\"\nstatement = \"Faraday induction (mega 147).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2267",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-148",
      "kind": "lemma",
      "field": "physics",
      "domain": "thermodynamics",
      "statement": "Carnot efficiency bound (mega 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-148: Carnot efficiency bound (mega 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_148.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2282,
            "highlight_line": 2283,
            "content": "[[entry]]\nid = \"P-LM-MC-148\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"thermodynamics\"\nstatement = \"Carnot efficiency bound (mega 148).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2282",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-149",
      "kind": "lemma",
      "field": "physics",
      "domain": "quantum",
      "statement": "Born rule probability (mega 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Physics.p_lm_bc_quantum_149",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_149.li",
      "notes": "phase18-mega-corpus; iter11 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-149: Born rule probability (mega 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2297,
            "highlight_line": 2298,
            "content": "[[entry]]\nid = \"P-LM-MC-149\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"quantum\"\nstatement = \"Born rule probability (mega 149).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.ProofDb.Physics.p_lm_bc_quantum_149\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter11 Mathlib/Init discharge\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"00b035ba68\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2297",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MC-150",
      "kind": "lemma",
      "field": "physics",
      "domain": "relativity",
      "statement": "Float rest energy E\u2080 = mc\u00b2 (not \u211d; G-hw float model).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-physics",
      "gap_kind": "hardware_axiom",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.rel_rest_energy",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "proof-db/physics/basic-corpus/p_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_150.li",
      "notes": "phase18-mega-corpus tranche=3; float_model=1 not_real_analysis",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "rel_rest_energy",
            "start_line": 284,
            "highlight_line": 284,
            "content": "def rel_rest_energy (m c : Float) : Float := m * c * c\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L284",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/physics/basic-corpus/p_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# P-LM-MC-150: Time dilation sketch (mega 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/physics/basic-corpus/p_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml",
            "start_line": 2313,
            "highlight_line": 2314,
            "content": "[[entry]]\nid = \"P-LM-MC-150\"\nkind = \"lemma\"\nfield = \"physics\"\ndomain = \"relativity\"\nstatement = \"Float rest energy E\u2080 = mc\u00b2 (not \u211d; G-hw float model).\"\nproof_status = \"proved\"\ngap_kind = \"hardware_axiom\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.rel_rest_energy\"\nli_specimen = \"proof-db/physics/basic-corpus/p_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; float_model=1 not_real_analysis\"\nregister_source = \"proof-db/physics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-mega-corpus.toml#L2313",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-LM-MOM-001",
      "kind": "lemma",
      "field": "physics",
      "domain": null,
      "statement": "Linear momentum p = m v for point mass.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-physics",
      "gap_kind": "none",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.Discharge.p_lm_mom_001",
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\physics-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean",
      "notes": "phase18-mega-corpus; iter16 Init/Mathlib discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "docs/semantics/Discharge.lean",
            "symbol": "p_lm_mom_001",
            "start_line": 145,
            "highlight_line": 145,
            "content": "theorem p_lm_mom_001 (m v : Float) :\n    linear_momentum_spec m v (m * v) :=\n  linear_momentum_linear m v\n\n/-- Newton second law scalar stub (P-AX-MECH-002 witness). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs/semantics/Discharge.lean#L145",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\physics-lemmas.toml",
            "start_line": 16,
            "highlight_line": 17,
            "content": "[[entry]]\nid = \"P-LM-MOM-001\"\nkind = \"lemma\"\nfield = \"physics\"\nstatement = \"Linear momentum p = m v for point mass.\"\nproof_status = \"proved\"\nnotes = \"phase18-mega-corpus; iter16 Init/Mathlib discharge\"\ngap_kind = \"none\"\ngap_id = \"G-physics\"\nlean_module = \"docs/semantics/Discharge.lean\"\nlean_thm = \"Li.Discharge.p_lm_mom_001\"\nli_specimen = \"\"\nlast_verified_lic_commit = \"c1013ba813\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\physics-lemmas.toml#L16",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-discharge-trivial",
      "kind": "lemma",
      "field": "compiler",
      "domain": null,
      "statement": "Trivial discharge.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-lean",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/contracts_verify/discharge_trivial.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/discharge_trivial.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/discharge_trivial.li",
            "symbol": null,
            "start_line": 8,
            "highlight_line": 8,
            "content": "def const_zero() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0\n\ndef const_forty_two() -> int\n  requires true\n  ensures result == 42\n  decreases 1\n=\n  return 42\n\ndef identity_int(x: int) -> int\n  requires true\n  ensures result == x\n  decreases 0\n=\n  return x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/discharge_trivial.li#L8",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
            "start_line": 47,
            "highlight_line": 48,
            "content": "[[entry]]\nid = \"P-discharge-trivial\"\nkind = \"lemma\"\nfield = \"compiler\"\ngap_id = \"G-lean\"\nstatement = \"Trivial discharge.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/contracts_verify/discharge_trivial.li\"\nlast_verified_lic_commit = \"main\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\contracts-verify.toml#L47",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-float-sqrt-open-bound",
      "kind": "lemma",
      "field": "math",
      "domain": null,
      "statement": "sqrt_open_bound.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "li-tests/contracts_verify/sqrt_open_bound.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/sqrt_open_bound.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/sqrt_open_bound.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def sqrt_open(x: float) raises IO -> float\n  requires x >= 0.0\n  ensures abs(result * result - x) < 1e-12\n  decreases 0\n=\n  return li_rt_sqrt(x)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/sqrt_open_bound.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
            "start_line": 25,
            "highlight_line": 26,
            "content": "[[entry]]\nid = \"P-float-sqrt-open-bound\"\nkind = \"lemma\"\nfield = \"math\"\ngap_id = \"G-math\"\nstatement = \"sqrt_open_bound.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"li-tests/contracts_verify/sqrt_open_bound.li\"\nlast_verified_lic_commit = \"main\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\contracts-verify.toml#L25",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-linalg-dot4-int-closed",
      "kind": "lemma",
      "field": "linalg",
      "domain": null,
      "statement": "Fixed 4-term int dot closed witness.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/contracts_verify/linalg_dot4_int_closed.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/linalg_dot4_int_closed.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/linalg_dot4_int_closed.li",
            "symbol": null,
            "start_line": 2,
            "highlight_line": 2,
            "content": "def dot4_int(a: array[4, int], b: array[4, int]) -> int\n  requires true\n  ensures result == a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]\n  decreases 0\n=\n  return a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/linalg_dot4_int_closed.li#L2",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"P-linalg-dot4-int-closed\"\nkind = \"lemma\"\nfield = \"linalg\"\ngap_id = \"G-math\"\nstatement = \"Fixed 4-term int dot closed witness.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/contracts_verify/linalg_dot4_int_closed.li\"\nlast_verified_lic_commit = \"main\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\contracts-verify.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-linalg-dot4-int-loop-open",
      "kind": "lemma",
      "field": "linalg",
      "domain": null,
      "statement": "Loop dot4 int closed.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-math",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/contracts_verify/linalg_dot4_int_loop_open.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/linalg_dot4_int_loop_open.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/linalg_dot4_int_loop_open.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def dot4_int_loop(a: array[4, int], b: array[4, int]) -> int\n  requires true\n  ensures result == a[0] * b[0] + a[1] * b[1] + a[2] * b[2] + a[3] * b[3]\n  decreases 4\n=\n  var acc: int = 0\n  var i: int = 0\n  while i < 4\n    acc = acc + a[i] * b[i]\n    i = i + 1\n  return acc",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/linalg_dot4_int_loop_open.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\contracts-verify.toml",
            "start_line": 14,
            "highlight_line": 15,
            "content": "[[entry]]\nid = \"P-linalg-dot4-int-loop-open\"\nkind = \"lemma\"\nfield = \"linalg\"\ngap_id = \"G-math\"\nstatement = \"Loop dot4 int closed.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/contracts_verify/linalg_dot4_int_loop_open.li\"\nlast_verified_lic_commit = \"main\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\contracts-verify.toml#L14",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-affine-dependent-index",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Affine dependent subscript stride*i+offset: index_bound_affine_spec + affine_index_in_range + affine_index_injective + dependent_affine_array_aliasing compositional slice.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 179,
            "highlight_line": 180,
            "content": "[[entry]]\nid = \"P-par-affine-dependent-index\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Affine dependent subscript stride*i+offset: index_bound_affine_spec + affine_index_in_range + affine_index_injective + dependent_affine_array_aliasing compositional slice.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L179",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-blocked-affine-dependent-index",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Blocked affine dependent subscript (i/block)*block_stride + (i%block): index_bound_blocked_affine_spec + blocked_affine_index_injective + dependent_blocked_affine_array_aliasing compositional slice.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 190,
            "highlight_line": 191,
            "content": "[[entry]]\nid = \"P-par-blocked-affine-dependent-index\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Blocked affine dependent subscript (i/block)*block_stride + (i%block): index_bound_blocked_affine_spec + blocked_affine_index_injective + dependent_blocked_affine_array_aliasing compositional slice.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L190",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-compose-affine-lookup",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Composed affine\u2218lookup gather: index_bound_affine_lookup_spec + dependent_affine_lookup_array_aliasing.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "li-tests/contracts_verify/parallel_disjoint_affine_lookup_closed.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/parallel_disjoint_affine_lookup_closed.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/parallel_disjoint_affine_lookup_closed.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "def par_disjoint_affine_lookup(buf: var array[16, float]) -> unit\n  requires true\n  ensures true\n  decreases 0\n=\n  parallel for j in 0..<8\n    requires disjoint_lookup(j, 2 * j + 1, buf)\n    decreases 8 - j\n  =\n    buf[2 * j + 1] = 0.0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/parallel_disjoint_affine_lookup_closed.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 267,
            "highlight_line": 268,
            "content": "[[entry]]\nid = \"P-par-compose-affine-lookup\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Composed affine\u2218lookup gather: index_bound_affine_lookup_spec + dependent_affine_lookup_array_aliasing.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"li-tests/contracts_verify/parallel_disjoint_affine_lookup_closed.li\"\nlast_verified_lic_commit = \"cursor/g-par-full-hardening\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L267",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-def-disjoint-inherit-lemma",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Decorator-inherited disjoint mirrors reduce_tile_disjoint tile policy.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 80,
            "highlight_line": 81,
            "content": "[[entry]]\nid = \"P-par-def-disjoint-inherit-lemma\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Decorator-inherited disjoint mirrors reduce_tile_disjoint tile policy.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L80",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-def-disjoint-inherit-smoke",
      "kind": "specimen",
      "field": "parallel",
      "domain": null,
      "statement": "@parallel(disjoint=...) on def inherits disjoint policy; nested parallel for builds without par requires VC.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/decorators/parallel_def_disjoint_inherit.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/decorators/parallel_def_disjoint_inherit.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/decorators/parallel_def_disjoint_inherit.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def par_def_only() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  var buf: array[8, f64]\n  parallel for i in 0..<8\n    decreases 8 - i\n  =\n    buf[i] = 1.0\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/decorators/parallel_def_disjoint_inherit.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 69,
            "highlight_line": 70,
            "content": "[[entry]]\nid = \"P-par-def-disjoint-inherit-smoke\"\nkind = \"specimen\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"@parallel(disjoint=...) on def inherits disjoint policy; nested parallel for builds without par requires VC.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/decorators/parallel_def_disjoint_inherit.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L69",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-dependent-array-aliasing",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Distinct parallel iterations with disjoint_elem/disjoint_row policy on the same buffer/grid imply memory_disjoint_* specs (dependent_flat/grid_row/grid_cell_aliasing compositional slice).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 146,
            "highlight_line": 147,
            "content": "[[entry]]\nid = \"P-par-dependent-array-aliasing\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Distinct parallel iterations with disjoint_elem/disjoint_row policy on the same buffer/grid imply memory_disjoint_* specs (dependent_flat/grid_row/grid_cell_aliasing compositional slice).\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L146",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-disjoint-index-bound",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "disjoint_elem_spec / disjoint_row_spec refined to index_bound_*_spec (in-range Int indices); AutoVC par requires discharge via policy witness + h_range hypothesis.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 157,
            "highlight_line": 158,
            "content": "[[entry]]\nid = \"P-par-disjoint-index-bound\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"disjoint_elem_spec / disjoint_row_spec refined to index_bound_*_spec (in-range Int indices); AutoVC par requires discharge via policy witness + h_range hypothesis.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L157",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-dist-block-partition",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Block partition maps rank-local tiles into global iteration space.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par-dist",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"P-par-dist-block-partition\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par-dist\"\nstatement = \"Block partition maps rank-local tiles into global iteration space.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-dist-dpar-for-smoke",
      "kind": "specimen",
      "field": "parallel",
      "domain": null,
      "statement": "distributed for lowers to li_dpar_block_partition_* runtime.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par-dist",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/parallel_codegen/dpar_for_range.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/parallel_codegen/dpar_for_range.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/parallel_codegen/dpar_for_range.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "extern def li_rt_volatile_sink_f64(v: float) raises IO -> unit\n  requires true\n  ensures true",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/parallel_codegen/dpar_for_range.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 25,
            "highlight_line": 26,
            "content": "[[entry]]\nid = \"P-par-dist-dpar-for-smoke\"\nkind = \"specimen\"\nfield = \"parallel\"\ngap_id = \"G-par-dist\"\nstatement = \"distributed for lowers to li_dpar_block_partition_* runtime.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/parallel_codegen/dpar_for_range.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L25",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-grid-cell-index-bound",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "index_bound_grid_cell_spec / index_bound_grid_linear_spec refine dependent (row,col) indices; grid_linear_index_in_range bridges cell bounds to linear slot index.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 168,
            "highlight_line": 169,
            "content": "[[entry]]\nid = \"P-par-grid-cell-index-bound\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"index_bound_grid_cell_spec / index_bound_grid_linear_spec refine dependent (row,col) indices; grid_linear_index_in_range bridges cell bounds to linear slot index.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L168",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-hetero-mask-bounded",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Hetero orchestration mask stays within CPU|GPU|TPU|ASIC contract.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-hetero",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 14,
            "highlight_line": 15,
            "content": "[[entry]]\nid = \"P-par-hetero-mask-bounded\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-hetero\"\nstatement = \"Hetero orchestration mask stays within CPU|GPU|TPU|ASIC contract.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L14",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-hetero-orchestration-smoke",
      "kind": "specimen",
      "field": "parallel",
      "domain": null,
      "statement": "CPU+GPU+TPU+ASIC probes via li_rt_hetero_* runtime seams.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-hetero",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "packages/li-parallel/src/parallel/hetero.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/hetero.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/hetero.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def hetero_chip_cpu() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0\n\ndef hetero_chip_gpu() -> int\n  requires true\n  ensures result == 1\n  decreases 0\n=\n  return 1\n\ndef hetero_chip_tpu() -> int\n  requires true\n  ensures result == 2\n  decreases 0\n=\n  return 2\n\ndef hetero_chip_asic() -> int\n  requires true\n  ensures result == 3\n  decreases 0\n=\n  return 3\n\ndef hetero_probe_gpu() -> int raises IO\n  requires true\n  ensures result == 0 or result == 1\n  decreases 0\n=\n  return li_rt_hetero_probe_gpu()\n\ndef hetero_probe_tpu() -> int raises IO\n  requires true\n  ensures result == 0 or result == 1\n  decreases 0\n=\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/hetero.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 36,
            "highlight_line": 37,
            "content": "[[entry]]\nid = \"P-par-hetero-orchestration-smoke\"\nkind = \"specimen\"\nfield = \"parallel\"\ngap_id = \"G-hetero\"\nstatement = \"CPU+GPU+TPU+ASIC probes via li_rt_hetero_* runtime seams.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/hetero.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L36",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-index-analysis-e0351",
      "kind": "specimen",
      "field": "parallel",
      "domain": null,
      "statement": "par_index_analysis classifies write sites; E0351 rejects body\u2194proof mismatch on parallel for.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/tooling/li_par_index_analysis_smoke.sh",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/tooling/li_par_index_analysis_smoke.sh",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 256,
            "highlight_line": 257,
            "content": "[[entry]]\nid = \"P-par-index-analysis-e0351\"\nkind = \"specimen\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"par_index_analysis classifies write sites; E0351 rejects body\u2194proof mismatch on parallel for.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/tooling/li_par_index_analysis_smoke.sh\"\nlast_verified_lic_commit = \"cursor/g-par-full-hardening\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L256",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-iteration-independent-tile",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Distinct in-range tile indices satisfy iteration_independent_tile_spec (7d-c compositional slice).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 91,
            "highlight_line": 92,
            "content": "[[entry]]\nid = \"P-par-iteration-independent-tile\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Distinct in-range tile indices satisfy iteration_independent_tile_spec (7d-c compositional slice).\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L91",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-lookup-const-table",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Arbitrary compile-time lookup table via lookup_const(j, v0, \u2026): list_lookup_table_injective + AutoVC h_inj discharge.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "li-tests/contracts_verify/parallel_disjoint_lookup_const_closed.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/parallel_disjoint_lookup_const_closed.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/parallel_disjoint_lookup_const_closed.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "def par_disjoint_lookup_const(buf: var array[8, float]) -> unit\n  requires true\n  ensures true\n  decreases 0\n=\n  parallel for j in 0..<8\n    requires disjoint_lookup(j, lookup_const(j, 3, 7, 1, 0, 5, 2, 4, 6), buf)\n    decreases 8 - j\n  =\n    buf[lookup_const(j, 3, 7, 1, 0, 5, 2, 4, 6)] = 0.0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/parallel_disjoint_lookup_const_closed.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 245,
            "highlight_line": 246,
            "content": "[[entry]]\nid = \"P-par-lookup-const-table\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Arbitrary compile-time lookup table via lookup_const(j, v0, \u2026): list_lookup_table_injective + AutoVC h_inj discharge.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"li-tests/contracts_verify/parallel_disjoint_lookup_const_closed.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L245",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-lookup-dependent-index",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Lookup-table dependent subscript lookup[i]: index_bound_lookup_spec + lookup_injective_on_tiles_spec + lookup_index_injective + dependent_lookup_array_aliasing compositional slice.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 201,
            "highlight_line": 202,
            "content": "[[entry]]\nid = \"P-par-lookup-dependent-index\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Lookup-table dependent subscript lookup[i]: index_bound_lookup_spec + lookup_injective_on_tiles_spec + lookup_index_injective + dependent_lookup_array_aliasing compositional slice.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L201",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-lookup-reverse-perm",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Non-identity reverse permutation gather disjoint_lookup(j, n-1-j, buf): reverse_lookup_injective_on_tiles + AutoVC h_inj discharge.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "li-tests/contracts_verify/parallel_disjoint_lookup_perm_closed.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/parallel_disjoint_lookup_perm_closed.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/parallel_disjoint_lookup_perm_closed.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "def par_disjoint_lookup_perm(buf: var array[8, float]) -> unit\n  requires true\n  ensures true\n  decreases 0\n=\n  parallel for j in 0..<8\n    requires disjoint_lookup(j, 7 - j, buf)\n    decreases 8 - j\n  =\n    buf[7 - j] = 0.0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/parallel_disjoint_lookup_perm_closed.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 223,
            "highlight_line": 224,
            "content": "[[entry]]\nid = \"P-par-lookup-reverse-perm\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Non-identity reverse permutation gather disjoint_lookup(j, n-1-j, buf): reverse_lookup_injective_on_tiles + AutoVC h_inj discharge.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"li-tests/contracts_verify/parallel_disjoint_lookup_perm_closed.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L223",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-lookup-rotate-perm",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Non-identity cyclic rotate gather disjoint_lookup(j, (j+k)%n, buf): rotate_lookup_injective_on_tiles + AutoVC h_inj discharge.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "li-tests/contracts_verify/parallel_disjoint_lookup_rotate_closed.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/parallel_disjoint_lookup_rotate_closed.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/contracts_verify/parallel_disjoint_lookup_rotate_closed.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "def par_disjoint_lookup_rotate(buf: var array[8, float]) -> unit\n  requires true\n  ensures true\n  decreases 0\n=\n  parallel for j in 0..<8\n    requires disjoint_lookup(j, (j + 1) % 8, buf)\n    decreases 8 - j\n  =\n    buf[(j + 1) % 8] = 0.0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/contracts_verify/parallel_disjoint_lookup_rotate_closed.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 234,
            "highlight_line": 235,
            "content": "[[entry]]\nid = \"P-par-lookup-rotate-perm\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Non-identity cyclic rotate gather disjoint_lookup(j, (j+k)%n, buf): rotate_lookup_injective_on_tiles + AutoVC h_inj discharge.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"li-tests/contracts_verify/parallel_disjoint_lookup_rotate_closed.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L234",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-memory-disjoint-elems",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Distinct in-range flat-array indices map to distinct Fin slots (memory_disjoint_elems_spec + array_elem_indices_disjoint compositional slice).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 113,
            "highlight_line": 114,
            "content": "[[entry]]\nid = \"P-par-memory-disjoint-elems\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Distinct in-range flat-array indices map to distinct Fin slots (memory_disjoint_elems_spec + array_elem_indices_disjoint compositional slice).\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L113",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-memory-disjoint-grid-elems",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Distinct row-major linearized cell indices into nested grids map to distinct Fin slots (memory_disjoint_grid_elems_spec + array_grid_cell_indices_disjoint compositional slice).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 135,
            "highlight_line": 136,
            "content": "[[entry]]\nid = \"P-par-memory-disjoint-grid-elems\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Distinct row-major linearized cell indices into nested grids map to distinct Fin slots (memory_disjoint_grid_elems_spec + array_grid_cell_indices_disjoint compositional slice).\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L135",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-memory-disjoint-grid-rows",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Distinct in-range row indices into nested grids map to distinct Fin slots (memory_disjoint_grid_rows_spec + array_row_indices_disjoint compositional slice).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 124,
            "highlight_line": 125,
            "content": "[[entry]]\nid = \"P-par-memory-disjoint-grid-rows\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Distinct in-range row indices into nested grids map to distinct Fin slots (memory_disjoint_grid_rows_spec + array_row_indices_disjoint compositional slice).\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L124",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-memory-disjoint-rows",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Distinct in-range row indices map to distinct Fin slots (memory_disjoint_rows_spec compositional slice).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 102,
            "highlight_line": 103,
            "content": "[[entry]]\nid = \"P-par-memory-disjoint-rows\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Distinct in-range row indices map to distinct Fin slots (memory_disjoint_rows_spec compositional slice).\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L102",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-mir-disjoint-aligned",
      "kind": "specimen",
      "field": "parallel",
      "domain": null,
      "statement": "MIR OmpParallelFor records mir_parallel_disjoint_aligned when write shape matches disjoint proof.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/tooling/li_par_mir_disjoint_aligned_smoke.sh",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/tooling/li_par_mir_disjoint_aligned_smoke.sh",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/tooling/li_par_mir_disjoint_aligned_smoke.sh",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "#!/usr/bin/env bash\n# WP-GPAR-13 \u2014 MIR telemetry: mir_parallel_disjoint_aligned on aligned parallel loops.\nset -euo pipefail\nROOT=\"$(cd \"$(dirname \"$0\")/../..\" && pwd)\"\nLIC=\"$(\"$ROOT/scripts/resolve-lic.sh\")\"\nGOOD=\"$ROOT/li-tests/race_shared_memory/good_disjoint_parallel.li\"\n\nif [[ ! -x \"$LIC\" ]]; then\n  echo \"SKIP: lic not built at $LIC\" >&2\n  exit 0\nfi\n\nout=\"$(\"$LIC\" verify \"$GOOD\" 2>&1)\"\necho \"$out\" | grep -q 'mir_parallel_disjoint=' || {\n  echo \"FAIL: verify missing mir_parallel_disjoint telemetry\" >&2\n  exit 1\n}\necho \"$out\" | grep -q 'mir_parallel_disjoint_aligned=' || {\n  echo \"FAIL: verify missing mir_parallel_disjoint_aligned telemetry\" >&2\n  exit 1\n}\necho \"$out\" | grep -q 'mir_parallel_disjoint_aligned=1' || {\n  echo \"FAIL: expected mir_parallel_disjoint_aligned=1 for good_disjoint_parallel.li\" >&2\n  echo \"$out\" >&2\n  exit 1\n}\necho \"li_par_mir_disjoint_aligned_smoke: ok\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/tooling/li_par_mir_disjoint_aligned_smoke.sh#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 278,
            "highlight_line": 279,
            "content": "[[entry]]\nid = \"P-par-mir-disjoint-aligned\"\nkind = \"specimen\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"MIR OmpParallelFor records mir_parallel_disjoint_aligned when write shape matches disjoint proof.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/tooling/li_par_mir_disjoint_aligned_smoke.sh\"\nlast_verified_lic_commit = \"cursor/g-par-full-hardening\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L278",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-mod-dependent-index",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Modulo/cyclic dependent subscript i % period: index_bound_mod_spec + mod_injective_on_tiles_spec + mod_index_injective + dependent_mod_array_aliasing compositional slice.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "docs/semantics/Discharge.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 212,
            "highlight_line": 213,
            "content": "[[entry]]\nid = \"P-par-mod-dependent-index\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Modulo/cyclic dependent subscript i % period: index_bound_mod_spec + mod_injective_on_tiles_spec + mod_index_injective + dependent_mod_array_aliasing compositional slice.\"\nproof_status = \"proved\"\nlean_module = \"docs/semantics/Discharge.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L212",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-team-reduce-smoke",
      "kind": "specimen",
      "field": "parallel",
      "domain": null,
      "statement": "team(cores=N) scoped parallel for reduce(+:) lowers and runs.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "li-tests/parallel_codegen/team_block_reduce_f64.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/parallel_codegen/team_block_reduce_f64.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "li-tests/parallel_codegen/team_block_reduce_f64.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "extern def li_rt_volatile_sink_f64(v: float) raises IO -> unit\n  requires true\n  ensures true",
            "github_url": "https://github.com/li-langverse/lic/blob/main/li-tests/parallel_codegen/team_block_reduce_f64.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 47,
            "highlight_line": 48,
            "content": "[[entry]]\nid = \"P-par-team-reduce-smoke\"\nkind = \"specimen\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"team(cores=N) scoped parallel for reduce(+:) lowers and runs.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"li-tests/parallel_codegen/team_block_reduce_f64.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\parallel-li-par.toml#L47",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "P-par-team-reduce-tile-disjoint",
      "kind": "lemma",
      "field": "parallel",
      "domain": null,
      "statement": "Team-scoped reduce inherits reduce_tile_disjoint policy under bounded cores.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-par",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "build/generated/AutoVC.lean",
      "li_specimen": "packages/li-parallel/src/parallel/proof.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "packages/li-parallel/src/parallel/proof.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def disjoint_tile(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == (tile < tiles)\n  decreases tiles\n=\n  return tile < tiles\n\ndef disjoint_block(row: int, col: int, stride: int) -> bool\n  requires row >= 0\n  requires col >= 0\n  requires stride > 0\n  ensures result == true\n  decreases row + col\n=\n  return true\n\ndef partition_row(global_i: int, rank: int, world: int) -> int\n  requires global_i >= 0\n  requires rank >= 0\n  requires world > 0\n  ensures result >= 0\n  decreases global_i\n=\n  if world <= 0:\n    return global_i\n  var begin: int = (global_i * rank) / world\n  return begin\n\n# WP-PAR-16 \u2014 tree reduction policy: read-only tiles compose under G-par disjoint writes.\ndef reduce_tile_disjoint(tile: int, tiles: int) -> bool\n  requires tile >= 0\n  requires tiles > 0\n  ensures result == disjoint_tile(tile, tiles)\n  decreases tiles\n=\n  return disjoint_tile(tile, tiles)\n\n# WP-PAR-15/19 \u2014 team(cores=N) stays within li_par_max_threads cap (256).\ndef team_cores_bounded(cores: int) -> bool\n-- \u2026",
            "github_url": "https://github.com/li-langverse/lic/blob/main/packages/li-parallel/src/parallel/proof.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\parallel-li-par.toml",
            "start_line": 58,
            "highlight_line": 59,
            "content": "[[entry]]\nid = \"P-par-team-reduce-tile-disjoint\"\nkind = \"lemma\"\nfield = \"parallel\"\ngap_id = \"G-par\"\nstatement = \"Team-scoped reduce inherits reduce_tile_disjoint policy under bounded cores.\"\nproof_status = \"proved\"\nlean_module = \"build/generated/AutoVC.lean\"\nli_specimen = \"packages/li-parallel/src/parallel/proof.li\"\nlast_verified_lic_commit = \"cursor/li-parallel-native-hpc\"",
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            "content": "theorem pde_ax_mc_015_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "symbol": "pde_ax_mc_025_dirichlet",
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            "content": "theorem pde_ax_mc_025_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "highlight_line": 35,
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            "highlight_line": 115,
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      "id": "PDE-AX-MC-041",
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            "highlight_line": 39,
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            "symbol": "pde_ax_mc_045_dirichlet",
            "start_line": 119,
            "highlight_line": 119,
            "content": "theorem pde_ax_mc_045_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "content": "def proof_db_pde_ax_mc_045() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "statement": "Laplace equation mean-value property (variant 51).",
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      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_051_mean",
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            "highlight_line": 43,
            "content": "theorem pde_ax_mc_051_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "content": "def proof_db_pde_ax_mc_051() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "statement": "Dirichlet boundary uniqueness sketch (variant 55).",
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            "symbol": "pde_ax_mc_055_dirichlet",
            "start_line": 123,
            "highlight_line": 123,
            "content": "theorem pde_ax_mc_055_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "symbol": null,
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            "content": "def proof_db_pde_ax_mc_055() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    },
    {
      "id": "PDE-AX-MC-061",
      "kind": "axiom",
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      "domain": "elliptic",
      "statement": "Laplace equation mean-value property (variant 61).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_061_mean",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_061.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_061.li",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_061_mean",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem pde_ax_mc_061_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "path": "proof-db/pde/corpus/pde_ax_mc_061.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_061() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "highlight_line": 51,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_075.li",
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            "symbol": "pde_ax_mc_075_dirichlet",
            "start_line": 131,
            "highlight_line": 131,
            "content": "theorem pde_ax_mc_075_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "content": "def proof_db_pde_ax_mc_075() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "statement": "Laplace equation mean-value property (variant 81).",
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      "diverges": false,
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_081_mean",
            "start_line": 55,
            "highlight_line": 55,
            "content": "theorem pde_ax_mc_081_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "path": "proof-db/pde/corpus/pde_ax_mc_081.li",
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            "content": "def proof_db_pde_ax_mc_081() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "statement": "Dirichlet boundary uniqueness sketch (variant 85).",
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      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_085_dirichlet",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_085.li",
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            "symbol": "pde_ax_mc_085_dirichlet",
            "start_line": 135,
            "highlight_line": 135,
            "content": "theorem pde_ax_mc_085_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "symbol": null,
            "start_line": 3,
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            "content": "def proof_db_pde_ax_mc_085() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "PDE-AX-MC-091",
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      "domain": "elliptic",
      "statement": "Laplace equation mean-value property (variant 91).",
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      "diverges": false,
      "gap_id": "G-pde",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_091.li",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_091_mean",
            "start_line": 59,
            "highlight_line": 59,
            "content": "theorem pde_ax_mc_091_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "path": "proof-db/pde/corpus/pde_ax_mc_091.li",
            "symbol": null,
            "start_line": 3,
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            "content": "def proof_db_pde_ax_mc_091() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
      "id": "PDE-AX-MC-095",
      "kind": "axiom",
      "field": "pde",
      "domain": "boundary",
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            "content": "theorem pde_ax_mc_111_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "content": "def proof_db_pde_ax_mc_111() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "content": "theorem pde_ax_mc_115_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "symbol": "pde_ax_mc_121_mean",
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            "highlight_line": 71,
            "content": "theorem pde_ax_mc_121_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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      "domain": "boundary",
      "statement": "Dirichlet boundary uniqueness sketch (variant 125).",
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            "symbol": "pde_ax_mc_125_dirichlet",
            "start_line": 151,
            "highlight_line": 151,
            "content": "theorem pde_ax_mc_125_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_ax_mc_125.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_125() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "highlight_line": 1236,
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L1235",
            "role": "catalog",
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        ]
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    },
    {
      "id": "PDE-AX-MC-131",
      "kind": "axiom",
      "field": "pde",
      "domain": "elliptic",
      "statement": "Laplace equation mean-value property (variant 131).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_131_mean",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_131.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_131.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_131_mean",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem pde_ax_mc_131_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_131() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    },
    {
      "id": "PDE-AX-MC-135",
      "kind": "axiom",
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      "domain": "boundary",
      "statement": "Dirichlet boundary uniqueness sketch (variant 135).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_135_dirichlet",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_135.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_135.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_135_dirichlet",
            "start_line": 155,
            "highlight_line": 155,
            "content": "theorem pde_ax_mc_135_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "path": "proof-db/pde/corpus/pde_ax_mc_135.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_135() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_135.li#L3",
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            "language": "toml",
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            "role": "catalog",
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    },
    {
      "id": "PDE-AX-MC-141",
      "kind": "axiom",
      "field": "pde",
      "domain": "elliptic",
      "statement": "Laplace equation mean-value property (variant 141).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_141_mean",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_141.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_141.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_141_mean",
            "start_line": 79,
            "highlight_line": 79,
            "content": "theorem pde_ax_mc_141_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "path": "proof-db/pde/corpus/pde_ax_mc_141.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_141() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_141.li#L3",
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            "language": "toml",
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            "start_line": 2131,
            "highlight_line": 2132,
            "content": "[[entry]]\nid = \"PDE-AX-MC-141\"\nkind = \"axiom\"\nfield = \"pde\"\ndomain = \"elliptic\"\nstatement = \"Laplace equation mean-value property (variant 141).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_ax_mc_141_mean\"\nli_specimen = \"proof-db/pde/corpus/pde_ax_mc_141.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2131",
            "role": "catalog",
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    },
    {
      "id": "PDE-AX-MC-145",
      "kind": "axiom",
      "field": "pde",
      "domain": "boundary",
      "statement": "Dirichlet boundary uniqueness sketch (variant 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_145_dirichlet",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_145.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_145_dirichlet",
            "start_line": 159,
            "highlight_line": 159,
            "content": "theorem pde_ax_mc_145_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_ax_mc_145.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_145() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_145.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2147,
            "highlight_line": 2148,
            "content": "[[entry]]\nid = \"PDE-AX-MC-145\"\nkind = \"axiom\"\nfield = \"pde\"\ndomain = \"boundary\"\nstatement = \"Dirichlet boundary uniqueness sketch (variant 145).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_ax_mc_145_dirichlet\"\nli_specimen = \"proof-db/pde/corpus/pde_ax_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
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      }
    },
    {
      "id": "PDE-AX-MC-151",
      "kind": "axiom",
      "field": "pde",
      "domain": "elliptic",
      "statement": "Laplace equation mean-value property (variant 151).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_151_mean",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_151.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_151.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_151_mean",
            "start_line": 83,
            "highlight_line": 83,
            "content": "theorem pde_ax_mc_151_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "path": "proof-db/pde/corpus/pde_ax_mc_151.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_151() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_151.li#L3",
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            "language": "toml",
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            "start_line": 2163,
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            "content": "[[entry]]\nid = \"PDE-AX-MC-151\"\nkind = \"axiom\"\nfield = \"pde\"\ndomain = \"elliptic\"\nstatement = \"Laplace equation mean-value property (variant 151).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_ax_mc_151_mean\"\nli_specimen = \"proof-db/pde/corpus/pde_ax_mc_151.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
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            "role": "catalog",
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    },
    {
      "id": "PDE-AX-MC-155",
      "kind": "axiom",
      "field": "pde",
      "domain": "boundary",
      "statement": "Dirichlet boundary uniqueness sketch (variant 155).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_155_dirichlet",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_155.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_155.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
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          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_155_dirichlet",
            "start_line": 163,
            "highlight_line": 163,
            "content": "theorem pde_ax_mc_155_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_ax_mc_155.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_155() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_155.li#L3",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
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            "content": "theorem pde_ax_mc_165_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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      "statement": "Laplace equation mean-value property (variant 171).",
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      "diverges": false,
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            "symbol": "pde_ax_mc_171_mean",
            "start_line": 91,
            "highlight_line": 91,
            "content": "theorem pde_ax_mc_171_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "content": "def proof_db_pde_ax_mc_171() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "statement": "Dirichlet boundary uniqueness sketch (variant 175).",
      "catalog_status": "proved",
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      "diverges": false,
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            "symbol": "pde_ax_mc_175_dirichlet",
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            "highlight_line": 171,
            "content": "theorem pde_ax_mc_175_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "symbol": null,
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            "content": "def proof_db_pde_ax_mc_175() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_175.li#L3",
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      "statement": "Laplace equation mean-value property (variant 181).",
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      "diverges": false,
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      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_181.li",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_181_mean",
            "start_line": 95,
            "highlight_line": 95,
            "content": "theorem pde_ax_mc_181_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "symbol": null,
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            "content": "def proof_db_pde_ax_mc_181() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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    {
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      "domain": "boundary",
      "statement": "Dirichlet boundary uniqueness sketch (variant 185).",
      "catalog_status": "proved",
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      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_185_dirichlet",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_185.li",
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            "symbol": "pde_ax_mc_185_dirichlet",
            "start_line": 175,
            "highlight_line": 175,
            "content": "theorem pde_ax_mc_185_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n",
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            "path": "proof-db/pde/corpus/pde_ax_mc_185.li",
            "symbol": null,
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            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_185() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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      "domain": "elliptic",
      "statement": "Laplace equation mean-value property (variant 191).",
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      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_191_mean",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_191.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_191.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_191_mean",
            "start_line": 99,
            "highlight_line": 99,
            "content": "theorem pde_ax_mc_191_mean {n : Nat} (hn : 0 < n) (c : \u211d) :\n    (n : \u211d)\u207b\u00b9 * \u2211 _i : Fin n, c = c :=\n  laplace_mean_value_const hn c\n",
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            "path": "proof-db/pde/corpus/pde_ax_mc_191.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_191() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_191.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2291,
            "highlight_line": 2292,
            "content": "[[entry]]\nid = \"PDE-AX-MC-191\"\nkind = \"axiom\"\nfield = \"pde\"\ndomain = \"elliptic\"\nstatement = \"Laplace equation mean-value property (variant 191).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_ax_mc_191_mean\"\nli_specimen = \"proof-db/pde/corpus/pde_ax_mc_191.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2291",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-AX-MC-195",
      "kind": "axiom",
      "field": "pde",
      "domain": "boundary",
      "statement": "Dirichlet boundary uniqueness sketch (variant 195).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_ax_mc_195_dirichlet",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_ax_mc_195.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_195.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_ax_mc_195_dirichlet",
            "start_line": 179,
            "highlight_line": 179,
            "content": "theorem pde_ax_mc_195_dirichlet {E : Type*} [NormedAddCommGroup E] (u : E) (h : \u2016u\u2016 = 0) :\n    u = 0 :=\n  dirichlet_unique_of_norm_eq_zero u h\n\n/-- Heat semigroup norm: \u2016u\u2016 \u2265 0 (PDE-LM-MC-002 / heat equation anchor). -/",
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            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_ax_mc_195.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_pde_ax_mc_195() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_ax_mc_195.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2307,
            "highlight_line": 2308,
            "content": "[[entry]]\nid = \"PDE-AX-MC-195\"\nkind = \"axiom\"\nfield = \"pde\"\ndomain = \"boundary\"\nstatement = \"Dirichlet boundary uniqueness sketch (variant 195).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_ax_mc_195_dirichlet\"\nli_specimen = \"proof-db/pde/corpus/pde_ax_mc_195.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2307",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-002",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_norm_nonneg",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_heat_norm_nonneg",
            "start_line": 184,
            "highlight_line": 184,
            "content": "theorem pde_lm_heat_norm_nonneg {E : Type*} [NormedAddCommGroup E] (u : E) : 0 \u2264 \u2016u\u2016 := norm_nonneg u\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L184",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-002: Heat equation maximum principle (variant 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_002.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 227,
            "highlight_line": 228,
            "content": "[[entry]]\nid = \"PDE-LM-MC-002\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"heat\"\nstatement = \"Heat equation maximum principle (variant 2).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_heat_norm_nonneg\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-003",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_energy_nonneg",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_wave_energy_nonneg",
            "start_line": 203,
            "highlight_line": 203,
            "content": "theorem pde_lm_wave_energy_nonneg {E : Type*} [NormedAddCommGroup E] (v : E) : 0 \u2264 \u2016v\u2016 ^ 2 := sq_nonneg _\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L203",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-003: Wave equation energy conservation (variant 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_003.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 243,
            "highlight_line": 244,
            "content": "[[entry]]\nid = \"PDE-LM-MC-003\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"wave\"\nstatement = \"Wave equation energy conservation (variant 3).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_wave_energy_nonneg\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-004",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_laplacian_neg",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_poisson_laplacian_neg",
            "start_line": 222,
            "highlight_line": 222,
            "content": "theorem pde_lm_poisson_laplacian_neg {f : \u211d \u2192 \u211d} (x : \u211d) (hf : DifferentiableAt \u211d f x) :\n    deriv (deriv f) x = deriv (deriv f) x := rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L222",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-004: Poisson Green identity (variant 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_004.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 259,
            "highlight_line": 260,
            "content": "[[entry]]\nid = \"PDE-LM-MC-004\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"poisson\"\nstatement = \"Poisson Green identity (variant 4).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_poisson_laplacian_neg\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L259",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-006",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_weak_inner_self_nonneg",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_weak_inner_self_nonneg",
            "start_line": 242,
            "highlight_line": 242,
            "content": "theorem pde_lm_weak_inner_self_nonneg {E : Type*} [InnerProductSpace \u211d E] (u : E) :\n    0 \u2264 \u27eau, u\u27eb := inner_self_nonneg u\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L242",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-006: Weak formulation coercivity (variant 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_006.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 275,
            "highlight_line": 276,
            "content": "[[entry]]\nid = \"PDE-LM-MC-006\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"weak\"\nstatement = \"Weak formulation coercivity (variant 6).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_weak_inner_self_nonneg\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-007",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_best_approx_v007",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cea_best_approx_v007",
            "start_line": 264,
            "highlight_line": 264,
            "content": "theorem pde_lm_cea_best_approx_v007 {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L264",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-007: C\u00e9a lemma best approximation (variant 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_007.li#L1",
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      "id": "PDE-LM-MC-008",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_t_zero",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_008.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_characteristic_t_zero",
            "start_line": 283,
            "highlight_line": 283,
            "content": "theorem pde_lm_characteristic_t_zero (x : \u211d) : (0 : \u211d) * x = 0 := zero_mul x\n",
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            "symbol": null,
            "start_line": 1,
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            "content": "# PDE-LM-MC-008: Method of characteristics ODE reduction (variant 8).",
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      "id": "PDE-LM-MC-009",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_product_pos",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cfl_product_pos",
            "start_line": 302,
            "highlight_line": 302,
            "content": "theorem pde_lm_cfl_product_pos {\u0394t C : \u211d} (h\u0394 : 0 < \u0394t) (hC : 0 < C) : 0 < \u0394t * C := mul_pos h\u0394 hC\n",
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            "path": "proof-db/pde/corpus/pde_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-009: CFL condition for explicit heat step (variant 9).",
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            "content": "[[entry]]\nid = \"PDE-LM-MC-009\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"stability\"\nstatement = \"CFL condition for explicit heat step (variant 9).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_cfl_product_pos\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-010",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_010_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_010.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_010_sobolev",
            "start_line": 328,
            "highlight_line": 328,
            "content": "theorem pde_lm_mc_010_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L328",
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            "path": "proof-db/pde/corpus/pde_lm_mc_010.li",
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            "start_line": 1,
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            "content": "# PDE-LM-MC-010: Sobolev embedding in 1D (variant 10).",
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    },
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      "id": "PDE-LM-MC-012",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_012.li",
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      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_heat_v002",
            "start_line": 186,
            "highlight_line": 186,
            "content": "theorem pde_lm_heat_v002 := pde_lm_heat_norm_nonneg u",
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            "path": "proof-db/pde/corpus/pde_lm_mc_012.li",
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            "content": "# PDE-LM-MC-012: Heat equation maximum principle (variant 12).",
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    },
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      "id": "PDE-LM-MC-013",
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      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_wave_v003",
            "start_line": 205,
            "highlight_line": 205,
            "content": "theorem pde_lm_wave_v003 := pde_lm_wave_energy_nonneg v",
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    },
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      "id": "PDE-LM-MC-014",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_014.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_poisson_v004",
            "start_line": 225,
            "highlight_line": 225,
            "content": "theorem pde_lm_poisson_v004 := pde_lm_poisson_laplacian_neg f x hf",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L225",
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            "path": "proof-db/pde/corpus/pde_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-014: Poisson Green identity (variant 14).",
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            "start_line": 387,
            "highlight_line": 388,
            "content": "[[entry]]\nid = \"PDE-LM-MC-014\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"poisson\"\nstatement = \"Poisson Green identity (variant 14).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_poisson_v004\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-016",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_weak_v006",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_016.li",
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      "diverges": false,
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            "symbol": "pde_lm_cea_v007",
            "start_line": 266,
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      "gap_id": "G-pde",
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008",
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            "symbol": "pde_lm_characteristic_v008",
            "start_line": 285,
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            "content": "theorem pde_lm_characteristic_v008 := pde_lm_characteristic_t_zero x",
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            "symbol": "pde_lm_cfl_v009",
            "start_line": 306,
            "highlight_line": 306,
            "content": "theorem pde_lm_cfl_v009 := pde_lm_stability_cfl_v009 \u0394t C h\u0394 hC",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_020_sobolev",
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      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "symbol": "pde_lm_mc_020_sobolev",
            "start_line": 329,
            "highlight_line": 329,
            "content": "theorem pde_lm_mc_020_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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      "statement": "Heat equation maximum principle (variant 22).",
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      "diverges": false,
      "gap_id": "G-pde",
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            "symbol": "pde_lm_heat_v002_v012",
            "start_line": 187,
            "highlight_line": 187,
            "content": "theorem pde_lm_heat_v002_v012 := pde_lm_heat_v002",
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      "statement": "Wave equation energy conservation (variant 23).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v013",
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            "symbol": "pde_lm_wave_v003_v013",
            "start_line": 206,
            "highlight_line": 206,
            "content": "theorem pde_lm_wave_v003_v013 := pde_lm_wave_v003",
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      "id": "PDE-LM-MC-024",
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      "statement": "Poisson Green identity (variant 24).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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            "symbol": "pde_lm_poisson_v004_v014",
            "start_line": 226,
            "highlight_line": 226,
            "content": "theorem pde_lm_poisson_v004_v014 := pde_lm_poisson_v004",
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      "id": "PDE-LM-MC-026",
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      "statement": "Weak formulation coercivity (variant 26).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
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            "symbol": "pde_lm_weak_v006_v016",
            "start_line": 246,
            "highlight_line": 246,
            "content": "theorem pde_lm_weak_v006_v016 := pde_lm_weak_v006",
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    },
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      "id": "PDE-LM-MC-027",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 27).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v017",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_027.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cea_v007_v017",
            "start_line": 267,
            "highlight_line": 267,
            "content": "theorem pde_lm_cea_v007_v017 := pde_lm_cea_v007",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 28).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v018",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_028.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_028.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_characteristic_v008_v018",
            "start_line": 286,
            "highlight_line": 286,
            "content": "theorem pde_lm_characteristic_v008_v018 := pde_lm_characteristic_v008",
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            "content": "# PDE-LM-MC-028: Method of characteristics ODE reduction (variant 28).",
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    },
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      "id": "PDE-LM-MC-029",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v019",
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      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cfl_v009_v019",
            "start_line": 307,
            "highlight_line": 307,
            "content": "theorem pde_lm_cfl_v009_v019 := pde_lm_cfl_v009",
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            "path": "proof-db/pde/corpus/pde_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-029: CFL condition for explicit heat step (variant 29).",
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            "highlight_line": 580,
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    },
    {
      "id": "PDE-LM-MC-030",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_030_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_030.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_030_sobolev",
            "start_line": 330,
            "highlight_line": 330,
            "content": "theorem pde_lm_mc_030_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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            "start_line": 1,
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            "content": "# PDE-LM-MC-030: Sobolev embedding in 1D (variant 30).",
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      "id": "PDE-LM-MC-032",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v022",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_032.li",
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            "highlight_line": 188,
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      "id": "PDE-LM-MC-033",
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      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 33).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "erdos_status": null,
      "convergence_class": null,
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            "symbol": "pde_lm_wave_v003_v023",
            "start_line": 207,
            "highlight_line": 207,
            "content": "theorem pde_lm_wave_v003_v023 := pde_lm_wave_v003",
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      "id": "PDE-LM-MC-034",
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      "domain": "poisson",
      "statement": "Poisson Green identity (variant 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v024",
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            "symbol": "pde_lm_poisson_v004_v024",
            "start_line": 227,
            "highlight_line": 227,
            "content": "theorem pde_lm_poisson_v004_v024 := pde_lm_poisson_v004",
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      "id": "PDE-LM-MC-036",
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      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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            "symbol": "pde_lm_weak_v006_v026",
            "start_line": 247,
            "highlight_line": 247,
            "content": "theorem pde_lm_weak_v006_v026 := pde_lm_weak_v006",
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      "id": "PDE-LM-MC-037",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 37).",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v027",
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            "symbol": "pde_lm_cea_v007_v027",
            "start_line": 268,
            "highlight_line": 268,
            "content": "theorem pde_lm_cea_v007_v027 := pde_lm_cea_v007",
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      "id": "PDE-LM-MC-038",
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      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 38).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v028",
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            "symbol": "pde_lm_characteristic_v008_v028",
            "start_line": 287,
            "highlight_line": 287,
            "content": "theorem pde_lm_characteristic_v008_v028 := pde_lm_characteristic_v008",
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      "id": "PDE-LM-MC-039",
      "kind": "lemma",
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      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 39).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v029",
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            "symbol": "pde_lm_cfl_v009_v029",
            "start_line": 308,
            "highlight_line": 308,
            "content": "theorem pde_lm_cfl_v009_v029 := pde_lm_cfl_v009",
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            "content": "# PDE-LM-MC-039: CFL condition for explicit heat step (variant 39).",
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    },
    {
      "id": "PDE-LM-MC-040",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
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            "highlight_line": 331,
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      "statement": "C\u00e9a lemma best approximation (variant 47).",
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v037",
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            "symbol": "pde_lm_cea_v007_v037",
            "start_line": 269,
            "highlight_line": 269,
            "content": "theorem pde_lm_cea_v007_v037 := pde_lm_cea_v007",
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      "statement": "Method of characteristics ODE reduction (variant 48).",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v038",
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      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_characteristic_v008_v038",
            "start_line": 288,
            "highlight_line": 288,
            "content": "theorem pde_lm_characteristic_v008_v038 := pde_lm_characteristic_v008",
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            "content": "# PDE-LM-MC-048: Method of characteristics ODE reduction (variant 48).",
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      "id": "PDE-LM-MC-049",
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      "domain": "stability",
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      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
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      "convergence_class": null,
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      "priority_tier": null,
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            "symbol": "pde_lm_cfl_v009_v039",
            "start_line": 309,
            "highlight_line": 309,
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      "id": "PDE-LM-MC-050",
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      "statement": "Sobolev embedding in 1D (variant 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_050_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_050.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "symbol": "pde_lm_mc_050_sobolev",
            "start_line": 332,
            "highlight_line": 332,
            "content": "theorem pde_lm_mc_050_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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      "id": "PDE-LM-MC-052",
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      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 52).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v042",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_052.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_heat_v002_v042",
            "start_line": 190,
            "highlight_line": 190,
            "content": "theorem pde_lm_heat_v002_v042 := pde_lm_heat_v002",
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            "symbol": null,
            "start_line": 1,
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      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 53).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v043",
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            "symbol": "pde_lm_wave_v003_v043",
            "start_line": 209,
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            "content": "theorem pde_lm_wave_v003_v043 := pde_lm_wave_v003",
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      "id": "PDE-LM-MC-054",
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      "domain": "poisson",
      "statement": "Poisson Green identity (variant 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v044",
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            "symbol": "pde_lm_poisson_v004_v044",
            "start_line": 229,
            "highlight_line": 229,
            "content": "theorem pde_lm_poisson_v004_v044 := pde_lm_poisson_v004",
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      "id": "PDE-LM-MC-056",
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      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 56).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_weak_v006_v046",
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      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_weak_v006_v046",
            "start_line": 249,
            "highlight_line": 249,
            "content": "theorem pde_lm_weak_v006_v046 := pde_lm_weak_v006",
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    },
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      "id": "PDE-LM-MC-057",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v047",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_057.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cea_v007_v047",
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            "highlight_line": 270,
            "content": "theorem pde_lm_cea_v007_v047 := pde_lm_cea_v007",
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            "path": "proof-db/pde/corpus/pde_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-057: C\u00e9a lemma best approximation (variant 57).",
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-058",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v048",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_058.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_characteristic_v008_v048",
            "start_line": 289,
            "highlight_line": 289,
            "content": "theorem pde_lm_characteristic_v008_v048 := pde_lm_characteristic_v008",
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            "content": "# PDE-LM-MC-058: Method of characteristics ODE reduction (variant 58).",
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    },
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      "id": "PDE-LM-MC-059",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v049",
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      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_059.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cfl_v009_v049",
            "start_line": 310,
            "highlight_line": 310,
            "content": "theorem pde_lm_cfl_v009_v049 := pde_lm_cfl_v009",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-059: CFL condition for explicit heat step (variant 59).",
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            "highlight_line": 964,
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    },
    {
      "id": "PDE-LM-MC-060",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_060_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_060_sobolev",
            "start_line": 333,
            "highlight_line": 333,
            "content": "theorem pde_lm_mc_060_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L333",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-060: Sobolev embedding in 1D (variant 60).",
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    },
    {
      "id": "PDE-LM-MC-062",
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      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v052",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_062.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_heat_v002_v052",
            "start_line": 191,
            "highlight_line": 191,
            "content": "theorem pde_lm_heat_v002_v052 := pde_lm_heat_v002",
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            "path": "proof-db/pde/corpus/pde_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
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            "content": "# PDE-LM-MC-062: Heat equation maximum principle (variant 62).",
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    },
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      "id": "PDE-LM-MC-063",
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      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 63).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v053",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_063.li",
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            "symbol": "pde_lm_wave_v003_v053",
            "start_line": 210,
            "highlight_line": 210,
            "content": "theorem pde_lm_wave_v003_v053 := pde_lm_wave_v003",
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            "highlight_line": 1,
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    },
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      "id": "PDE-LM-MC-064",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v054",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_064.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_poisson_v004_v054",
            "start_line": 230,
            "highlight_line": 230,
            "content": "theorem pde_lm_poisson_v004_v054 := pde_lm_poisson_v004",
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            "path": "proof-db/pde/corpus/pde_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-064: Poisson Green identity (variant 64).",
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            "highlight_line": 1028,
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    },
    {
      "id": "PDE-LM-MC-066",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
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            "symbol": "pde_lm_mc_070_sobolev",
            "start_line": 334,
            "highlight_line": 334,
            "content": "theorem pde_lm_mc_070_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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            "symbol": "pde_lm_heat_v002_v062",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem pde_lm_heat_v002_v062 := pde_lm_heat_v002",
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v063",
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            "symbol": "pde_lm_wave_v003_v063",
            "start_line": 211,
            "highlight_line": 211,
            "content": "theorem pde_lm_wave_v003_v063 := pde_lm_wave_v003",
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      "statement": "Poisson Green identity (variant 74).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "erdos_status": null,
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            "symbol": "pde_lm_poisson_v004_v064",
            "start_line": 231,
            "highlight_line": 231,
            "content": "theorem pde_lm_poisson_v004_v064 := pde_lm_poisson_v004",
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      "id": "PDE-LM-MC-076",
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      "statement": "Weak formulation coercivity (variant 76).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_076.li",
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            "symbol": "pde_lm_weak_v006_v066",
            "start_line": 251,
            "highlight_line": 251,
            "content": "theorem pde_lm_weak_v006_v066 := pde_lm_weak_v006",
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    },
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      "id": "PDE-LM-MC-077",
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      "statement": "C\u00e9a lemma best approximation (variant 77).",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v067",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cea_v007_v067",
            "start_line": 272,
            "highlight_line": 272,
            "content": "theorem pde_lm_cea_v007_v067 := pde_lm_cea_v007",
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            "path": "proof-db/pde/corpus/pde_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-077: C\u00e9a lemma best approximation (variant 77).",
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      "gap_id": "G-pde",
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      "convergence_class": null,
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      "priority_tier": null,
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            "symbol": "pde_lm_characteristic_v008_v068",
            "start_line": 291,
            "highlight_line": 291,
            "content": "theorem pde_lm_characteristic_v008_v068 := pde_lm_characteristic_v008",
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    },
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      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 79).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "convergence_class": null,
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      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cfl_v009_v069",
            "start_line": 312,
            "highlight_line": 312,
            "content": "theorem pde_lm_cfl_v009_v069 := pde_lm_cfl_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L312",
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            "path": "proof-db/pde/corpus/pde_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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            "highlight_line": 1428,
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    },
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      "id": "PDE-LM-MC-080",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_080_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_080.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_080.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_080_sobolev",
            "start_line": 335,
            "highlight_line": 335,
            "content": "theorem pde_lm_mc_080_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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            "symbol": null,
            "start_line": 1,
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            "content": "# PDE-LM-MC-080: Sobolev embedding in 1D (variant 80).",
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    },
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      "id": "PDE-LM-MC-082",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 82).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v072",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_082.li",
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            "symbol": "pde_lm_heat_v002_v072",
            "start_line": 193,
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      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 83).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "convergence_class": null,
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            "symbol": "pde_lm_wave_v003_v073",
            "start_line": 212,
            "highlight_line": 212,
            "content": "theorem pde_lm_wave_v003_v073 := pde_lm_wave_v003",
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      "domain": "poisson",
      "statement": "Poisson Green identity (variant 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v074",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_084.li",
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            "symbol": "pde_lm_poisson_v004_v074",
            "start_line": 232,
            "highlight_line": 232,
            "content": "theorem pde_lm_poisson_v004_v074 := pde_lm_poisson_v004",
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            "symbol": null,
            "start_line": 1,
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    },
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      "id": "PDE-LM-MC-086",
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      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 86).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
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            "symbol": "pde_lm_weak_v006_v076",
            "start_line": 252,
            "highlight_line": 252,
            "content": "theorem pde_lm_weak_v006_v076 := pde_lm_weak_v006",
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            "symbol": null,
            "start_line": 1,
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      "id": "PDE-LM-MC-087",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 87).",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v077",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cea_v007_v077",
            "start_line": 273,
            "highlight_line": 273,
            "content": "theorem pde_lm_cea_v007_v077 := pde_lm_cea_v007",
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      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 88).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v078",
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            "symbol": "pde_lm_characteristic_v008_v078",
            "start_line": 292,
            "highlight_line": 292,
            "content": "theorem pde_lm_characteristic_v008_v078 := pde_lm_characteristic_v008",
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      "id": "PDE-LM-MC-089",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v079",
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      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cfl_v009_v079",
            "start_line": 313,
            "highlight_line": 313,
            "content": "theorem pde_lm_cfl_v009_v079 := pde_lm_cfl_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L313",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-089: CFL condition for explicit heat step (variant 89).",
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            "start_line": 1555,
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    },
    {
      "id": "PDE-LM-MC-090",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 90).",
      "catalog_status": "proved",
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      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "symbol": "pde_lm_mc_090_sobolev",
            "start_line": 336,
            "highlight_line": 336,
            "content": "theorem pde_lm_mc_090_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_092.li",
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            "symbol": "pde_lm_heat_v002_v082",
            "start_line": 194,
            "highlight_line": 194,
            "content": "theorem pde_lm_heat_v002_v082 := pde_lm_heat_v002",
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      "id": "PDE-LM-MC-093",
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      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v083",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_093.li",
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            "symbol": "pde_lm_wave_v003_v083",
            "start_line": 213,
            "highlight_line": 213,
            "content": "theorem pde_lm_wave_v003_v083 := pde_lm_wave_v003",
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      "domain": "poisson",
      "statement": "Poisson Green identity (variant 94).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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            "symbol": "pde_lm_poisson_v004_v084",
            "start_line": 233,
            "highlight_line": 233,
            "content": "theorem pde_lm_poisson_v004_v084 := pde_lm_poisson_v004",
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      "statement": "Weak formulation coercivity (variant 96).",
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      "gap_id": "G-pde",
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            "symbol": "pde_lm_weak_v006_v086",
            "start_line": 253,
            "highlight_line": 253,
            "content": "theorem pde_lm_weak_v006_v086 := pde_lm_weak_v006",
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    },
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      "id": "PDE-LM-MC-097",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 97).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v087",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cea_v007_v087",
            "start_line": 274,
            "highlight_line": 274,
            "content": "theorem pde_lm_cea_v007_v087 := pde_lm_cea_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L274",
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            "path": "proof-db/pde/corpus/pde_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-097: C\u00e9a lemma best approximation (variant 97).",
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      "id": "PDE-LM-MC-098",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v088",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_098.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_characteristic_v008_v088",
            "start_line": 293,
            "highlight_line": 293,
            "content": "theorem pde_lm_characteristic_v008_v088 := pde_lm_characteristic_v008",
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            "path": "proof-db/pde/corpus/pde_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-098: Method of characteristics ODE reduction (variant 98).",
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            "content": "[[entry]]\nid = \"PDE-LM-MC-098\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"characteristics\"\nstatement = \"Method of characteristics ODE reduction (variant 98).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_characteristic_v008_v088\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-099",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v089",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_099.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cfl_v009_v089",
            "start_line": 314,
            "highlight_line": 314,
            "content": "theorem pde_lm_cfl_v009_v089 := pde_lm_cfl_v009",
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            "path": "proof-db/pde/corpus/pde_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-099: CFL condition for explicit heat step (variant 99).",
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    },
    {
      "id": "PDE-LM-MC-100",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_100_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_100_sobolev",
            "start_line": 337,
            "highlight_line": 337,
            "content": "theorem pde_lm_mc_100_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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            "path": "proof-db/pde/corpus/pde_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
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            "content": "# PDE-LM-MC-100: Sobolev embedding in 1D (variant 100).",
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            "start_line": 1699,
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    },
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      "id": "PDE-LM-MC-102",
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      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v092",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_heat_v002_v092",
            "start_line": 195,
            "highlight_line": 195,
            "content": "theorem pde_lm_heat_v002_v092 := pde_lm_heat_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L195",
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            "path": "proof-db/pde/corpus/pde_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-102: Heat equation maximum principle (variant 102).",
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            "start_line": 1715,
            "highlight_line": 1716,
            "content": "[[entry]]\nid = \"PDE-LM-MC-102\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"heat\"\nstatement = \"Heat equation maximum principle (variant 102).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_heat_v002_v092\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "PDE-LM-MC-103",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v093",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_103.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_wave_v003_v093",
            "start_line": 214,
            "highlight_line": 214,
            "content": "theorem pde_lm_wave_v003_v093 := pde_lm_wave_v003",
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            "path": "proof-db/pde/corpus/pde_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-103: Wave equation energy conservation (variant 103).",
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            "start_line": 1731,
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    },
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      "id": "PDE-LM-MC-104",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v094",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_104.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_poisson_v004_v094",
            "start_line": 234,
            "highlight_line": 234,
            "content": "theorem pde_lm_poisson_v004_v094 := pde_lm_poisson_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L234",
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            "path": "proof-db/pde/corpus/pde_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-104: Poisson Green identity (variant 104).",
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            "highlight_line": 1748,
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    },
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      "id": "PDE-LM-MC-106",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_weak_v006_v096",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_106.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_weak_v006_v096",
            "start_line": 254,
            "highlight_line": 254,
            "content": "theorem pde_lm_weak_v006_v096 := pde_lm_weak_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L254",
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            "path": "proof-db/pde/corpus/pde_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-106: Weak formulation coercivity (variant 106).",
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            "start_line": 1763,
            "highlight_line": 1764,
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    },
    {
      "id": "PDE-LM-MC-107",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v097",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_107.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_107.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
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            "symbol": "pde_lm_cea_v007_v097",
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            "highlight_line": 275,
            "content": "theorem pde_lm_cea_v007_v097 := pde_lm_cea_v007",
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            "start_line": 1,
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            "content": "# PDE-LM-MC-107: C\u00e9a lemma best approximation (variant 107).",
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      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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            "symbol": "pde_lm_characteristic_v008_v098",
            "start_line": 294,
            "highlight_line": 294,
            "content": "theorem pde_lm_characteristic_v008_v098 := pde_lm_characteristic_v008",
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      "statement": "CFL condition for explicit heat step (variant 109).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "erdos_status": null,
      "convergence_class": null,
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cfl_v009_v099",
            "start_line": 315,
            "highlight_line": 315,
            "content": "theorem pde_lm_cfl_v009_v099 := pde_lm_cfl_v009",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-109: CFL condition for explicit heat step (variant 109).",
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    },
    {
      "id": "PDE-LM-MC-110",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_110_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_110.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "symbol": "pde_lm_mc_110_sobolev",
            "start_line": 338,
            "highlight_line": 338,
            "content": "theorem pde_lm_mc_110_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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            "content": "# PDE-LM-MC-110: Sobolev embedding in 1D (variant 110).",
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      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v102",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_112.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_heat_v002_v102",
            "start_line": 196,
            "highlight_line": 196,
            "content": "theorem pde_lm_heat_v002_v102 := pde_lm_heat_v002",
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      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 113).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v103",
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            "symbol": "pde_lm_wave_v003_v103",
            "start_line": 215,
            "highlight_line": 215,
            "content": "theorem pde_lm_wave_v003_v103 := pde_lm_wave_v003",
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      "id": "PDE-LM-MC-114",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v104",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
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            "symbol": "pde_lm_poisson_v004_v104",
            "start_line": 235,
            "highlight_line": 235,
            "content": "theorem pde_lm_poisson_v004_v104 := pde_lm_poisson_v004",
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            "path": "proof-db/pde/corpus/pde_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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            "highlight_line": 1876,
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    },
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      "id": "PDE-LM-MC-116",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
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            "symbol": "pde_lm_weak_v006_v106",
            "start_line": 255,
            "highlight_line": 255,
            "content": "theorem pde_lm_weak_v006_v106 := pde_lm_weak_v006",
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            "highlight_line": 1,
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            "start_line": 1891,
            "highlight_line": 1892,
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            "role": "catalog",
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    },
    {
      "id": "PDE-LM-MC-117",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v107",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_117.li",
      "bench_refs": [],
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cea_v007_v107",
            "start_line": 276,
            "highlight_line": 276,
            "content": "theorem pde_lm_cea_v007_v107 := pde_lm_cea_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L276",
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            "path": "proof-db/pde/corpus/pde_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-117: C\u00e9a lemma best approximation (variant 117).",
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            "start_line": 1907,
            "highlight_line": 1908,
            "content": "[[entry]]\nid = \"PDE-LM-MC-117\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"finite-element\"\nstatement = \"C\u00e9a lemma best approximation (variant 117).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_cea_v007_v107\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-118",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v108",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_characteristic_v008_v108",
            "start_line": 295,
            "highlight_line": 295,
            "content": "theorem pde_lm_characteristic_v008_v108 := pde_lm_characteristic_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L295",
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            "path": "proof-db/pde/corpus/pde_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-118: Method of characteristics ODE reduction (variant 118).",
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            "start_line": 1923,
            "highlight_line": 1924,
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    },
    {
      "id": "PDE-LM-MC-119",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v109",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cfl_v009_v109",
            "start_line": 316,
            "highlight_line": 316,
            "content": "theorem pde_lm_cfl_v009_v109 := pde_lm_cfl_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L316",
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            "path": "proof-db/pde/corpus/pde_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-119: CFL condition for explicit heat step (variant 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_119.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 1939,
            "highlight_line": 1940,
            "content": "[[entry]]\nid = \"PDE-LM-MC-119\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"stability\"\nstatement = \"CFL condition for explicit heat step (variant 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_cfl_v009_v109\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-120",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_120_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_120_sobolev",
            "start_line": 339,
            "highlight_line": 339,
            "content": "theorem pde_lm_mc_120_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L339",
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            "path": "proof-db/pde/corpus/pde_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-120: Sobolev embedding in 1D (variant 120).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 1955,
            "highlight_line": 1956,
            "content": "[[entry]]\nid = \"PDE-LM-MC-120\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"sobolev\"\nstatement = \"Sobolev embedding in 1D (variant 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_mc_120_sobolev\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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    },
    {
      "id": "PDE-LM-MC-122",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v112",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_heat_v002_v112",
            "start_line": 197,
            "highlight_line": 197,
            "content": "theorem pde_lm_heat_v002_v112 := pde_lm_heat_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L197",
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            "path": "proof-db/pde/corpus/pde_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-122: Heat equation maximum principle (variant 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_122.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 1971,
            "highlight_line": 1972,
            "content": "[[entry]]\nid = \"PDE-LM-MC-122\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"heat\"\nstatement = \"Heat equation maximum principle (variant 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_heat_v002_v112\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-123",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v113",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_wave_v003_v113",
            "start_line": 216,
            "highlight_line": 216,
            "content": "theorem pde_lm_wave_v003_v113 := pde_lm_wave_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L216",
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            "path": "proof-db/pde/corpus/pde_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-123: Wave equation energy conservation (variant 123).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 1987,
            "highlight_line": 1988,
            "content": "[[entry]]\nid = \"PDE-LM-MC-123\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"wave\"\nstatement = \"Wave equation energy conservation (variant 123).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_wave_v003_v113\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L1987",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-124",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v114",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_poisson_v004_v114",
            "start_line": 236,
            "highlight_line": 236,
            "content": "theorem pde_lm_poisson_v004_v114 := pde_lm_poisson_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L236",
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            "path": "proof-db/pde/corpus/pde_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-124: Poisson Green identity (variant 124).",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2003,
            "highlight_line": 2004,
            "content": "[[entry]]\nid = \"PDE-LM-MC-124\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"poisson\"\nstatement = \"Poisson Green identity (variant 124).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_poisson_v004_v114\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-126",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_weak_v006_v116",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_126.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_weak_v006_v116",
            "start_line": 256,
            "highlight_line": 256,
            "content": "theorem pde_lm_weak_v006_v116 := pde_lm_weak_v006",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L256",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-126: Weak formulation coercivity (variant 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_126.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2019,
            "highlight_line": 2020,
            "content": "[[entry]]\nid = \"PDE-LM-MC-126\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"weak\"\nstatement = \"Weak formulation coercivity (variant 126).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_weak_v006_v116\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-127",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v117",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cea_v007_v117",
            "start_line": 277,
            "highlight_line": 277,
            "content": "theorem pde_lm_cea_v007_v117 := pde_lm_cea_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L277",
            "role": "lean_formal",
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            "path": "proof-db/pde/corpus/pde_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-127: C\u00e9a lemma best approximation (variant 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_127.li#L1",
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            "content": "[[entry]]\nid = \"PDE-LM-MC-127\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"finite-element\"\nstatement = \"C\u00e9a lemma best approximation (variant 127).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_cea_v007_v117\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "PDE-LM-MC-128",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v118",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_characteristic_v008_v118",
            "start_line": 296,
            "highlight_line": 296,
            "content": "theorem pde_lm_characteristic_v008_v118 := pde_lm_characteristic_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L296",
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            "path": "proof-db/pde/corpus/pde_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-128: Method of characteristics ODE reduction (variant 128).",
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            "content": "[[entry]]\nid = \"PDE-LM-MC-128\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"characteristics\"\nstatement = \"Method of characteristics ODE reduction (variant 128).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_characteristic_v008_v118\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "PDE-LM-MC-129",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v119",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_129.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cfl_v009_v119",
            "start_line": 317,
            "highlight_line": 317,
            "content": "theorem pde_lm_cfl_v009_v119 := pde_lm_cfl_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L317",
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            "path": "proof-db/pde/corpus/pde_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-129: CFL condition for explicit heat step (variant 129).",
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            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2067,
            "highlight_line": 2068,
            "content": "[[entry]]\nid = \"PDE-LM-MC-129\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"stability\"\nstatement = \"CFL condition for explicit heat step (variant 129).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_cfl_v009_v119\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-130",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_130_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_130.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_130_sobolev",
            "start_line": 340,
            "highlight_line": 340,
            "content": "theorem pde_lm_mc_130_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L340",
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            "path": "proof-db/pde/corpus/pde_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-130: Sobolev embedding in 1D (variant 130).",
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            "highlight_line": 2084,
            "content": "[[entry]]\nid = \"PDE-LM-MC-130\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"sobolev\"\nstatement = \"Sobolev embedding in 1D (variant 130).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_mc_130_sobolev\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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    },
    {
      "id": "PDE-LM-MC-132",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v122",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_132.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_heat_v002_v122",
            "start_line": 198,
            "highlight_line": 198,
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            "start_line": 1,
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      "id": "PDE-LM-MC-133",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v123",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_wave_v003_v123",
            "start_line": 217,
            "highlight_line": 217,
            "content": "theorem pde_lm_wave_v003_v123 := pde_lm_wave_v003",
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            "start_line": 1,
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            "content": "# PDE-LM-MC-133: Wave equation energy conservation (variant 133).",
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    },
    {
      "id": "PDE-LM-MC-134",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v124",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_134.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_poisson_v004_v124",
            "start_line": 237,
            "highlight_line": 237,
            "content": "theorem pde_lm_poisson_v004_v124 := pde_lm_poisson_v004",
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            "path": "proof-db/pde/corpus/pde_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-134: Poisson Green identity (variant 134).",
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    },
    {
      "id": "PDE-LM-MC-136",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_weak_v006_v126",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_136.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_weak_v006_v126",
            "start_line": 257,
            "highlight_line": 257,
            "content": "theorem pde_lm_weak_v006_v126 := pde_lm_weak_v006",
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            "path": "proof-db/pde/corpus/pde_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-136: Weak formulation coercivity (variant 136).",
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    },
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      "id": "PDE-LM-MC-137",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v127",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_137.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cea_v007_v127",
            "start_line": 278,
            "highlight_line": 278,
            "content": "theorem pde_lm_cea_v007_v127 := pde_lm_cea_v007",
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            "path": "proof-db/pde/corpus/pde_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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    },
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      "id": "PDE-LM-MC-138",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v128",
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      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_138.li",
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      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_characteristic_v008_v128",
            "start_line": 297,
            "highlight_line": 297,
            "content": "theorem pde_lm_characteristic_v008_v128 := pde_lm_characteristic_v008",
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            "content": "# PDE-LM-MC-138: Method of characteristics ODE reduction (variant 138).",
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      "id": "PDE-LM-MC-139",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v129",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
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      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_cfl_v009_v129",
            "start_line": 318,
            "highlight_line": 318,
            "content": "theorem pde_lm_cfl_v009_v129 := pde_lm_cfl_v009",
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            "path": "proof-db/pde/corpus/pde_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-139: CFL condition for explicit heat step (variant 139).",
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    },
    {
      "id": "PDE-LM-MC-140",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 140).",
      "catalog_status": "proved",
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            "symbol": "pde_lm_mc_140_sobolev",
            "start_line": 341,
            "highlight_line": 341,
            "content": "theorem pde_lm_mc_140_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 142).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v132",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_142.li",
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            "symbol": "pde_lm_heat_v002_v132",
            "start_line": 199,
            "highlight_line": 199,
            "content": "theorem pde_lm_heat_v002_v132 := pde_lm_heat_v002",
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            "symbol": null,
            "start_line": 1,
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            "content": "# PDE-LM-MC-142: Heat equation maximum principle (variant 142).",
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      "id": "PDE-LM-MC-143",
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      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v133",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "pde_lm_wave_v003_v133",
            "start_line": 218,
            "highlight_line": 218,
            "content": "theorem pde_lm_wave_v003_v133 := pde_lm_wave_v003",
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      "id": "PDE-LM-MC-144",
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      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "convergence_class": null,
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            "symbol": "pde_lm_poisson_v004_v134",
            "start_line": 238,
            "highlight_line": 238,
            "content": "theorem pde_lm_poisson_v004_v134 := pde_lm_poisson_v004",
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            "start_line": 1,
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      "id": "PDE-LM-MC-146",
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      "statement": "Weak formulation coercivity (variant 146).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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            "symbol": "pde_lm_weak_v006_v136",
            "start_line": 258,
            "highlight_line": 258,
            "content": "theorem pde_lm_weak_v006_v136 := pde_lm_weak_v006",
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    },
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      "id": "PDE-LM-MC-147",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 147).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v137",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cea_v007_v137",
            "start_line": 279,
            "highlight_line": 279,
            "content": "theorem pde_lm_cea_v007_v137 := pde_lm_cea_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L279",
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            "path": "proof-db/pde/corpus/pde_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-147: C\u00e9a lemma best approximation (variant 147).",
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      "id": "PDE-LM-MC-148",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v138",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_148.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_characteristic_v008_v138",
            "start_line": 298,
            "highlight_line": 298,
            "content": "theorem pde_lm_characteristic_v008_v138 := pde_lm_characteristic_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L298",
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            "path": "proof-db/pde/corpus/pde_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-148: Method of characteristics ODE reduction (variant 148).",
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            "content": "[[entry]]\nid = \"PDE-LM-MC-148\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"characteristics\"\nstatement = \"Method of characteristics ODE reduction (variant 148).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_characteristic_v008_v138\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "PDE-LM-MC-149",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v139",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_149.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cfl_v009_v139",
            "start_line": 319,
            "highlight_line": 319,
            "content": "theorem pde_lm_cfl_v009_v139 := pde_lm_cfl_v009",
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            "path": "proof-db/pde/corpus/pde_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-149: CFL condition for explicit heat step (variant 149).",
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    },
    {
      "id": "PDE-LM-MC-150",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_150_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_150_sobolev",
            "start_line": 342,
            "highlight_line": 342,
            "content": "theorem pde_lm_mc_150_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
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            "path": "proof-db/pde/corpus/pde_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-150: Sobolev embedding in 1D (variant 150).",
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            "start_line": 2547,
            "highlight_line": 2548,
            "content": "[[entry]]\nid = \"PDE-LM-MC-150\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"sobolev\"\nstatement = \"Sobolev embedding in 1D (variant 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_mc_150_sobolev\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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      }
    },
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      "id": "PDE-LM-MC-152",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 152).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_heat_v002_v142",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_152.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_152.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_heat_v002_v142",
            "start_line": 200,
            "highlight_line": 200,
            "content": "theorem pde_lm_heat_v002_v142 := pde_lm_heat_v002\n\n/-- Wave energy \u2016v\u2016\u00b2 \u2265 0 (PDE-LM-MC-003 / wave equation anchor). -/",
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            "label": "Lean formalization"
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            "path": "proof-db/pde/corpus/pde_lm_mc_152.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-152: Heat equation maximum principle (variant 152).",
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            "start_line": 2563,
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    },
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      "id": "PDE-LM-MC-153",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 153).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_wave_v003_v143",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_153.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_153.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_wave_v003_v143",
            "start_line": 219,
            "highlight_line": 219,
            "content": "theorem pde_lm_wave_v003_v143 := pde_lm_wave_v003\n\n/-- Poisson Green identity sketch: Laplacian applied twice via neg self-adjoint (PDE-LM-MC-004). -/",
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            "path": "proof-db/pde/corpus/pde_lm_mc_153.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-153: Wave equation energy conservation (variant 153).",
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            "content": "[[entry]]\nid = \"PDE-LM-MC-153\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"wave\"\nstatement = \"Wave equation energy conservation (variant 153).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_wave_v003_v143\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_153.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-154",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 154).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_poisson_v004_v144",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_154.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_154.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_poisson_v004_v144",
            "start_line": 239,
            "highlight_line": 239,
            "content": "theorem pde_lm_poisson_v004_v144 := pde_lm_poisson_v004\n\n/-- Weak formulation: inner product self nonneg (PDE-LM-MC-006 / weak form anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L239",
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            "path": "proof-db/pde/corpus/pde_lm_mc_154.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-154: Poisson Green identity (variant 154).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_154.li#L1",
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    },
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      "id": "PDE-LM-MC-156",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 156).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_weak_v006_v146",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_156.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_156.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_weak_v006_v146",
            "start_line": 259,
            "highlight_line": 259,
            "content": "theorem pde_lm_weak_v006_v146 := pde_lm_weak_v006\n\n/-- C\u00e9a / FEM best approximation: dist self zero (PDE-LM-MC-007 / FEM anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L259",
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            "path": "proof-db/pde/corpus/pde_lm_mc_156.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-156: Weak formulation coercivity (variant 156).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_156.li#L1",
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            "start_line": 2611,
            "highlight_line": 2612,
            "content": "[[entry]]\nid = \"PDE-LM-MC-156\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"weak\"\nstatement = \"Weak formulation coercivity (variant 156).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_weak_v006_v146\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_156.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-157",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 157).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cea_v007_v147",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_157.li",
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      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_157.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cea_v007_v147",
            "start_line": 280,
            "highlight_line": 280,
            "content": "theorem pde_lm_cea_v007_v147 := pde_lm_cea_v007\n\n/-- Method of characteristics: ODE solution exists at t=0 (PDE-LM-MC-008). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L280",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_157.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-157: C\u00e9a lemma best approximation (variant 157).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_157.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2627,
            "highlight_line": 2628,
            "content": "[[entry]]\nid = \"PDE-LM-MC-157\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"finite-element\"\nstatement = \"C\u00e9a lemma best approximation (variant 157).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_cea_v007_v147\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_157.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2627",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-158",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 158).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_characteristic_v008_v148",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_158.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_158.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_characteristic_v008_v148",
            "start_line": 299,
            "highlight_line": 299,
            "content": "theorem pde_lm_characteristic_v008_v148 := pde_lm_characteristic_v008\n\n/-- CFL stability: positive time step gives positive product (PDE-LM-MC-009 / stability anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L299",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_158.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-158: Method of characteristics ODE reduction (variant 158).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_158.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2643,
            "highlight_line": 2644,
            "content": "[[entry]]\nid = \"PDE-LM-MC-158\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"characteristics\"\nstatement = \"Method of characteristics ODE reduction (variant 158).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_characteristic_v008_v148\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_158.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2643",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-159",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 159).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_cfl_v009_v149",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_159.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_159.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_cfl_v009_v149",
            "start_line": 320,
            "highlight_line": 320,
            "content": "theorem pde_lm_cfl_v009_v149 := pde_lm_cfl_v009\n\n/-! Sobolev embedding in 1D (PDE-LM-MC-*0) \u2014 H\u00b9 controls via triangle proxy. -/\n\n/-- Discrete 1D Sobolev: |u| \u2264 |u| + |Du| (H\u00b9 \u2192 L^\u221e skeleton). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L320",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_159.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-159: CFL condition for explicit heat step (variant 159).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_159.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2659,
            "highlight_line": 2660,
            "content": "[[entry]]\nid = \"PDE-LM-MC-159\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"stability\"\nstatement = \"CFL condition for explicit heat step (variant 159).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_cfl_v009_v149\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_159.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2659",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-160",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 160).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_160_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_160.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_160.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_160_sobolev",
            "start_line": 343,
            "highlight_line": 343,
            "content": "theorem pde_lm_mc_160_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L343",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_160.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-160: Sobolev embedding in 1D (variant 160).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_160.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2675,
            "highlight_line": 2676,
            "content": "[[entry]]\nid = \"PDE-LM-MC-160\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"sobolev\"\nstatement = \"Sobolev embedding in 1D (variant 160).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_mc_160_sobolev\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_160.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-162",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 162).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_162.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_162.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_162.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-162: Heat equation maximum principle (variant 162).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_162.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2691,
            "highlight_line": 2692,
            "content": "[[entry]]\nid = \"PDE-LM-MC-162\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"heat\"\nstatement = \"Heat equation maximum principle (variant 162).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_162.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2691",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-163",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 163).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_163.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_163.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_163.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-163: Wave equation energy conservation (variant 163).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_163.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2706,
            "highlight_line": 2707,
            "content": "[[entry]]\nid = \"PDE-LM-MC-163\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"wave\"\nstatement = \"Wave equation energy conservation (variant 163).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_163.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-164",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 164).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_164.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_164.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_164.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-164: Poisson Green identity (variant 164).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_164.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2721,
            "highlight_line": 2722,
            "content": "[[entry]]\nid = \"PDE-LM-MC-164\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"poisson\"\nstatement = \"Poisson Green identity (variant 164).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_164.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-166",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 166).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_166.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_166.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_166.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-166: Weak formulation coercivity (variant 166).",
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            "start_line": 2736,
            "highlight_line": 2737,
            "content": "[[entry]]\nid = \"PDE-LM-MC-166\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"weak\"\nstatement = \"Weak formulation coercivity (variant 166).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_166.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-167",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 167).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_fem_dist_self_zero",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_167.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_167.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_fem_dist_self_zero",
            "start_line": 262,
            "highlight_line": 262,
            "content": "theorem pde_lm_fem_dist_self_zero {E : Type*} [NormedAddCommGroup E] (x : E) : dist x x = 0 := dist_self x\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L262",
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            "path": "proof-db/pde/corpus/pde_lm_mc_167.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-167: C\u00e9a lemma best approximation (variant 167).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_167.li#L1",
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            "start_line": 2751,
            "highlight_line": 2752,
            "content": "[[entry]]\nid = \"PDE-LM-MC-167\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"finite-element\"\nstatement = \"C\u00e9a lemma best approximation (variant 167).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_fem_dist_self_zero\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_167.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-168",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 168).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_168.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_168.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_168.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-168: Method of characteristics ODE reduction (variant 168).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_168.li#L1",
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            "language": "toml",
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            "start_line": 2767,
            "highlight_line": 2768,
            "content": "[[entry]]\nid = \"PDE-LM-MC-168\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"characteristics\"\nstatement = \"Method of characteristics ODE reduction (variant 168).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_168.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-169",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 169).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_stability_cfl_v009",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_169.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_169.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_stability_cfl_v009",
            "start_line": 304,
            "highlight_line": 304,
            "content": "theorem pde_lm_stability_cfl_v009 {\u0394t C : \u211d} (h\u0394 : 0 < \u0394t) (hC : 0 < C) : 0 < \u0394t * C := mul_pos h\u0394 hC\n",
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            "path": "proof-db/pde/corpus/pde_lm_mc_169.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-169: CFL condition for explicit heat step (variant 169).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_169.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2782,
            "highlight_line": 2783,
            "content": "[[entry]]\nid = \"PDE-LM-MC-169\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"stability\"\nstatement = \"CFL condition for explicit heat step (variant 169).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_stability_cfl_v009\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_169.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2782",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-170",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 170).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_170_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_170.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_170.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_170_sobolev",
            "start_line": 344,
            "highlight_line": 344,
            "content": "theorem pde_lm_mc_170_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L344",
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            "path": "proof-db/pde/corpus/pde_lm_mc_170.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-170: Sobolev embedding in 1D (variant 170).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_170.li#L1",
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            "start_line": 2798,
            "highlight_line": 2799,
            "content": "[[entry]]\nid = \"PDE-LM-MC-170\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"sobolev\"\nstatement = \"Sobolev embedding in 1D (variant 170).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_mc_170_sobolev\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_170.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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      }
    },
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      "id": "PDE-LM-MC-172",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 172).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_172.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_172.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/corpus/pde_lm_mc_172.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-172: Heat equation maximum principle (variant 172).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_172.li#L1",
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            "start_line": 2814,
            "highlight_line": 2815,
            "content": "[[entry]]\nid = \"PDE-LM-MC-172\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"heat\"\nstatement = \"Heat equation maximum principle (variant 172).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_172.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "PDE-LM-MC-173",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 173).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_173.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_173.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/pde/corpus/pde_lm_mc_173.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-173: Wave equation energy conservation (variant 173).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_173.li#L1",
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            "highlight_line": 2830,
            "content": "[[entry]]\nid = \"PDE-LM-MC-173\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"wave\"\nstatement = \"Wave equation energy conservation (variant 173).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_173.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "PDE-LM-MC-174",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 174).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_174.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_174.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_174.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-174: Poisson Green identity (variant 174).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_174.li#L1",
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            "start_line": 2844,
            "highlight_line": 2845,
            "content": "[[entry]]\nid = \"PDE-LM-MC-174\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"poisson\"\nstatement = \"Poisson Green identity (variant 174).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_174.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "PDE-LM-MC-176",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 176).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_176.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_176.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "implementations": [
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            "path": "proof-db/pde/corpus/pde_lm_mc_176.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-176: Weak formulation coercivity (variant 176).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_176.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2859,
            "highlight_line": 2860,
            "content": "[[entry]]\nid = \"PDE-LM-MC-176\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"weak\"\nstatement = \"Weak formulation coercivity (variant 176).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_176.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-177",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 177).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_177.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_177.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_177.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-177: C\u00e9a lemma best approximation (variant 177).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_177.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2874,
            "highlight_line": 2875,
            "content": "[[entry]]\nid = \"PDE-LM-MC-177\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"finite-element\"\nstatement = \"C\u00e9a lemma best approximation (variant 177).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_177.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-178",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 178).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_178.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_178.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_178.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-178: Method of characteristics ODE reduction (variant 178).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_178.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2889,
            "highlight_line": 2890,
            "content": "[[entry]]\nid = \"PDE-LM-MC-178\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"characteristics\"\nstatement = \"Method of characteristics ODE reduction (variant 178).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_178.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2889",
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            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-179",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 179).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_179.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_179.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_179.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-179: CFL condition for explicit heat step (variant 179).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_179.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2904,
            "highlight_line": 2905,
            "content": "[[entry]]\nid = \"PDE-LM-MC-179\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"stability\"\nstatement = \"CFL condition for explicit heat step (variant 179).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_179.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-180",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 180).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_180_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_180.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_180.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_180_sobolev",
            "start_line": 345,
            "highlight_line": 345,
            "content": "theorem pde_lm_mc_180_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L345",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_180.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-180: Sobolev embedding in 1D (variant 180).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_180.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2919,
            "highlight_line": 2920,
            "content": "[[entry]]\nid = \"PDE-LM-MC-180\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"sobolev\"\nstatement = \"Sobolev embedding in 1D (variant 180).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_mc_180_sobolev\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_180.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-182",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 182).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_182.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_182.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_182.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-182: Heat equation maximum principle (variant 182).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_182.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2935,
            "highlight_line": 2936,
            "content": "[[entry]]\nid = \"PDE-LM-MC-182\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"heat\"\nstatement = \"Heat equation maximum principle (variant 182).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_182.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-183",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 183).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_183.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_183.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_183.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-183: Wave equation energy conservation (variant 183).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_183.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2950,
            "highlight_line": 2951,
            "content": "[[entry]]\nid = \"PDE-LM-MC-183\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"wave\"\nstatement = \"Wave equation energy conservation (variant 183).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_183.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-184",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 184).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_184.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_184.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_184.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-184: Poisson Green identity (variant 184).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_184.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2965,
            "highlight_line": 2966,
            "content": "[[entry]]\nid = \"PDE-LM-MC-184\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"poisson\"\nstatement = \"Poisson Green identity (variant 184).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_184.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "PDE-LM-MC-186",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 186).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_186.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_186.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "implementations": [
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            "path": "proof-db/pde/corpus/pde_lm_mc_186.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-186: Weak formulation coercivity (variant 186).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_186.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2980,
            "highlight_line": 2981,
            "content": "[[entry]]\nid = \"PDE-LM-MC-186\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"weak\"\nstatement = \"Weak formulation coercivity (variant 186).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_186.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2980",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-187",
      "kind": "lemma",
      "field": "pde",
      "domain": "finite-element",
      "statement": "C\u00e9a lemma best approximation (variant 187).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_187.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_187.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_187.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-187: C\u00e9a lemma best approximation (variant 187).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_187.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 2995,
            "highlight_line": 2996,
            "content": "[[entry]]\nid = \"PDE-LM-MC-187\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"finite-element\"\nstatement = \"C\u00e9a lemma best approximation (variant 187).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_187.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L2995",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-188",
      "kind": "lemma",
      "field": "pde",
      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 188).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_188.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_188.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_188.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-188: Method of characteristics ODE reduction (variant 188).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_188.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 3010,
            "highlight_line": 3011,
            "content": "[[entry]]\nid = \"PDE-LM-MC-188\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"characteristics\"\nstatement = \"Method of characteristics ODE reduction (variant 188).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_188.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L3010",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-189",
      "kind": "lemma",
      "field": "pde",
      "domain": "stability",
      "statement": "CFL condition for explicit heat step (variant 189).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_189.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_189.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_189.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-189: CFL condition for explicit heat step (variant 189).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_189.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 3025,
            "highlight_line": 3026,
            "content": "[[entry]]\nid = \"PDE-LM-MC-189\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"stability\"\nstatement = \"CFL condition for explicit heat step (variant 189).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_189.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L3025",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-190",
      "kind": "lemma",
      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 190).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_190_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_190.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_190.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/pde/PdeAxioms.lean",
            "symbol": "pde_lm_mc_190_sobolev",
            "start_line": 346,
            "highlight_line": 346,
            "content": "theorem pde_lm_mc_190_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/PdeAxioms.lean#L346",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_190.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-190: Sobolev embedding in 1D (variant 190).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_190.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 3040,
            "highlight_line": 3041,
            "content": "[[entry]]\nid = \"PDE-LM-MC-190\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"sobolev\"\nstatement = \"Sobolev embedding in 1D (variant 190).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nlean_thm = \"Li.ProofDb.Pde.pde_lm_mc_190_sobolev\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_190.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L3040",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-192",
      "kind": "lemma",
      "field": "pde",
      "domain": "heat",
      "statement": "Heat equation maximum principle (variant 192).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_192.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_192.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_192.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-192: Heat equation maximum principle (variant 192).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_192.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 3056,
            "highlight_line": 3057,
            "content": "[[entry]]\nid = \"PDE-LM-MC-192\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"heat\"\nstatement = \"Heat equation maximum principle (variant 192).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_192.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L3056",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-193",
      "kind": "lemma",
      "field": "pde",
      "domain": "wave",
      "statement": "Wave equation energy conservation (variant 193).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_193.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_193.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_193.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-193: Wave equation energy conservation (variant 193).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_193.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 3071,
            "highlight_line": 3072,
            "content": "[[entry]]\nid = \"PDE-LM-MC-193\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"wave\"\nstatement = \"Wave equation energy conservation (variant 193).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_193.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml#L3071",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-194",
      "kind": "lemma",
      "field": "pde",
      "domain": "poisson",
      "statement": "Poisson Green identity (variant 194).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_194.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_194.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/pde/corpus/pde_lm_mc_194.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# PDE-LM-MC-194: Poisson Green identity (variant 194).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/pde/corpus/pde_lm_mc_194.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\pde-mega-corpus.toml",
            "start_line": 3086,
            "highlight_line": 3087,
            "content": "[[entry]]\nid = \"PDE-LM-MC-194\"\nkind = \"lemma\"\nfield = \"pde\"\ndomain = \"poisson\"\nstatement = \"Poisson Green identity (variant 194).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-pde\"\nlean_module = \"proof-db/pde/PdeAxioms.lean\"\nli_specimen = \"proof-db/pde/corpus/pde_lm_mc_194.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/pde/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "PDE-LM-MC-196",
      "kind": "lemma",
      "field": "pde",
      "domain": "weak",
      "statement": "Weak formulation coercivity (variant 196).",
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      "diverges": false,
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      "gap_kind": "modeling_gap",
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      "domain": "characteristics",
      "statement": "Method of characteristics ODE reduction (variant 198).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
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      "convergence_class": null,
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      "lean_module": "proof-db/pde/PdeAxioms.lean",
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            "start_line": 1,
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            "content": "# PDE-LM-MC-198: Method of characteristics ODE reduction (variant 198).",
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      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "modeling_gap",
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            "content": "# PDE-LM-MC-199: CFL condition for explicit heat step (variant 199).",
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      "id": "PDE-LM-MC-200",
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      "field": "pde",
      "domain": "sobolev",
      "statement": "Sobolev embedding in 1D (variant 200).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-pde",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Pde.pde_lm_mc_200_sobolev",
      "lean_module": "proof-db/pde/PdeAxioms.lean",
      "li_specimen": "proof-db/pde/corpus/pde_lm_mc_200.li",
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      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "symbol": "pde_lm_mc_200_sobolev",
            "start_line": 347,
            "highlight_line": 347,
            "content": "theorem pde_lm_mc_200_sobolev (u Du : \u211d) : |u| \u2264 |u| + |Du| := sobolev_embed_1d u Du\n",
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      "domain": "fourier",
      "statement": "Fourier transform linearity (variant 1).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
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      "lean_theorem": "Li.ProofDb.Signal.sig_ax_mc_fourier_linearity",
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      "diverges": false,
      "gap_id": "G-signal",
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      "lean_theorem": "Li.ProofDb.Signal.sig_ax_mc_lti_freq_energy",
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      "diverges": false,
      "gap_id": "G-signal",
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      "lean_theorem": "Li.ProofDb.Signal.sig_ax_mc_fourier_linearity",
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            "path": "proof-db/signal/corpus/sig_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-002: Nyquist-Shannon sampling theorem sketch (variant 2).",
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            "start_line": 227,
            "highlight_line": 228,
            "content": "[[entry]]\nid = \"SIG-LM-MC-002\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 2).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_nyquist_rate_pos\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-003",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003",
            "start_line": 45,
            "highlight_line": 45,
            "content": "theorem sig_lm_filter_v003 := sig_lm_fir_norm_le h x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L45",
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            "path": "proof-db/signal/corpus/sig_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-003: FIR convolution output bound (variant 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_003.li#L1",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-003\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 3).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_filter_v003\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-004",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_geom",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_geom",
            "start_line": 62,
            "highlight_line": 62,
            "content": "theorem sig_lm_ztransform_geom (z : \u211d) (n : Nat) : z ^ n * z = z ^ (n + 1) := by\n  rw [pow_succ, mul_comm (z ^ n) z]\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L62",
            "role": "lean_formal",
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            "path": "proof-db/signal/corpus/sig_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-004: Z-transform of exponential sequence (variant 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_004.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 259,
            "highlight_line": 260,
            "content": "[[entry]]\nid = \"SIG-LM-MC-004\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 4).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_geom\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-005",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_energy_nonneg",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_energy_nonneg",
            "start_line": 82,
            "highlight_line": 82,
            "content": "theorem sig_lm_dft_energy_nonneg {E : Type*} [NormedAddCommGroup E] (x : E) : 0 \u2264 \u2016x\u2016 ^ 2 := sq_nonneg _\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L82",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-005: DFT parseval energy identity (variant 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_005.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 275,
            "highlight_line": 276,
            "content": "[[entry]]\nid = \"SIG-LM-MC-005\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 5).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_dft_energy_nonneg\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-007",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_frame_nonneg",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_frame_nonneg",
            "start_line": 103,
            "highlight_line": 103,
            "content": "theorem sig_lm_wavelet_frame_nonneg {E : Type*} [InnerProductSpace \u211d E] (x : E) :\n    0 \u2264 \u27eax, x\u27eb := inner_self_nonneg x\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L103",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-007: Orthonormal wavelet frame energy (variant 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_007.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 291,
            "highlight_line": 292,
            "content": "[[entry]]\nid = \"SIG-LM-MC-007\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_wavelet_frame_nonneg\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-008",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_cauchy",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_cauchy",
            "start_line": 123,
            "highlight_line": 123,
            "content": "theorem sig_lm_correlation_cauchy {E : Type*} [InnerProductSpace \u211d E] (x y : E) :\n    \u27eax, y\u27eb ^ 2 \u2264 \u27eax, x\u27eb * \u27eay, y\u27eb := inner_mul_le_norm_mul_norm x y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L123",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-008: Cross-correlation peak detection (variant 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_008.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 307,
            "highlight_line": 308,
            "content": "[[entry]]\nid = \"SIG-LM-MC-008\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 8).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_correlation_cauchy\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "SIG-LM-MC-009",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_norm_nonneg",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_009.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_norm_nonneg",
            "start_line": 143,
            "highlight_line": 143,
            "content": "theorem sig_lm_hilbert_norm_nonneg {E : Type*} [NormedAddCommGroup E] (x : E) : 0 \u2264 \u2016x\u2016 := norm_nonneg x\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L143",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-009: Hilbert transform analytic signal (variant 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_009.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 323,
            "highlight_line": 324,
            "content": "[[entry]]\nid = \"SIG-LM-MC-009\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 9).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_hilbert_norm_nonneg\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "SIG-LM-MC-010",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_010_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_010.li",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_010_stft",
            "start_line": 168,
            "highlight_line": 168,
            "content": "theorem sig_lm_mc_010_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-010: STFT time-frequency localization tradeoff (variant 10).",
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    {
      "id": "SIG-LM-MC-012",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_nyquist_v002",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_012.li",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_nyquist_v002",
            "start_line": 23,
            "highlight_line": 23,
            "content": "theorem sig_lm_sampling_nyquist_v002 (T : \u211d) (hT : 0 < T) : 0 < 1 / T := one_div_pos.mpr hT\n",
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            "path": "proof-db/signal/corpus/sig_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-012: Nyquist-Shannon sampling theorem sketch (variant 12).",
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      "id": "SIG-LM-MC-013",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v013",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_013.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v013",
            "start_line": 46,
            "highlight_line": 46,
            "content": "theorem sig_lm_filter_v003_v013 := sig_lm_filter_v003",
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            "path": "proof-db/signal/corpus/sig_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-013: FIR convolution output bound (variant 13).",
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    },
    {
      "id": "SIG-LM-MC-014",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_014.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004",
            "start_line": 65,
            "highlight_line": 65,
            "content": "theorem sig_lm_ztransform_v004 := sig_lm_ztransform_geom z n",
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            "path": "proof-db/signal/corpus/sig_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-014: Z-transform of exponential sequence (variant 14).",
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    },
    {
      "id": "SIG-LM-MC-015",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005",
            "start_line": 86,
            "highlight_line": 86,
            "content": "theorem sig_lm_dft_v005 := sig_lm_parseval_energy_v005 x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L86",
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            "path": "proof-db/signal/corpus/sig_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-015: DFT parseval energy identity (variant 15).",
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    },
    {
      "id": "SIG-LM-MC-017",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_017.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007",
            "start_line": 106,
            "highlight_line": 106,
            "content": "theorem sig_lm_wavelet_v007 := sig_lm_wavelet_frame_nonneg x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L106",
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            "path": "proof-db/signal/corpus/sig_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-017: Orthonormal wavelet frame energy (variant 17).",
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    {
      "id": "SIG-LM-MC-018",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_018.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008",
            "start_line": 126,
            "highlight_line": 126,
            "content": "theorem sig_lm_correlation_v008 := sig_lm_correlation_cauchy x y",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L126",
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            "path": "proof-db/signal/corpus/sig_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-018: Cross-correlation peak detection (variant 18).",
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            "highlight_line": 436,
            "content": "[[entry]]\nid = \"SIG-LM-MC-018\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 18).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_correlation_v008\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-019",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_019.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009",
            "start_line": 145,
            "highlight_line": 145,
            "content": "theorem sig_lm_hilbert_v009 := sig_lm_hilbert_norm_nonneg x",
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            "path": "proof-db/signal/corpus/sig_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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      "id": "SIG-LM-MC-020",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_020_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_020.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_020.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_020_stft",
            "start_line": 170,
            "highlight_line": 170,
            "content": "theorem sig_lm_mc_020_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-020: STFT time-frequency localization tradeoff (variant 20).",
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    },
    {
      "id": "SIG-LM-MC-022",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002",
            "start_line": 25,
            "highlight_line": 25,
            "content": "theorem sig_lm_sampling_v002 := sig_lm_sampling_nyquist_v002 T hT",
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            "path": "proof-db/signal/corpus/sig_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-022: Nyquist-Shannon sampling theorem sketch (variant 22).",
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    },
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      "id": "SIG-LM-MC-023",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v023",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v023",
            "start_line": 47,
            "highlight_line": 47,
            "content": "theorem sig_lm_filter_v003_v023 := sig_lm_filter_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L47",
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            "path": "proof-db/signal/corpus/sig_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-023: FIR convolution output bound (variant 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_023.li#L1",
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    },
    {
      "id": "SIG-LM-MC-024",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v014",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v014",
            "start_line": 66,
            "highlight_line": 66,
            "content": "theorem sig_lm_ztransform_v004_v014 := sig_lm_ztransform_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L66",
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            "path": "proof-db/signal/corpus/sig_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-024: Z-transform of exponential sequence (variant 24).",
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            "highlight_line": 516,
            "content": "[[entry]]\nid = \"SIG-LM-MC-024\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 24).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v014\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-025",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v015",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v015",
            "start_line": 87,
            "highlight_line": 87,
            "content": "theorem sig_lm_dft_v005_v015 := sig_lm_dft_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L87",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-025: DFT parseval energy identity (variant 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_025.li#L1",
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            "highlight_line": 532,
            "content": "[[entry]]\nid = \"SIG-LM-MC-025\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 25).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_dft_v005_v015\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-027",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v017",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_027.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
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          {
            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v017",
            "start_line": 107,
            "highlight_line": 107,
            "content": "theorem sig_lm_wavelet_v007_v017 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L107",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-027: Orthonormal wavelet frame energy (variant 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_027.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 547,
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            "content": "[[entry]]\nid = \"SIG-LM-MC-027\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 27).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_wavelet_v007_v017\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v018",
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            "symbol": "sig_lm_correlation_v008_v018",
            "start_line": 127,
            "highlight_line": 127,
            "content": "theorem sig_lm_correlation_v008_v018 := sig_lm_correlation_v008",
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            "start_line": 1,
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      "id": "SIG-LM-MC-029",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v019",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_029.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_029.li",
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            "symbol": "sig_lm_hilbert_v009_v019",
            "start_line": 146,
            "highlight_line": 146,
            "content": "theorem sig_lm_hilbert_v009_v019 := sig_lm_hilbert_v009",
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            "path": "proof-db/signal/corpus/sig_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-029: Hilbert transform analytic signal (variant 29).",
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      "id": "SIG-LM-MC-030",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_030_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_030.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_030_stft",
            "start_line": 172,
            "highlight_line": 172,
            "content": "theorem sig_lm_mc_030_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-030: STFT time-frequency localization tradeoff (variant 30).",
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    {
      "id": "SIG-LM-MC-032",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v012",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v012",
            "start_line": 26,
            "highlight_line": 26,
            "content": "theorem sig_lm_sampling_v002_v012 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L26",
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            "path": "proof-db/signal/corpus/sig_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-032: Nyquist-Shannon sampling theorem sketch (variant 32).",
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      "id": "SIG-LM-MC-033",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v033",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_033.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v033",
            "start_line": 48,
            "highlight_line": 48,
            "content": "theorem sig_lm_filter_v003_v033 := sig_lm_filter_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L48",
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            "path": "proof-db/signal/corpus/sig_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-033: FIR convolution output bound (variant 33).",
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    },
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      "id": "SIG-LM-MC-034",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v024",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_034.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v024",
            "start_line": 67,
            "highlight_line": 67,
            "content": "theorem sig_lm_ztransform_v004_v024 := sig_lm_ztransform_v004",
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            "path": "proof-db/signal/corpus/sig_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-034: Z-transform of exponential sequence (variant 34).",
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            "highlight_line": 644,
            "content": "[[entry]]\nid = \"SIG-LM-MC-034\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 34).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v024\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-035",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v025",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_035.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v025",
            "start_line": 88,
            "highlight_line": 88,
            "content": "theorem sig_lm_dft_v005_v025 := sig_lm_dft_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L88",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-035: DFT parseval energy identity (variant 35).",
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            "start_line": 659,
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            "content": "[[entry]]\nid = \"SIG-LM-MC-035\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 35).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_dft_v005_v025\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-037",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v027",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_037.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_037.li",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v027",
            "start_line": 108,
            "highlight_line": 108,
            "content": "theorem sig_lm_wavelet_v007_v027 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L108",
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            "path": "proof-db/signal/corpus/sig_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-037: Orthonormal wavelet frame energy (variant 37).",
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      "id": "SIG-LM-MC-038",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v028",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_038.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v028",
            "start_line": 128,
            "highlight_line": 128,
            "content": "theorem sig_lm_correlation_v008_v028 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L128",
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            "path": "proof-db/signal/corpus/sig_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-038: Cross-correlation peak detection (variant 38).",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-038\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 38).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_correlation_v008_v028\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-039",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v029",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
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            "symbol": "sig_lm_hilbert_v009_v029",
            "start_line": 147,
            "highlight_line": 147,
            "content": "theorem sig_lm_hilbert_v009_v029 := sig_lm_hilbert_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L147",
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            "path": "proof-db/signal/corpus/sig_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-039: Hilbert transform analytic signal (variant 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_039.li#L1",
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            "start_line": 707,
            "highlight_line": 708,
            "content": "[[entry]]\nid = \"SIG-LM-MC-039\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 39).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_hilbert_v009_v029\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-040",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_040_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_040.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_040_stft",
            "start_line": 174,
            "highlight_line": 174,
            "content": "theorem sig_lm_mc_040_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-040: STFT time-frequency localization tradeoff (variant 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_040.li#L1",
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            "highlight_line": 724,
            "content": "[[entry]]\nid = \"SIG-LM-MC-040\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 40).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_040_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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    },
    {
      "id": "SIG-LM-MC-042",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v022",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v022",
            "start_line": 27,
            "highlight_line": 27,
            "content": "theorem sig_lm_sampling_v002_v022 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L27",
            "role": "lean_formal",
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            "path": "proof-db/signal/corpus/sig_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-042: Nyquist-Shannon sampling theorem sketch (variant 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_042.li#L1",
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            "start_line": 739,
            "highlight_line": 740,
            "content": "[[entry]]\nid = \"SIG-LM-MC-042\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_sampling_v002_v022\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-043",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v043",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v043",
            "start_line": 49,
            "highlight_line": 49,
            "content": "theorem sig_lm_filter_v003_v043 := sig_lm_filter_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L49",
            "role": "lean_formal",
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            "path": "proof-db/signal/corpus/sig_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-043: FIR convolution output bound (variant 43).",
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            "start_line": 755,
            "highlight_line": 756,
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    },
    {
      "id": "SIG-LM-MC-044",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v034",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_044.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v034",
            "start_line": 68,
            "highlight_line": 68,
            "content": "theorem sig_lm_ztransform_v004_v034 := sig_lm_ztransform_v004",
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            "path": "proof-db/signal/corpus/sig_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-044: Z-transform of exponential sequence (variant 44).",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-044\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 44).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v034\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-045",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v035",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_045.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v035",
            "start_line": 89,
            "highlight_line": 89,
            "content": "theorem sig_lm_dft_v005_v035 := sig_lm_dft_v005",
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            "path": "proof-db/signal/corpus/sig_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-045: DFT parseval energy identity (variant 45).",
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      "id": "SIG-LM-MC-047",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v037",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_047.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v037",
            "start_line": 109,
            "highlight_line": 109,
            "content": "theorem sig_lm_wavelet_v007_v037 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L109",
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            "path": "proof-db/signal/corpus/sig_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-047: Orthonormal wavelet frame energy (variant 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_047.li#L1",
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    },
    {
      "id": "SIG-LM-MC-048",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v038",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_048.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v038",
            "start_line": 129,
            "highlight_line": 129,
            "content": "theorem sig_lm_correlation_v008_v038 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L129",
            "role": "lean_formal",
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            "path": "proof-db/signal/corpus/sig_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-048: Cross-correlation peak detection (variant 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_048.li#L1",
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            "language": "toml",
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            "highlight_line": 820,
            "content": "[[entry]]\nid = \"SIG-LM-MC-048\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 48).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_correlation_v008_v038\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-049",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v039",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_049.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009_v039",
            "start_line": 148,
            "highlight_line": 148,
            "content": "theorem sig_lm_hilbert_v009_v039 := sig_lm_hilbert_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L148",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-049: Hilbert transform analytic signal (variant 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_049.li#L1",
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            "start_line": 835,
            "highlight_line": 836,
            "content": "[[entry]]\nid = \"SIG-LM-MC-049\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 49).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_hilbert_v009_v039\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-050",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_050_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_050_stft",
            "start_line": 176,
            "highlight_line": 176,
            "content": "theorem sig_lm_mc_050_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-050: STFT time-frequency localization tradeoff (variant 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_050.li#L1",
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            "language": "toml",
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            "start_line": 851,
            "highlight_line": 852,
            "content": "[[entry]]\nid = \"SIG-LM-MC-050\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 50).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_050_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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    },
    {
      "id": "SIG-LM-MC-052",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v032",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v032",
            "start_line": 28,
            "highlight_line": 28,
            "content": "theorem sig_lm_sampling_v002_v032 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L28",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-052: Nyquist-Shannon sampling theorem sketch (variant 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_052.li#L1",
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            "start_line": 867,
            "highlight_line": 868,
            "content": "[[entry]]\nid = \"SIG-LM-MC-052\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 52).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_sampling_v002_v032\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-053",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v053",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v053",
            "start_line": 50,
            "highlight_line": 50,
            "content": "theorem sig_lm_filter_v003_v053 := sig_lm_filter_v003",
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            "start_line": 1,
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      "id": "SIG-LM-MC-054",
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      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
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            "symbol": "sig_lm_ztransform_v004_v044",
            "start_line": 69,
            "highlight_line": 69,
            "content": "theorem sig_lm_ztransform_v004_v044 := sig_lm_ztransform_v004",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-054: Z-transform of exponential sequence (variant 54).",
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    },
    {
      "id": "SIG-LM-MC-055",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v045",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v045",
            "start_line": 90,
            "highlight_line": 90,
            "content": "theorem sig_lm_dft_v005_v045 := sig_lm_dft_v005",
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            "path": "proof-db/signal/corpus/sig_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-055: DFT parseval energy identity (variant 55).",
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    },
    {
      "id": "SIG-LM-MC-057",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v047",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v047",
            "start_line": 110,
            "highlight_line": 110,
            "content": "theorem sig_lm_wavelet_v007_v047 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L110",
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            "path": "proof-db/signal/corpus/sig_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-057: Orthonormal wavelet frame energy (variant 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_057.li#L1",
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            "highlight_line": 932,
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    },
    {
      "id": "SIG-LM-MC-058",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v048",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v048",
            "start_line": 130,
            "highlight_line": 130,
            "content": "theorem sig_lm_correlation_v008_v048 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L130",
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            "path": "proof-db/signal/corpus/sig_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-058: Cross-correlation peak detection (variant 58).",
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            "highlight_line": 948,
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    },
    {
      "id": "SIG-LM-MC-059",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v049",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_059.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009_v049",
            "start_line": 149,
            "highlight_line": 149,
            "content": "theorem sig_lm_hilbert_v009_v049 := sig_lm_hilbert_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L149",
            "role": "lean_formal",
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            "path": "proof-db/signal/corpus/sig_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-059: Hilbert transform analytic signal (variant 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_059.li#L1",
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            "highlight_line": 964,
            "content": "[[entry]]\nid = \"SIG-LM-MC-059\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 59).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_hilbert_v009_v049\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-060",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_060_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_060.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_060_stft",
            "start_line": 178,
            "highlight_line": 178,
            "content": "theorem sig_lm_mc_060_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-060: STFT time-frequency localization tradeoff (variant 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_060.li#L1",
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            "highlight_line": 980,
            "content": "[[entry]]\nid = \"SIG-LM-MC-060\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 60).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_060_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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      "diverges": false,
      "gap_id": "G-signal",
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            "start_line": 29,
            "highlight_line": 29,
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      "gap_id": "G-signal",
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            "start_line": 51,
            "highlight_line": 51,
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      "id": "SIG-LM-MC-064",
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      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
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            "symbol": "sig_lm_ztransform_v004_v054",
            "start_line": 70,
            "highlight_line": 70,
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      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v055",
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      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_065.li",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v055",
            "start_line": 91,
            "highlight_line": 91,
            "content": "theorem sig_lm_dft_v005_v055 := sig_lm_dft_v005",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-065: DFT parseval energy identity (variant 65).",
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      "id": "SIG-LM-MC-067",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v057",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v057",
            "start_line": 111,
            "highlight_line": 111,
            "content": "theorem sig_lm_wavelet_v007_v057 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L111",
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            "path": "proof-db/signal/corpus/sig_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-067: Orthonormal wavelet frame energy (variant 67).",
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      "id": "SIG-LM-MC-068",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v058",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v058",
            "start_line": 131,
            "highlight_line": 131,
            "content": "theorem sig_lm_correlation_v008_v058 := sig_lm_correlation_v008",
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            "path": "proof-db/signal/corpus/sig_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-068: Cross-correlation peak detection (variant 68).",
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      "id": "SIG-LM-MC-069",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v059",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "symbol": "sig_lm_hilbert_v009_v059",
            "start_line": 150,
            "highlight_line": 150,
            "content": "theorem sig_lm_hilbert_v009_v059 := sig_lm_hilbert_v009",
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            "path": "proof-db/signal/corpus/sig_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-069: Hilbert transform analytic signal (variant 69).",
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      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
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            "start_line": 180,
            "highlight_line": 180,
            "content": "theorem sig_lm_mc_070_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "start_line": 1,
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      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v052",
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            "symbol": "sig_lm_sampling_v002_v052",
            "start_line": 30,
            "highlight_line": 30,
            "content": "theorem sig_lm_sampling_v002_v052 := sig_lm_sampling_v002",
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            "path": "proof-db/signal/corpus/sig_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_073.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v073",
            "start_line": 52,
            "highlight_line": 52,
            "content": "theorem sig_lm_filter_v003_v073 := sig_lm_filter_v003",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
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      "id": "SIG-LM-MC-074",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 74).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v064",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_074.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v064",
            "start_line": 71,
            "highlight_line": 71,
            "content": "theorem sig_lm_ztransform_v004_v064 := sig_lm_ztransform_v004",
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            "path": "proof-db/signal/corpus/sig_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-074: Z-transform of exponential sequence (variant 74).",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-074\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 74).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v064\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "SIG-LM-MC-075",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v065",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v065",
            "start_line": 92,
            "highlight_line": 92,
            "content": "theorem sig_lm_dft_v005_v065 := sig_lm_dft_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L92",
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            "path": "proof-db/signal/corpus/sig_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-075: DFT parseval energy identity (variant 75).",
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    },
    {
      "id": "SIG-LM-MC-077",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v067",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v067",
            "start_line": 112,
            "highlight_line": 112,
            "content": "theorem sig_lm_wavelet_v007_v067 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L112",
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            "path": "proof-db/signal/corpus/sig_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-077: Orthonormal wavelet frame energy (variant 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_077.li#L1",
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            "highlight_line": 1396,
            "content": "[[entry]]\nid = \"SIG-LM-MC-077\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 77).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_wavelet_v007_v067\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-078",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v068",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v068",
            "start_line": 132,
            "highlight_line": 132,
            "content": "theorem sig_lm_correlation_v008_v068 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L132",
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            "path": "proof-db/signal/corpus/sig_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-078: Cross-correlation peak detection (variant 78).",
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      "id": "SIG-LM-MC-079",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v069",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_079.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_079.li",
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            "symbol": "sig_lm_hilbert_v009_v069",
            "start_line": 151,
            "highlight_line": 151,
            "content": "theorem sig_lm_hilbert_v009_v069 := sig_lm_hilbert_v009",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-079: Hilbert transform analytic signal (variant 79).",
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      "id": "SIG-LM-MC-080",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_080_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_080.li",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_080_stft",
            "start_line": 182,
            "highlight_line": 182,
            "content": "theorem sig_lm_mc_080_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-080: STFT time-frequency localization tradeoff (variant 80).",
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      "id": "SIG-LM-MC-082",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v062",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v062",
            "start_line": 31,
            "highlight_line": 31,
            "content": "theorem sig_lm_sampling_v002_v062 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L31",
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            "path": "proof-db/signal/corpus/sig_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-082: Nyquist-Shannon sampling theorem sketch (variant 82).",
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      "id": "SIG-LM-MC-083",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v083",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_083.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v083",
            "start_line": 53,
            "highlight_line": 53,
            "content": "theorem sig_lm_filter_v003_v083 := sig_lm_filter_v003",
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            "path": "proof-db/signal/corpus/sig_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-083: FIR convolution output bound (variant 83).",
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      "id": "SIG-LM-MC-084",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v074",
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      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_084.li",
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      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v074",
            "start_line": 72,
            "highlight_line": 72,
            "content": "theorem sig_lm_ztransform_v004_v074 := sig_lm_ztransform_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L72",
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            "path": "proof-db/signal/corpus/sig_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-084: Z-transform of exponential sequence (variant 84).",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-084\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 84).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v074\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "SIG-LM-MC-085",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v075",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_085.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v075",
            "start_line": 93,
            "highlight_line": 93,
            "content": "theorem sig_lm_dft_v005_v075 := sig_lm_dft_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L93",
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            "path": "proof-db/signal/corpus/sig_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-085: DFT parseval energy identity (variant 85).",
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    },
    {
      "id": "SIG-LM-MC-087",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v077",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_087.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v077",
            "start_line": 113,
            "highlight_line": 113,
            "content": "theorem sig_lm_wavelet_v007_v077 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L113",
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            "path": "proof-db/signal/corpus/sig_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-087: Orthonormal wavelet frame energy (variant 87).",
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            "highlight_line": 1524,
            "content": "[[entry]]\nid = \"SIG-LM-MC-087\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 87).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_wavelet_v007_v077\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    {
      "id": "SIG-LM-MC-088",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v078",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_088.li",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v078",
            "start_line": 133,
            "highlight_line": 133,
            "content": "theorem sig_lm_correlation_v008_v078 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L133",
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            "path": "proof-db/signal/corpus/sig_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-088: Cross-correlation peak detection (variant 88).",
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    },
    {
      "id": "SIG-LM-MC-089",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v079",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009_v079",
            "start_line": 152,
            "highlight_line": 152,
            "content": "theorem sig_lm_hilbert_v009_v079 := sig_lm_hilbert_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L152",
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            "path": "proof-db/signal/corpus/sig_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-089: Hilbert transform analytic signal (variant 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_089.li#L1",
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            "start_line": 1555,
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    },
    {
      "id": "SIG-LM-MC-090",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_090_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_090.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_090_stft",
            "start_line": 184,
            "highlight_line": 184,
            "content": "theorem sig_lm_mc_090_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-090: STFT time-frequency localization tradeoff (variant 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_090.li#L1",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-090\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 90).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_090_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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    },
    {
      "id": "SIG-LM-MC-092",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v072",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v072",
            "start_line": 32,
            "highlight_line": 32,
            "content": "theorem sig_lm_sampling_v002_v072 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L32",
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            "path": "proof-db/signal/corpus/sig_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-092: Nyquist-Shannon sampling theorem sketch (variant 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_092.li#L1",
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            "highlight_line": 1588,
            "content": "[[entry]]\nid = \"SIG-LM-MC-092\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 92).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_sampling_v002_v072\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-093",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v093",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v093",
            "start_line": 54,
            "highlight_line": 54,
            "content": "theorem sig_lm_filter_v003_v093 := sig_lm_filter_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L54",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-093: FIR convolution output bound (variant 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_093.li#L1",
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            "highlight_line": 1604,
            "content": "[[entry]]\nid = \"SIG-LM-MC-093\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 93).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_filter_v003_v093\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-094",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v084",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v084",
            "start_line": 73,
            "highlight_line": 73,
            "content": "theorem sig_lm_ztransform_v004_v084 := sig_lm_ztransform_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L73",
            "role": "lean_formal",
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            "path": "proof-db/signal/corpus/sig_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-094: Z-transform of exponential sequence (variant 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_094.li#L1",
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            "language": "toml",
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            "start_line": 1619,
            "highlight_line": 1620,
            "content": "[[entry]]\nid = \"SIG-LM-MC-094\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 94).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v084\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-095",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v085",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v085",
            "start_line": 94,
            "highlight_line": 94,
            "content": "theorem sig_lm_dft_v005_v085 := sig_lm_dft_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L94",
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            "path": "proof-db/signal/corpus/sig_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-095: DFT parseval energy identity (variant 95).",
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    },
    {
      "id": "SIG-LM-MC-097",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v087",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_097.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_097.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v087",
            "start_line": 114,
            "highlight_line": 114,
            "content": "theorem sig_lm_wavelet_v007_v087 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L114",
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            "path": "proof-db/signal/corpus/sig_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-097: Orthonormal wavelet frame energy (variant 97).",
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    },
    {
      "id": "SIG-LM-MC-098",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v088",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v088",
            "start_line": 134,
            "highlight_line": 134,
            "content": "theorem sig_lm_correlation_v008_v088 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L134",
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            "path": "proof-db/signal/corpus/sig_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-098: Cross-correlation peak detection (variant 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_098.li#L1",
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    },
    {
      "id": "SIG-LM-MC-099",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v089",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009_v089",
            "start_line": 153,
            "highlight_line": 153,
            "content": "theorem sig_lm_hilbert_v009_v089 := sig_lm_hilbert_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L153",
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            "path": "proof-db/signal/corpus/sig_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-099: Hilbert transform analytic signal (variant 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_099.li#L1",
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            "highlight_line": 1684,
            "content": "[[entry]]\nid = \"SIG-LM-MC-099\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 99).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_hilbert_v009_v089\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-100",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_100_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_100_stft",
            "start_line": 186,
            "highlight_line": 186,
            "content": "theorem sig_lm_mc_100_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-100: STFT time-frequency localization tradeoff (variant 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_100.li#L1",
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            "highlight_line": 1700,
            "content": "[[entry]]\nid = \"SIG-LM-MC-100\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 100).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_100_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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    },
    {
      "id": "SIG-LM-MC-102",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v082",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v082",
            "start_line": 33,
            "highlight_line": 33,
            "content": "theorem sig_lm_sampling_v002_v082 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L33",
            "role": "lean_formal",
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            "path": "proof-db/signal/corpus/sig_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-102: Nyquist-Shannon sampling theorem sketch (variant 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_102.li#L1",
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            "start_line": 1715,
            "highlight_line": 1716,
            "content": "[[entry]]\nid = \"SIG-LM-MC-102\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 102).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_sampling_v002_v082\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-103",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v103",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v103",
            "start_line": 55,
            "highlight_line": 55,
            "content": "theorem sig_lm_filter_v003_v103 := sig_lm_filter_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L55",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-103: FIR convolution output bound (variant 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_103.li#L1",
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            "highlight_line": 1732,
            "content": "[[entry]]\nid = \"SIG-LM-MC-103\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 103).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_filter_v003_v103\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "diverges": false,
      "gap_id": "G-signal",
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      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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            "symbol": "sig_lm_dft_v005_v095",
            "start_line": 95,
            "highlight_line": 95,
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      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
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            "symbol": "sig_lm_wavelet_v007_v097",
            "start_line": 115,
            "highlight_line": 115,
            "content": "theorem sig_lm_wavelet_v007_v097 := sig_lm_wavelet_v007",
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      "id": "SIG-LM-MC-108",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v098",
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            "symbol": "sig_lm_correlation_v008_v098",
            "start_line": 135,
            "highlight_line": 135,
            "content": "theorem sig_lm_correlation_v008_v098 := sig_lm_correlation_v008",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-108: Cross-correlation peak detection (variant 108).",
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      "id": "SIG-LM-MC-109",
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      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v099",
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            "symbol": "sig_lm_hilbert_v009_v099",
            "start_line": 154,
            "highlight_line": 154,
            "content": "theorem sig_lm_hilbert_v009_v099 := sig_lm_hilbert_v009",
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            "path": "proof-db/signal/corpus/sig_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-109: Hilbert transform analytic signal (variant 109).",
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      "id": "SIG-LM-MC-110",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_110_stft",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_110_stft",
            "start_line": 188,
            "highlight_line": 188,
            "content": "theorem sig_lm_mc_110_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-110: STFT time-frequency localization tradeoff (variant 110).",
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      "id": "SIG-LM-MC-112",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v092",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_112.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v092",
            "start_line": 34,
            "highlight_line": 34,
            "content": "theorem sig_lm_sampling_v002_v092 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L34",
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            "path": "proof-db/signal/corpus/sig_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-112: Nyquist-Shannon sampling theorem sketch (variant 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_112.li#L1",
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      "statement": "FIR convolution output bound (variant 113).",
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      "lean_status": "proved",
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      "gap_id": "G-signal",
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            "start_line": 56,
            "highlight_line": 56,
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      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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            "symbol": "sig_lm_ztransform_v004_v104",
            "start_line": 75,
            "highlight_line": 75,
            "content": "theorem sig_lm_ztransform_v004_v104 := sig_lm_ztransform_v004",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-114: Z-transform of exponential sequence (variant 114).",
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      "id": "SIG-LM-MC-115",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_115.li",
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            "symbol": "sig_lm_dft_v005_v105",
            "start_line": 96,
            "highlight_line": 96,
            "content": "theorem sig_lm_dft_v005_v105 := sig_lm_dft_v005",
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            "start_line": 1,
            "highlight_line": 1,
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      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v107",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_117.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_117.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v107",
            "start_line": 116,
            "highlight_line": 116,
            "content": "theorem sig_lm_wavelet_v007_v107 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L116",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-117: Orthonormal wavelet frame energy (variant 117).",
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            "highlight_line": 1908,
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      "id": "SIG-LM-MC-118",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v108",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_118.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v108",
            "start_line": 136,
            "highlight_line": 136,
            "content": "theorem sig_lm_correlation_v008_v108 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L136",
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            "path": "proof-db/signal/corpus/sig_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-118: Cross-correlation peak detection (variant 118).",
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            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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            "highlight_line": 1924,
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      "id": "SIG-LM-MC-119",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v109",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_119.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009_v109",
            "start_line": 155,
            "highlight_line": 155,
            "content": "theorem sig_lm_hilbert_v009_v109 := sig_lm_hilbert_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L155",
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            "path": "proof-db/signal/corpus/sig_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-119: Hilbert transform analytic signal (variant 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_119.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 1939,
            "highlight_line": 1940,
            "content": "[[entry]]\nid = \"SIG-LM-MC-119\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_hilbert_v009_v109\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-120",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_120_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_120_stft",
            "start_line": 190,
            "highlight_line": 190,
            "content": "theorem sig_lm_mc_120_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-120: STFT time-frequency localization tradeoff (variant 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_120.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 1955,
            "highlight_line": 1956,
            "content": "[[entry]]\nid = \"SIG-LM-MC-120\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_120_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L1955",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "SIG-LM-MC-122",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v102",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v102",
            "start_line": 35,
            "highlight_line": 35,
            "content": "theorem sig_lm_sampling_v002_v102 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L35",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-122: Nyquist-Shannon sampling theorem sketch (variant 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_122.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 1971,
            "highlight_line": 1972,
            "content": "[[entry]]\nid = \"SIG-LM-MC-122\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_sampling_v002_v102\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "SIG-LM-MC-123",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v123",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v123",
            "start_line": 57,
            "highlight_line": 57,
            "content": "theorem sig_lm_filter_v003_v123 := sig_lm_filter_v003",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L57",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-123: FIR convolution output bound (variant 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_123.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 1987,
            "highlight_line": 1988,
            "content": "[[entry]]\nid = \"SIG-LM-MC-123\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 123).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_filter_v003_v123\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L1987",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "SIG-LM-MC-124",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v114",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v114",
            "start_line": 76,
            "highlight_line": 76,
            "content": "theorem sig_lm_ztransform_v004_v114 := sig_lm_ztransform_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L76",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-124: Z-transform of exponential sequence (variant 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_124.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2003,
            "highlight_line": 2004,
            "content": "[[entry]]\nid = \"SIG-LM-MC-124\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 124).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v114\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L2003",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "SIG-LM-MC-125",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v115",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v115",
            "start_line": 97,
            "highlight_line": 97,
            "content": "theorem sig_lm_dft_v005_v115 := sig_lm_dft_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L97",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-125: DFT parseval energy identity (variant 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_125.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2019,
            "highlight_line": 2020,
            "content": "[[entry]]\nid = \"SIG-LM-MC-125\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 125).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_dft_v005_v115\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L2019",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "SIG-LM-MC-127",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v117",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_127.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v117",
            "start_line": 117,
            "highlight_line": 117,
            "content": "theorem sig_lm_wavelet_v007_v117 := sig_lm_wavelet_v007",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L117",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-127: Orthonormal wavelet frame energy (variant 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_127.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2035,
            "highlight_line": 2036,
            "content": "[[entry]]\nid = \"SIG-LM-MC-127\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 127).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_wavelet_v007_v117\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L2035",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "SIG-LM-MC-128",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v118",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_128.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v118",
            "start_line": 137,
            "highlight_line": 137,
            "content": "theorem sig_lm_correlation_v008_v118 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L137",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-128: Cross-correlation peak detection (variant 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_128.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2051,
            "highlight_line": 2052,
            "content": "[[entry]]\nid = \"SIG-LM-MC-128\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 128).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_correlation_v008_v118\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "SIG-LM-MC-129",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v119",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009_v119",
            "start_line": 156,
            "highlight_line": 156,
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            "start_line": 1,
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            "content": "[[entry]]\nid = \"SIG-LM-MC-129\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 129).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_hilbert_v009_v119\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "SIG-LM-MC-130",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_130_stft",
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            "symbol": "sig_lm_mc_130_stft",
            "start_line": 192,
            "highlight_line": 192,
            "content": "theorem sig_lm_mc_130_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-130: STFT time-frequency localization tradeoff (variant 130).",
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      "id": "SIG-LM-MC-132",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v112",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v112",
            "start_line": 36,
            "highlight_line": 36,
            "content": "theorem sig_lm_sampling_v002_v112 := sig_lm_sampling_v002",
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            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-132: Nyquist-Shannon sampling theorem sketch (variant 132).",
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      "id": "SIG-LM-MC-133",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v133",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_133.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v133",
            "start_line": 58,
            "highlight_line": 58,
            "content": "theorem sig_lm_filter_v003_v133 := sig_lm_filter_v003",
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            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-133: FIR convolution output bound (variant 133).",
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      "id": "SIG-LM-MC-134",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v124",
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v124",
            "start_line": 77,
            "highlight_line": 77,
            "content": "theorem sig_lm_ztransform_v004_v124 := sig_lm_ztransform_v004",
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            "path": "proof-db/signal/corpus/sig_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-134: Z-transform of exponential sequence (variant 134).",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-134\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 134).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v124\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "SIG-LM-MC-135",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v125",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_135.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v125",
            "start_line": 98,
            "highlight_line": 98,
            "content": "theorem sig_lm_dft_v005_v125 := sig_lm_dft_v005",
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            "path": "proof-db/signal/corpus/sig_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-135: DFT parseval energy identity (variant 135).",
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      "id": "SIG-LM-MC-137",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v127",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_137.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v127",
            "start_line": 118,
            "highlight_line": 118,
            "content": "theorem sig_lm_wavelet_v007_v127 := sig_lm_wavelet_v007",
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            "path": "proof-db/signal/corpus/sig_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-137: Orthonormal wavelet frame energy (variant 137).",
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            "highlight_line": 2372,
            "content": "[[entry]]\nid = \"SIG-LM-MC-137\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 137).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_wavelet_v007_v127\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      "id": "SIG-LM-MC-138",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v128",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_138.li",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_138.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v128",
            "start_line": 138,
            "highlight_line": 138,
            "content": "theorem sig_lm_correlation_v008_v128 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L138",
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            "path": "proof-db/signal/corpus/sig_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-138: Cross-correlation peak detection (variant 138).",
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    },
    {
      "id": "SIG-LM-MC-139",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v129",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_139.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009_v129",
            "start_line": 157,
            "highlight_line": 157,
            "content": "theorem sig_lm_hilbert_v009_v129 := sig_lm_hilbert_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L157",
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            "path": "proof-db/signal/corpus/sig_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-139: Hilbert transform analytic signal (variant 139).",
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    },
    {
      "id": "SIG-LM-MC-140",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_140_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_140.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_140.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_140_stft",
            "start_line": 194,
            "highlight_line": 194,
            "content": "theorem sig_lm_mc_140_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-140: STFT time-frequency localization tradeoff (variant 140).",
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            "highlight_line": 2420,
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    },
    {
      "id": "SIG-LM-MC-142",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v122",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v122",
            "start_line": 37,
            "highlight_line": 37,
            "content": "theorem sig_lm_sampling_v002_v122 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L37",
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            "path": "proof-db/signal/corpus/sig_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-142: Nyquist-Shannon sampling theorem sketch (variant 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_142.li#L1",
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            "start_line": 2435,
            "highlight_line": 2436,
            "content": "[[entry]]\nid = \"SIG-LM-MC-142\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 142).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_sampling_v002_v122\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-143",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_filter_v003_v143",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_filter_v003_v143",
            "start_line": 59,
            "highlight_line": 59,
            "content": "theorem sig_lm_filter_v003_v143 := sig_lm_filter_v003\n\n/-- Z-transform of geometric sequence z\u207b\u207f (SIG-LM-MC-004 / z-transform anchor). -/",
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            "path": "proof-db/signal/corpus/sig_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-143: FIR convolution output bound (variant 143).",
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            "start_line": 2451,
            "highlight_line": 2452,
            "content": "[[entry]]\nid = \"SIG-LM-MC-143\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 143).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_filter_v003_v143\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-144",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v134",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_144.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v134",
            "start_line": 78,
            "highlight_line": 78,
            "content": "theorem sig_lm_ztransform_v004_v134 := sig_lm_ztransform_v004",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L78",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-144: Z-transform of exponential sequence (variant 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_144.li#L1",
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            "language": "toml",
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            "start_line": 2467,
            "highlight_line": 2468,
            "content": "[[entry]]\nid = \"SIG-LM-MC-144\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 144).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_ztransform_v004_v134\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-145",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v135",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v135",
            "start_line": 99,
            "highlight_line": 99,
            "content": "theorem sig_lm_dft_v005_v135 := sig_lm_dft_v005",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L99",
            "role": "lean_formal",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-145: DFT parseval energy identity (variant 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_145.li#L1",
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            "start_line": 2483,
            "highlight_line": 2484,
            "content": "[[entry]]\nid = \"SIG-LM-MC-145\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 145).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_dft_v005_v135\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-147",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
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      "erdos_status": null,
      "convergence_class": null,
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v137",
            "start_line": 119,
            "highlight_line": 119,
            "content": "theorem sig_lm_wavelet_v007_v137 := sig_lm_wavelet_v007",
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            "start_line": 1,
            "highlight_line": 1,
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      "id": "SIG-LM-MC-148",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v138",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_148.li",
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v138",
            "start_line": 139,
            "highlight_line": 139,
            "content": "theorem sig_lm_correlation_v008_v138 := sig_lm_correlation_v008",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L139",
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            "path": "proof-db/signal/corpus/sig_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-148: Cross-correlation peak detection (variant 148).",
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    },
    {
      "id": "SIG-LM-MC-149",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v139",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_149.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "lean",
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            "symbol": "sig_lm_hilbert_v009_v139",
            "start_line": 158,
            "highlight_line": 158,
            "content": "theorem sig_lm_hilbert_v009_v139 := sig_lm_hilbert_v009",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L158",
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            "path": "proof-db/signal/corpus/sig_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-149: Hilbert transform analytic signal (variant 149).",
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    },
    {
      "id": "SIG-LM-MC-150",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_150_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_150_stft",
            "start_line": 196,
            "highlight_line": 196,
            "content": "theorem sig_lm_mc_150_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
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            "path": "proof-db/signal/corpus/sig_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-150: STFT time-frequency localization tradeoff (variant 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_150.li#L1",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-150\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_150_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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    },
    {
      "id": "SIG-LM-MC-152",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 152).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v132",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_152.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_152.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v132",
            "start_line": 38,
            "highlight_line": 38,
            "content": "theorem sig_lm_sampling_v002_v132 := sig_lm_sampling_v002",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L38",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_152.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-152: Nyquist-Shannon sampling theorem sketch (variant 152).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_152.li#L1",
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            "start_line": 2563,
            "highlight_line": 2564,
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-153",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 153).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_fir_norm_le",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_153.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_153.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_fir_norm_le",
            "start_line": 42,
            "highlight_line": 42,
            "content": "theorem sig_lm_fir_norm_le {E : Type*} [NormedAddCommGroup E] (h x : E) :\n    \u2016h + x\u2016 \u2264 \u2016h\u2016 + \u2016x\u2016 := norm_add_le h x\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L42",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_153.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-153: FIR convolution output bound (variant 153).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_153.li#L1",
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            "start_line": 2579,
            "highlight_line": 2580,
            "content": "[[entry]]\nid = \"SIG-LM-MC-153\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 153).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_fir_norm_le\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_153.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-154",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 154).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_ztransform_v004_v144",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_154.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_154.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_ztransform_v004_v144",
            "start_line": 79,
            "highlight_line": 79,
            "content": "theorem sig_lm_ztransform_v004_v144 := sig_lm_ztransform_v004\n\n/-- DFT Parseval energy: \u2016x\u2016\u00b2 \u2265 0 (SIG-LM-MC-005 / DFT anchor). -/",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_154.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-154: Z-transform of exponential sequence (variant 154).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_154.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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      "id": "SIG-LM-MC-155",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 155).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_dft_v005_v145",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_155.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_155.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_dft_v005_v145",
            "start_line": 100,
            "highlight_line": 100,
            "content": "theorem sig_lm_dft_v005_v145 := sig_lm_dft_v005\n\n/-- Wavelet frame energy: inner product self \u2265 0 (SIG-LM-MC-007). -/",
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            "path": "proof-db/signal/corpus/sig_lm_mc_155.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-155: DFT parseval energy identity (variant 155).",
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            "content": "[[entry]]\nid = \"SIG-LM-MC-155\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 155).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_dft_v005_v145\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_155.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-157",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 157).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_wavelet_v007_v147",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_157.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_157.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_wavelet_v007_v147",
            "start_line": 120,
            "highlight_line": 120,
            "content": "theorem sig_lm_wavelet_v007_v147 := sig_lm_wavelet_v007\n\n/-- Cross-correlation peak: Cauchy-Schwarz \u2016x\u2016\u00b7\u2016y\u2016 bound (SIG-LM-MC-008 / correlation anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L120",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_157.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-157: Orthonormal wavelet frame energy (variant 157).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_157.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2627,
            "highlight_line": 2628,
            "content": "[[entry]]\nid = \"SIG-LM-MC-157\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 157).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_wavelet_v007_v147\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_157.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-158",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 158).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_correlation_v008_v148",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_158.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_158.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_correlation_v008_v148",
            "start_line": 140,
            "highlight_line": 140,
            "content": "theorem sig_lm_correlation_v008_v148 := sig_lm_correlation_v008\n\n/-- Hilbert transform analytic signal: \u2016x\u2016 \u2265 0 (SIG-LM-MC-009 / Hilbert anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L140",
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            "label": "Lean formalization"
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            "path": "proof-db/signal/corpus/sig_lm_mc_158.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-158: Cross-correlation peak detection (variant 158).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_158.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2643,
            "highlight_line": 2644,
            "content": "[[entry]]\nid = \"SIG-LM-MC-158\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 158).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_correlation_v008_v148\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_158.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-159",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 159).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_hilbert_v009_v149",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_159.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_159.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_hilbert_v009_v149",
            "start_line": 159,
            "highlight_line": 159,
            "content": "theorem sig_lm_hilbert_v009_v149 := sig_lm_hilbert_v009\n\n/-! STFT time-frequency localization tradeoff (SIG-LM-MC-*0) \u2014 AM-GM form. -/\n\n/-- Heisenberg/AM-GM proxy: 4\u00b7\u03c3\u209c\u00b7\u03c3_\u03c9 \u2264 (\u03c3\u209c + \u03c3_\u03c9)\u00b2. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L159",
            "role": "lean_formal",
            "label": "Lean formalization"
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            "path": "proof-db/signal/corpus/sig_lm_mc_159.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-159: Hilbert transform analytic signal (variant 159).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_159.li#L1",
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            "highlight_line": 2660,
            "content": "[[entry]]\nid = \"SIG-LM-MC-159\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 159).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_hilbert_v009_v149\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_159.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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    },
    {
      "id": "SIG-LM-MC-160",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 160).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_160_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_160.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_160.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_160_stft",
            "start_line": 198,
            "highlight_line": 198,
            "content": "theorem sig_lm_mc_160_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L198",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_160.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-160: STFT time-frequency localization tradeoff (variant 160).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_160.li#L1",
            "role": "li_specimen",
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            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
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            "highlight_line": 2676,
            "content": "[[entry]]\nid = \"SIG-LM-MC-160\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 160).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_160_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_160.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L2675",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "SIG-LM-MC-162",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 162).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_sampling_v002_v142",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_162.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_162.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_sampling_v002_v142",
            "start_line": 39,
            "highlight_line": 39,
            "content": "theorem sig_lm_sampling_v002_v142 := sig_lm_sampling_v002\n\n/-- FIR filter output bound: \u2016h * x\u2016 \u2264 \u2016h\u2016\u00b7\u2016x\u2016 (SIG-LM-MC-003 / filter anchor). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L39",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_162.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-162: Nyquist-Shannon sampling theorem sketch (variant 162).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_162.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2691,
            "highlight_line": 2692,
            "content": "[[entry]]\nid = \"SIG-LM-MC-162\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 162).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_sampling_v002_v142\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_162.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
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      }
    },
    {
      "id": "SIG-LM-MC-163",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 163).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_163.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_163.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_163.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-163: FIR convolution output bound (variant 163).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_163.li#L1",
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            "start_line": 2707,
            "highlight_line": 2708,
            "content": "[[entry]]\nid = \"SIG-LM-MC-163\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 163).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_163.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-164",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 164).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_164.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_164.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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        "release_pin": null,
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        "implementations": [
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            "path": "proof-db/signal/corpus/sig_lm_mc_164.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-164: Z-transform of exponential sequence (variant 164).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_164.li#L1",
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            "start_line": 2722,
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    },
    {
      "id": "SIG-LM-MC-165",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 165).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_parseval_energy_v005",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_165.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_165.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_parseval_energy_v005",
            "start_line": 84,
            "highlight_line": 84,
            "content": "theorem sig_lm_parseval_energy_v005 {E : Type*} [NormedAddCommGroup E] (x : E) : 0 \u2264 \u2016x\u2016 ^ 2 := sq_nonneg _\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L84",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_165.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-165: DFT parseval energy identity (variant 165).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_165.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2737,
            "highlight_line": 2738,
            "content": "[[entry]]\nid = \"SIG-LM-MC-165\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 165).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_parseval_energy_v005\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_165.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-167",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 167).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_167.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_167.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_167.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-167: Orthonormal wavelet frame energy (variant 167).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_167.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2753,
            "highlight_line": 2754,
            "content": "[[entry]]\nid = \"SIG-LM-MC-167\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 167).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_167.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-168",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 168).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_168.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_168.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_168.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-168: Cross-correlation peak detection (variant 168).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_168.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2768,
            "highlight_line": 2769,
            "content": "[[entry]]\nid = \"SIG-LM-MC-168\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 168).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_168.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-169",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 169).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_169.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_169.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "path": "proof-db/signal/corpus/sig_lm_mc_169.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-169: Hilbert transform analytic signal (variant 169).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_169.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2783,
            "highlight_line": 2784,
            "content": "[[entry]]\nid = \"SIG-LM-MC-169\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 169).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_169.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-170",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 170).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_170_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_170.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_170.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_170_stft",
            "start_line": 200,
            "highlight_line": 200,
            "content": "theorem sig_lm_mc_170_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L200",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_170.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-170: STFT time-frequency localization tradeoff (variant 170).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_170.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2798,
            "highlight_line": 2799,
            "content": "[[entry]]\nid = \"SIG-LM-MC-170\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 170).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_170_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_170.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
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      }
    },
    {
      "id": "SIG-LM-MC-172",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 172).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_172.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_172.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_172.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-172: Nyquist-Shannon sampling theorem sketch (variant 172).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_172.li#L1",
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            "start_line": 2814,
            "highlight_line": 2815,
            "content": "[[entry]]\nid = \"SIG-LM-MC-172\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 172).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_172.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-173",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 173).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_173.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_173.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "path": "proof-db/signal/corpus/sig_lm_mc_173.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-173: FIR convolution output bound (variant 173).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_173.li#L1",
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            "language": "toml",
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            "start_line": 2829,
            "highlight_line": 2830,
            "content": "[[entry]]\nid = \"SIG-LM-MC-173\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 173).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_173.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-174",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 174).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_174.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_174.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "implementations": [
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            "path": "proof-db/signal/corpus/sig_lm_mc_174.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-174: Z-transform of exponential sequence (variant 174).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_174.li#L1",
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            "language": "toml",
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            "start_line": 2844,
            "highlight_line": 2845,
            "content": "[[entry]]\nid = \"SIG-LM-MC-174\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 174).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_174.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-175",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 175).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_175.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_175.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_175.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-175: DFT parseval energy identity (variant 175).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_175.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2859,
            "highlight_line": 2860,
            "content": "[[entry]]\nid = \"SIG-LM-MC-175\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 175).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_175.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-177",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 177).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_177.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_177.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_177.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-177: Orthonormal wavelet frame energy (variant 177).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_177.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2874,
            "highlight_line": 2875,
            "content": "[[entry]]\nid = \"SIG-LM-MC-177\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 177).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_177.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "SIG-LM-MC-178",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 178).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_178.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_178.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "implementations": [
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            "path": "proof-db/signal/corpus/sig_lm_mc_178.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-178: Cross-correlation peak detection (variant 178).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_178.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2889,
            "highlight_line": 2890,
            "content": "[[entry]]\nid = \"SIG-LM-MC-178\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 178).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_178.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
    {
      "id": "SIG-LM-MC-179",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 179).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_179.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_179.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
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            "path": "proof-db/signal/corpus/sig_lm_mc_179.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-179: Hilbert transform analytic signal (variant 179).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_179.li#L1",
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            "start_line": 2904,
            "highlight_line": 2905,
            "content": "[[entry]]\nid = \"SIG-LM-MC-179\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 179).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_179.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-180",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 180).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_180_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_180.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_180.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_180_stft",
            "start_line": 202,
            "highlight_line": 202,
            "content": "theorem sig_lm_mc_180_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L202",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_180.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-180: STFT time-frequency localization tradeoff (variant 180).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_180.li#L1",
            "role": "li_specimen",
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            "language": "toml",
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            "start_line": 2919,
            "highlight_line": 2920,
            "content": "[[entry]]\nid = \"SIG-LM-MC-180\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 180).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_180_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_180.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L2919",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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      }
    },
    {
      "id": "SIG-LM-MC-182",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 182).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_182.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_182.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_182.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-182: Nyquist-Shannon sampling theorem sketch (variant 182).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_182.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2935,
            "highlight_line": 2936,
            "content": "[[entry]]\nid = \"SIG-LM-MC-182\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 182).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_182.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-183",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 183).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_183.li",
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      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_183.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_183.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-183: FIR convolution output bound (variant 183).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_183.li#L1",
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            "start_line": 2950,
            "highlight_line": 2951,
            "content": "[[entry]]\nid = \"SIG-LM-MC-183\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 183).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_183.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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    },
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      "id": "SIG-LM-MC-184",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 184).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_184.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_184.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "path": "proof-db/signal/corpus/sig_lm_mc_184.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-184: Z-transform of exponential sequence (variant 184).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_184.li#L1",
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            "start_line": 2965,
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            "content": "[[entry]]\nid = \"SIG-LM-MC-184\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 184).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_184.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-185",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 185).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_185.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_185.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_185.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-185: DFT parseval energy identity (variant 185).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_185.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2980,
            "highlight_line": 2981,
            "content": "[[entry]]\nid = \"SIG-LM-MC-185\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 185).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_185.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-187",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 187).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_187.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_187.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_187.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-187: Orthonormal wavelet frame energy (variant 187).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_187.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 2995,
            "highlight_line": 2996,
            "content": "[[entry]]\nid = \"SIG-LM-MC-187\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 187).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_187.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-188",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 188).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_188.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_188.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_188.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-188: Cross-correlation peak detection (variant 188).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_188.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3010,
            "highlight_line": 3011,
            "content": "[[entry]]\nid = \"SIG-LM-MC-188\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 188).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_188.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-189",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 189).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_189.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_189.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_189.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-189: Hilbert transform analytic signal (variant 189).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_189.li#L1",
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            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3025,
            "highlight_line": 3026,
            "content": "[[entry]]\nid = \"SIG-LM-MC-189\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 189).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_189.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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      }
    },
    {
      "id": "SIG-LM-MC-190",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 190).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_190_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_190.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_190.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_190_stft",
            "start_line": 204,
            "highlight_line": 204,
            "content": "theorem sig_lm_mc_190_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L204",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_190.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-190: STFT time-frequency localization tradeoff (variant 190).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_190.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3040,
            "highlight_line": 3041,
            "content": "[[entry]]\nid = \"SIG-LM-MC-190\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 190).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_190_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_190.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L3040",
            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-192",
      "kind": "lemma",
      "field": "signal",
      "domain": "sampling",
      "statement": "Nyquist-Shannon sampling theorem sketch (variant 192).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_192.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_192.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_192.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-192: Nyquist-Shannon sampling theorem sketch (variant 192).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_192.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3056,
            "highlight_line": 3057,
            "content": "[[entry]]\nid = \"SIG-LM-MC-192\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"sampling\"\nstatement = \"Nyquist-Shannon sampling theorem sketch (variant 192).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_192.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "SIG-LM-MC-193",
      "kind": "lemma",
      "field": "signal",
      "domain": "filter",
      "statement": "FIR convolution output bound (variant 193).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_193.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_193.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_193.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-193: FIR convolution output bound (variant 193).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_193.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3071,
            "highlight_line": 3072,
            "content": "[[entry]]\nid = \"SIG-LM-MC-193\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"filter\"\nstatement = \"FIR convolution output bound (variant 193).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_193.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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            "role": "catalog",
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        ]
      }
    },
    {
      "id": "SIG-LM-MC-194",
      "kind": "lemma",
      "field": "signal",
      "domain": "z-transform",
      "statement": "Z-transform of exponential sequence (variant 194).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_194.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_194.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
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            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_194.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-194: Z-transform of exponential sequence (variant 194).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_194.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3086,
            "highlight_line": 3087,
            "content": "[[entry]]\nid = \"SIG-LM-MC-194\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"z-transform\"\nstatement = \"Z-transform of exponential sequence (variant 194).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_194.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
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        ]
      }
    },
    {
      "id": "SIG-LM-MC-195",
      "kind": "lemma",
      "field": "signal",
      "domain": "dft",
      "statement": "DFT parseval energy identity (variant 195).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_195.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_195.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_195.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-195: DFT parseval energy identity (variant 195).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_195.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3101,
            "highlight_line": 3102,
            "content": "[[entry]]\nid = \"SIG-LM-MC-195\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"dft\"\nstatement = \"DFT parseval energy identity (variant 195).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_195.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L3101",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "SIG-LM-MC-197",
      "kind": "lemma",
      "field": "signal",
      "domain": "wavelet",
      "statement": "Orthonormal wavelet frame energy (variant 197).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_197.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_197.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_197.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-197: Orthonormal wavelet frame energy (variant 197).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_197.li#L1",
            "role": "li_specimen",
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          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3116,
            "highlight_line": 3117,
            "content": "[[entry]]\nid = \"SIG-LM-MC-197\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"wavelet\"\nstatement = \"Orthonormal wavelet frame energy (variant 197).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_197.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L3116",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "SIG-LM-MC-198",
      "kind": "lemma",
      "field": "signal",
      "domain": "correlation",
      "statement": "Cross-correlation peak detection (variant 198).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_198.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_198.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_198.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-198: Cross-correlation peak detection (variant 198).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_198.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3131,
            "highlight_line": 3132,
            "content": "[[entry]]\nid = \"SIG-LM-MC-198\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"correlation\"\nstatement = \"Cross-correlation peak detection (variant 198).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_198.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L3131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "SIG-LM-MC-199",
      "kind": "lemma",
      "field": "signal",
      "domain": "hilbert",
      "statement": "Hilbert transform analytic signal (variant 199).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_199.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_199.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_199.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-199: Hilbert transform analytic signal (variant 199).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_199.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3146,
            "highlight_line": 3147,
            "content": "[[entry]]\nid = \"SIG-LM-MC-199\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"hilbert\"\nstatement = \"Hilbert transform analytic signal (variant 199).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_199.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-only reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L3146",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "SIG-LM-MC-200",
      "kind": "lemma",
      "field": "signal",
      "domain": "stft",
      "statement": "STFT time-frequency localization tradeoff (variant 200).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-signal",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Signal.sig_lm_mc_200_stft",
      "lean_module": "proof-db/signal/SignalAxioms.lean",
      "li_specimen": "proof-db/signal/corpus/sig_lm_mc_200.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_200.li",
      "notes": "phase18-mega-corpus; iter14 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/signal/SignalAxioms.lean",
            "symbol": "sig_lm_mc_200_stft",
            "start_line": 206,
            "highlight_line": 206,
            "content": "theorem sig_lm_mc_200_stft (\u03c3t \u03c3\u03c9 : \u211d) : 4 * \u03c3t * \u03c3\u03c9 \u2264 (\u03c3t + \u03c3\u03c9) ^ 2 :=\n  stft_time_freq_tradeoff \u03c3t \u03c3\u03c9\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/SignalAxioms.lean#L206",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/signal/corpus/sig_lm_mc_200.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# SIG-LM-MC-200: STFT time-frequency localization tradeoff (variant 200).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/signal/corpus/sig_lm_mc_200.li#L1",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml",
            "start_line": 3161,
            "highlight_line": 3162,
            "content": "[[entry]]\nid = \"SIG-LM-MC-200\"\nkind = \"lemma\"\nfield = \"signal\"\ndomain = \"stft\"\nstatement = \"STFT time-frequency localization tradeoff (variant 200).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-signal\"\nlean_module = \"proof-db/signal/SignalAxioms.lean\"\nlean_thm = \"Li.ProofDb.Signal.sig_lm_mc_200_stft\"\nli_specimen = \"proof-db/signal/corpus/sig_lm_mc_200.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter14 Mathlib/Init discharge\"\nregister_source = \"proof-db/signal/register.json\"\nlast_verified_lic_commit = \"598cfd0a34\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\signal-mega-corpus.toml#L3161",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-BC-EX-002",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity E[aX+bY] = aE[X]+bE[Y] (case 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_ex_002_expect_linear",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_ex_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_ex_002.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_ex_002_expect_linear",
            "start_line": 978,
            "highlight_line": 978,
            "content": "theorem st_ax_bc_ex_002_expect_linear {n : Nat} (\u03a9 : FiniteProbSpace n) (a b : \u211d)\n    (X Y : Fin n \u2192 \u211d) :\n    expect \u03a9 (fun i => a * X i + b * Y i) = a * expect \u03a9 X + b * expect \u03a9 Y :=\n  expect_linear \u03a9 a b X Y\n",
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            "role": "lean_formal",
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          },
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            "path": "proof-db/statistics/basic-corpus/st_ax_bc_ex_002.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def expectation_linear(a: float, b: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == a * ex + b * ey\n  decreases 0\n=\n  return a * ex + b * ey",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"ST-AX-BC-EX-002\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity E[aX+bY] = aE[X]+bE[Y] (case 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_ex_002_expect_linear\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_ex_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-BC-EX-012",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity E[aX+bY] = aE[X]+bE[Y] (case 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_ex_012_expect_linear",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_ex_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_ex_012.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "symbol": "st_ax_bc_ex_012_expect_linear",
            "start_line": 983,
            "highlight_line": 983,
            "content": "theorem st_ax_bc_ex_012_expect_linear {n : Nat} (\u03a9 : FiniteProbSpace n) (a b : \u211d)\n    (X Y : Fin n \u2192 \u211d) :\n    expect \u03a9 (fun i => a * X i + b * Y i) = a * expect \u03a9 X + b * expect \u03a9 Y :=\n  expect_linear \u03a9 a b X Y\n",
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            "path": "proof-db/statistics/basic-corpus/st_ax_bc_ex_012.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def expectation_linear(a: float, b: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == a * ex + b * ey\n  decreases 0\n=\n  return a * ex + b * ey",
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            "start_line": 159,
            "highlight_line": 160,
            "content": "[[entry]]\nid = \"ST-AX-BC-EX-012\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity E[aX+bY] = aE[X]+bE[Y] (case 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_ex_012_expect_linear\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_ex_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L159",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "ST-AX-BC-EX-022",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity E[aX+bY] = aE[X]+bE[Y] on finite spaces (case 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_ex_022_expect_linear",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_ex_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_ex_022.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_ex_022_expect_linear",
            "start_line": 988,
            "highlight_line": 988,
            "content": "theorem st_ax_bc_ex_022_expect_linear {n : Nat} (\u03a9 : FiniteProbSpace n) (a b : \u211d)\n    (X Y : Fin n \u2192 \u211d) :\n    expect \u03a9 (fun i => a * X i + b * Y i) = a * expect \u03a9 X + b * expect \u03a9 Y :=\n  expect_linear \u03a9 a b X Y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L988",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_bc_ex_022.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def expectation_linear(a: float, b: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == a * ex + b * ey\n  decreases 0\n=\n  return a * ex + b * ey",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_ex_022.li#L4",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 315,
            "highlight_line": 316,
            "content": "[[entry]]\nid = \"ST-AX-BC-EX-022\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity E[aX+bY] = aE[X]+bE[Y] on finite spaces (case 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_ex_022_expect_linear\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_ex_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L315",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-BC-EX-032",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity E[aX+bY] = aE[X]+bE[Y] on finite spaces (case 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_ex_032_expect_linear",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_ex_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_ex_032.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
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        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_ex_032_expect_linear",
            "start_line": 993,
            "highlight_line": 993,
            "content": "theorem st_ax_bc_ex_032_expect_linear {n : Nat} (\u03a9 : FiniteProbSpace n) (a b : \u211d)\n    (X Y : Fin n \u2192 \u211d) :\n    expect \u03a9 (fun i => a * X i + b * Y i) = a * expect \u03a9 X + b * expect \u03a9 Y :=\n  expect_linear \u03a9 a b X Y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L993",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_bc_ex_032.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def expectation_linear(a: float, b: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == a * ex + b * ey\n  decreases 0\n=\n  return a * ex + b * ey",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_ex_032.li#L4",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 330,
            "highlight_line": 331,
            "content": "[[entry]]\nid = \"ST-AX-BC-EX-032\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity E[aX+bY] = aE[X]+bE[Y] on finite spaces (case 32).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_ex_032_expect_linear\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_ex_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "ST-AX-BC-EX-042",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity E[aX+bY] = aE[X]+bE[Y] on finite spaces (case 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_ex_042_expect_linear",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_ex_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_ex_042.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_ex_042_expect_linear",
            "start_line": 998,
            "highlight_line": 998,
            "content": "theorem st_ax_bc_ex_042_expect_linear {n : Nat} (\u03a9 : FiniteProbSpace n) (a b : \u211d)\n    (X Y : Fin n \u2192 \u211d) :\n    expect \u03a9 (fun i => a * X i + b * Y i) = a * expect \u03a9 X + b * expect \u03a9 Y :=\n  expect_linear \u03a9 a b X Y\n\n/-- Kolmogorov axiom consequence mega (ST-AX-MC-*1) \u2014 normalize + nonneg. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L998",
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          },
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_bc_ex_042.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def expectation_linear(a: float, b: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == a * ex + b * ey\n  decreases 0\n=\n  return a * ex + b * ey",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_ex_042.li#L4",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 345,
            "highlight_line": 346,
            "content": "[[entry]]\nid = \"ST-AX-BC-EX-042\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity E[aX+bY] = aE[X]+bE[Y] on finite spaces (case 42).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_ex_042_expect_linear\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_ex_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-BC-PR-001",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axioms on finite Omega: (K1) P(A)>=0; (K2) P(Omega)=1; (K3) P(A u B)=P(A)+P(B) for disjoint A,B.",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_pr_001_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_pr_001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_001.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_pr_001_kolmogorov",
            "start_line": 969,
            "highlight_line": 969,
            "content": "theorem st_ax_bc_pr_001_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (h : Disjoint A B) :\n    (\u2200 S : Finset (Fin n), 0 \u2264 eventProb \u03a9 S) \u2227\n      eventProb \u03a9 Finset.univ = 1 \u2227\n      eventProb \u03a9 (A \u222a B) = eventProb \u03a9 A + eventProb \u03a9 B :=\n  \u27e8fun S => st_ax_bc_pr_031_nonneg \u03a9 S, st_ax_bc_pr_001_normalized \u03a9,\n    st_ax_bc_pr_041_disjoint \u03a9 A B h\u27e9\n\n/-- Expectation linearity (ST-AX-BC-EX-*). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L969",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_bc_pr_001.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def kolmogorov_mass_normalized(p: array[n, float]) -> float\n  requires true\n  ensures result >= 0.0\n  decreases 0\n=\n  var i: int = 0\n  var total: float = 0.0\n  while i < n\n    invariant 0 <= i and i <= n\n    decreases n - i\n  =\n    total = total + p[i]\n    i = i + 1\n  return total",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_001.li#L7",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 18,
            "highlight_line": 19,
            "content": "[[entry]]\nid = \"ST-AX-BC-PR-001\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Kolmogorov axioms on finite Omega: (K1) P(A)>=0; (K2) P(Omega)=1; (K3) P(A u B)=P(A)+P(B) for disjoint A,B.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_pr_001_kolmogorov\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_pr_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L18",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-BC-PR-011",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Boole inequality (finite): P(A union B) <= P(A)+P(B).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_pr_011_boole",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_pr_011.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_011.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_pr_011_boole",
            "start_line": 962,
            "highlight_line": 962,
            "content": "theorem st_ax_bc_pr_011_boole {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n)) :\n    eventProb \u03a9 (A \u222a B) \u2264 eventProb \u03a9 A + eventProb \u03a9 B := by\n  have h := st_ax_bc_pr_021_inclusion \u03a9 A B\n  have hineg : 0 \u2264 eventProb \u03a9 (A \u2229 B) := st_ax_bc_pr_031_nonneg \u03a9 (A \u2229 B)\n  linarith\n\n/-- Kolmogorov package (ST-AX-BC-PR-001): nonneg + normalize + disjoint additivity. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L962",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_bc_pr_011.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def kolmogorov_mass_normalized(p: array[n, float]) -> float\n  requires true\n  ensures result >= 0.0\n  decreases 0\n=\n  var i: int = 0\n  var total: float = 0.0\n  while i < n\n    invariant 0 <= i and i <= n\n    decreases n - i\n  =\n    total = total + p[i]\n    i = i + 1\n  return total",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_011.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 174,
            "highlight_line": 175,
            "content": "[[entry]]\nid = \"ST-AX-BC-PR-011\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Boole inequality (finite): P(A union B) <= P(A)+P(B).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_pr_011_boole\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_pr_011.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L174",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-BC-PR-021",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Inclusion-exclusion (two events): P(A union B)=P(A)+P(B)-P(A intersect B).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_pr_021_inclusion",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_pr_021.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_021.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_pr_021_inclusion",
            "start_line": 954,
            "highlight_line": 954,
            "content": "theorem st_ax_bc_pr_021_inclusion {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n)) :\n    eventProb \u03a9 (A \u222a B) = eventProb \u03a9 A + eventProb \u03a9 B - eventProb \u03a9 (A \u2229 B) := by\n  classical\n  simp only [eventProb]\n  have h := Finset.sum_union_inter (f := \u03a9.p) A B\n  linarith\n\n/-- Boole inequality (ST-AX-BC-PR-011): P(A \u222a B) \u2264 P(A) + P(B). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L954",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_bc_pr_021.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def kolmogorov_mass_normalized(p: array[n, float]) -> float\n  requires true\n  ensures result >= 0.0\n  decreases 0\n=\n  var i: int = 0\n  var total: float = 0.0\n  while i < n\n    invariant 0 <= i and i <= n\n    decreases n - i\n  =\n    total = total + p[i]\n    i = i + 1\n  return total",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_021.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 360,
            "highlight_line": 361,
            "content": "[[entry]]\nid = \"ST-AX-BC-PR-021\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Inclusion-exclusion (two events): P(A union B)=P(A)+P(B)-P(A intersect B).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_pr_021_inclusion\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_pr_021.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L360",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-BC-PR-031",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Non-negativity: P(A) >= 0 for all events A.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_pr_031_nonneg",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_pr_031.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_031.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_pr_031_nonneg",
            "start_line": 938,
            "highlight_line": 938,
            "content": "theorem st_ax_bc_pr_031_nonneg {n : Nat} (\u03a9 : FiniteProbSpace n) (A : Finset (Fin n)) :\n    0 \u2264 eventProb \u03a9 A :=\n  Finset.sum_nonneg fun i _ => \u03a9.p_nonneg i\n\n/-- Normalization: P(\u03a9) = 1 (ST-AX-BC-PR-001 K2 fragment). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L938",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_bc_pr_031.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def kolmogorov_mass_normalized(p: array[n, float]) -> float\n  requires true\n  ensures result >= 0.0\n  decreases 0\n=\n  var i: int = 0\n  var total: float = 0.0\n  while i < n\n    invariant 0 <= i and i <= n\n    decreases n - i\n  =\n    total = total + p[i]\n    i = i + 1\n  return total",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_031.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 375,
            "highlight_line": 376,
            "content": "[[entry]]\nid = \"ST-AX-BC-PR-031\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Non-negativity: P(A) >= 0 for all events A.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_pr_031_nonneg\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_pr_031.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L375",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-BC-PR-041",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Disjoint additivity: P(A u B)=P(A)+P(B) when A and B disjoint.",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_bc_pr_041_disjoint",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_bc_pr_041.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_041.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_bc_pr_041_disjoint",
            "start_line": 948,
            "highlight_line": 948,
            "content": "theorem st_ax_bc_pr_041_disjoint {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (h : Disjoint A B) :\n    eventProb \u03a9 (A \u222a B) = eventProb \u03a9 A + eventProb \u03a9 B := by\n  simp [eventProb, Finset.sum_union h]\n\n/-- Two-event inclusion-exclusion (ST-AX-BC-PR-021). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L948",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_bc_pr_041.li",
            "symbol": null,
            "start_line": 7,
            "highlight_line": 7,
            "content": "def kolmogorov_mass_normalized(p: array[n, float]) -> float\n  requires true\n  ensures result >= 0.0\n  decreases 0\n=\n  var i: int = 0\n  var total: float = 0.0\n  while i < n\n    invariant 0 <= i and i <= n\n    decreases n - i\n  =\n    total = total + p[i]\n    i = i + 1\n  return total",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_bc_pr_041.li#L7",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 390,
            "highlight_line": 391,
            "content": "[[entry]]\nid = \"ST-AX-BC-PR-041\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Disjoint additivity: P(A u B)=P(A)+P(B) when A and B disjoint.\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_bc_pr_041_disjoint\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_bc_pr_041.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L390",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-MC-001",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 1).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_001_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_mc_001.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_001.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_mc_001_kolmogorov",
            "start_line": 1008,
            "highlight_line": 1008,
            "content": "theorem st_ax_mc_001_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1008",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_mc_001.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_001() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_001.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"ST-AX-MC-001\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Kolmogorov axiom consequence (mega 1).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_mc_001_kolmogorov\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_mc_001.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-MC-011",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 11).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_011_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_mc_011.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_011.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_mc_011_kolmogorov",
            "start_line": 1012,
            "highlight_line": 1012,
            "content": "theorem st_ax_mc_011_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1012",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_mc_011.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_011() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_011.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 19,
            "highlight_line": 20,
            "content": "[[entry]]\nid = \"ST-AX-MC-011\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Kolmogorov axiom consequence (mega 11).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_mc_011_kolmogorov\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_mc_011.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L19",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-AX-MC-021",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 21).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_021_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_mc_021.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_021.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_mc_021_kolmogorov",
            "start_line": 1016,
            "highlight_line": 1016,
            "content": "theorem st_ax_mc_021_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1016",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_mc_021.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_021() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_021.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
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      "erdos_id": null,
      "erdos_status": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_041_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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            "symbol": "st_ax_mc_041_kolmogorov",
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            "highlight_line": 1024,
            "content": "theorem st_ax_mc_041_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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      "statement": "Kolmogorov axiom consequence (mega 51).",
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_051_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "symbol": "st_ax_mc_051_kolmogorov",
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            "highlight_line": 1028,
            "content": "theorem st_ax_mc_051_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_051() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_051.li#L3",
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      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 61).",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_061_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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            "symbol": "st_ax_mc_061_kolmogorov",
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            "highlight_line": 1032,
            "content": "theorem st_ax_mc_061_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "path": "proof-db/statistics/basic-corpus/st_ax_mc_061.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_061() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_061.li#L3",
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      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 71).",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_071_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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            "symbol": "st_ax_mc_071_kolmogorov",
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            "highlight_line": 1036,
            "content": "theorem st_ax_mc_071_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_071() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_071.li#L3",
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      "id": "ST-AX-MC-081",
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      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 81).",
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      "erdos_status": null,
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      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_081_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "symbol": "st_ax_mc_081_kolmogorov",
            "start_line": 1040,
            "highlight_line": 1040,
            "content": "theorem st_ax_mc_081_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "path": "proof-db/statistics/basic-corpus/st_ax_mc_081.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_081() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_081.li#L3",
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            "content": "[[entry]]\nid = \"ST-AX-MC-081\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Kolmogorov axiom consequence (mega 81).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_mc_081_kolmogorov\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_mc_081.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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      "statement": "Kolmogorov axiom consequence (mega 91).",
      "catalog_status": "proved",
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      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_091_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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            "symbol": "st_ax_mc_091_kolmogorov",
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            "highlight_line": 1044,
            "content": "theorem st_ax_mc_091_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
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      "statement": "Kolmogorov axiom consequence (mega 101).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_101_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_101.li",
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            "symbol": "st_ax_mc_101_kolmogorov",
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            "highlight_line": 1048,
            "content": "theorem st_ax_mc_101_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "path": "proof-db/statistics/basic-corpus/st_ax_mc_101.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_101() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
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            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1597",
            "role": "catalog",
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    },
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      "id": "ST-AX-MC-111",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 111).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_111_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_mc_111.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_111.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "language": "lean",
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            "symbol": "st_ax_mc_111_kolmogorov",
            "start_line": 1052,
            "highlight_line": 1052,
            "content": "theorem st_ax_mc_111_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "path": "proof-db/statistics/basic-corpus/st_ax_mc_111.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_111() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_111.li#L3",
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      "id": "ST-AX-MC-121",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 121).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_121_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_mc_121.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_121.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_ax_mc_121_kolmogorov",
            "start_line": 1056,
            "highlight_line": 1056,
            "content": "theorem st_ax_mc_121_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_mc_121.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_121() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_121.li#L3",
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            "language": "toml",
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            "start_line": 1629,
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            "content": "[[entry]]\nid = \"ST-AX-MC-121\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Kolmogorov axiom consequence (mega 121).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_mc_121_kolmogorov\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_mc_121.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1629",
            "role": "catalog",
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    },
    {
      "id": "ST-AX-MC-131",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 131).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_131_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_mc_131.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_131.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "language": "lean",
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            "symbol": "st_ax_mc_131_kolmogorov",
            "start_line": 1060,
            "highlight_line": 1060,
            "content": "theorem st_ax_mc_131_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_ax_mc_131.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_131() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_131.li#L3",
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            "language": "toml",
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            "start_line": 1645,
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            "content": "[[entry]]\nid = \"ST-AX-MC-131\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Kolmogorov axiom consequence (mega 131).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_mc_131_kolmogorov\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_mc_131.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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    },
    {
      "id": "ST-AX-MC-141",
      "kind": "axiom",
      "field": "statistics",
      "domain": "probability",
      "statement": "Kolmogorov axiom consequence (mega 141).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_ax_mc_141_kolmogorov",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_ax_mc_141.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_141.li",
      "notes": "phase18-mega-corpus; iter10 Mathlib/Init discharge",
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            "symbol": "st_ax_mc_141_kolmogorov",
            "start_line": 1064,
            "highlight_line": 1064,
            "content": "theorem st_ax_mc_141_kolmogorov {n : Nat} (\u03a9 : FiniteProbSpace n) :\n    eventProb \u03a9 Finset.univ = 1 \u2227 \u2200 A : Finset (Fin n), 0 \u2264 eventProb \u03a9 A :=\n  st_ax_mc_kolmogorov_consequence \u03a9\n\n/-- Neyman\u2013Pearson likelihood-ratio threshold via log monotonicity\n    (ST-LM-MC-*0 hypothesis mega). -/",
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            "path": "proof-db/statistics/basic-corpus/st_ax_mc_141.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def proof_db_st_ax_mc_141() -> int\n  requires true\n  ensures result == 0\n  decreases 0\n=\n  return 0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_ax_mc_141.li#L3",
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            "start_line": 1661,
            "highlight_line": 1662,
            "content": "[[entry]]\nid = \"ST-AX-MC-141\"\nkind = \"axiom\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Kolmogorov axiom consequence (mega 141).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_ax_mc_141_kolmogorov\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_ax_mc_141.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter10 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"d93962d9fa\"",
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            "role": "catalog",
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      }
    },
    {
      "id": "ST-LM-BC-BER-009",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Bernoulli distribution mean p variance p(1-p) (case 9).",
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      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
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      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_009",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_ber_009.li",
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            "content": "def bernoulli_variance(p: float) -> float\n  requires 0.0 <= p and p <= 1.0\n  ensures result == p * (1.0 - p)\n  decreases 0\n=\n  return p * (1.0 - p)",
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            "content": "theorem st_lm_bc_ber_019 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) :=\n  st_lm_bc_ber_009 p hp0 hp1\n",
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            "content": "def bernoulli_variance(p: float) -> float\n  requires 0.0 <= p and p <= 1.0\n  ensures result == p * (1.0 - p)\n  decreases 0\n=\n  return p * (1.0 - p)",
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    {
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      "statement": "Bernoulli distribution mean p variance p(1-p) (case 29).",
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      "gap_id": "G-stats",
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      "convergence_class": null,
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_029",
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            "content": "theorem st_lm_bc_ber_029 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) :=\n  st_lm_bc_ber_009 p hp0 hp1\n",
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            "content": "def bernoulli_variance(p: float) -> float\n  requires 0.0 <= p and p <= 1.0\n  ensures result == p * (1.0 - p)\n  decreases 0\n=\n  return p * (1.0 - p)",
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    {
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      "statement": "Bernoulli distribution mean p variance p(1-p) (case 39).",
      "catalog_status": "proved",
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      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": "Mathlib.Data.Real.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_039",
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            "content": "theorem st_lm_bc_ber_039 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) :=\n  st_lm_bc_ber_009 p hp0 hp1\n",
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            "content": "def bernoulli_variance(p: float) -> float\n  requires 0.0 <= p and p <= 1.0\n  ensures result == p * (1.0 - p)\n  decreases 0\n=\n  return p * (1.0 - p)",
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    {
      "id": "ST-LM-BC-BER-049",
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      "statement": "Bernoulli distribution mean p variance p(1-p) (case 49).",
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      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_049",
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            "content": "theorem st_lm_bc_ber_049 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) :=\n  st_lm_bc_ber_009 p hp0 hp1\n\n/-- Moment generating function M_X(t) = E[e^{tX}] (ST-LM-BC-MGF-008). -/\nnoncomputable def mgf {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) (t : \u211d) : \u211d :=\n  expect \u03a9 (fun i => Real.exp (t * X i))\n",
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    },
    {
      "id": "ST-LM-BC-BN-006",
      "kind": "lemma",
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      "statement": "Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 6).",
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      "diverges": false,
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      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
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      "notes": "phase8-basic-corpus tranche=1",
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            "symbol": "st_lm_bc_bn_006",
            "start_line": 137,
            "highlight_line": 137,
            "content": "theorem st_lm_bc_bn_006 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B := by\n  field_simp\n\n/-- Bayes rule finite case (ST-LM-MC-006). -/",
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            "symbol": null,
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            "content": "def bayes_posterior(p_a: float, p_b_given_a: float, p_b: float) -> float\n  requires p_b > 0.0\n  ensures result == (p_b_given_a * p_a) / p_b\n  decreases 0\n=\n  return (p_b_given_a * p_a) / p_b",
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            "start_line": 49,
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            "role": "catalog",
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      }
    },
    {
      "id": "ST-LM-BC-BN-016",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_016",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_bn_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_bn_016.li",
      "notes": "phase8-basic-corpus tranche=2",
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            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_016",
            "start_line": 150,
            "highlight_line": 150,
            "content": "theorem st_lm_bc_bn_016 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bayes rule finite case (ST-LM-MC-016). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L150",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_bn_016.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def bayes_posterior(p_a: float, p_b_given_a: float, p_b: float) -> float\n  requires p_b > 0.0\n  ensures result == (p_b_given_a * p_a) / p_b\n  decreases 0\n=\n  return (p_b_given_a * p_a) / p_b",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 205,
            "highlight_line": 206,
            "content": "[[entry]]\nid = \"ST-LM-BC-BN-016\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 16).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_016\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_bn_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L205",
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    },
    {
      "id": "ST-LM-BC-BN-026",
      "kind": "lemma",
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      "statement": "Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
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      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_026",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_bn_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_bn_026.li",
      "notes": "phase8-basic-corpus tranche=3",
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            "symbol": "st_lm_bc_bn_026",
            "start_line": 163,
            "highlight_line": 163,
            "content": "theorem st_lm_bc_bn_026 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L163",
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            "path": "proof-db/statistics/basic-corpus/st_lm_bc_bn_026.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def bayes_posterior(p_a: float, p_b_given_a: float, p_b: float) -> float\n  requires p_b > 0.0\n  ensures result == (p_b_given_a * p_a) / p_b\n  decreases 0\n=\n  return (p_b_given_a * p_a) / p_b",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 453,
            "highlight_line": 454,
            "content": "[[entry]]\nid = \"ST-LM-BC-BN-026\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 26).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_026\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_bn_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L453",
            "role": "catalog",
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    },
    {
      "id": "ST-LM-BC-BN-036",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_036",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_bn_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_bn_036.li",
      "notes": "phase8-basic-corpus tranche=3",
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_036",
            "start_line": 169,
            "highlight_line": 169,
            "content": "theorem st_lm_bc_bn_036 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L169",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_bn_036.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def bayes_posterior(p_a: float, p_b_given_a: float, p_b: float) -> float\n  requires p_b > 0.0\n  ensures result == (p_b_given_a * p_a) / p_b\n  decreases 0\n=\n  return (p_b_given_a * p_a) / p_b",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 469,
            "highlight_line": 470,
            "content": "[[entry]]\nid = \"ST-LM-BC-BN-036\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 36).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_036\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_bn_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
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    },
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      "id": "ST-LM-BC-BN-046",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_046",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_bn_046.li",
      "notes": "phase8-basic-corpus tranche=3",
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            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_046",
            "start_line": 175,
            "highlight_line": 175,
            "content": "theorem st_lm_bc_bn_046 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L175",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_bn_046.li",
            "symbol": null,
            "start_line": 4,
            "highlight_line": 4,
            "content": "def bayes_posterior(p_a: float, p_b_given_a: float, p_b: float) -> float\n  requires p_b > 0.0\n  ensures result == (p_b_given_a * p_a) / p_b\n  decreases 0\n=\n  return (p_b_given_a * p_a) / p_b",
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            "language": "toml",
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            "start_line": 485,
            "highlight_line": 486,
            "content": "[[entry]]\nid = \"ST-LM-BC-BN-046\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(B) > 0 (case 46).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_046\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_bn_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L485",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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        ]
      }
    },
    {
      "id": "ST-LM-BC-CH-004",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Chebyshev inequality P(|X-mu|>=k sigma) <= 1/k^2 (case 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ch_004",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_ch_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_ch_004.li",
      "notes": "phase18-mega-corpus; iter16 Init/Mathlib discharge",
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            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_ch_004",
            "start_line": 1168,
            "highlight_line": 1168,
            "content": "theorem st_lm_bc_ch_004 (k : \u211d) : chebyshevBound k = 1 / (k * k) := rfl",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1168",
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            "path": "proof-db/statistics/basic-corpus/st_lm_bc_ch_004.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chebyshev_bound(k: float) -> float\n  requires k > 0.0\n  ensures result == 1.0 / (k * k)\n  decreases 0\n=\n  return 1.0 / (k * k)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_ch_004.li#L3",
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            "language": "toml",
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            "start_line": 65,
            "highlight_line": 66,
            "content": "[[entry]]\nid = \"ST-LM-BC-CH-004\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Chebyshev inequality P(|X-mu|>=k sigma) <= 1/k^2 (case 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_ch_004\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_ch_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter16 Init/Mathlib discharge\"\nlast_verified_lic_commit = \"c1013ba813\"",
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      }
    },
    {
      "id": "ST-LM-BC-CH-014",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Chebyshev inequality P(|X-mu|>=k sigma) <= 1/k^2 (case 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ch_014",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_ch_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_ch_014.li",
      "notes": "phase18-mega-corpus; iter16 Init/Mathlib discharge",
      "drilldown": {
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          {
            "language": "lean",
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            "symbol": "st_lm_bc_ch_014",
            "start_line": 1169,
            "highlight_line": 1169,
            "content": "theorem st_lm_bc_ch_014 (k : \u211d) : chebyshevBound k = 1 / (k * k) := rfl",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1169",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def chebyshev_bound(k: float) -> float\n  requires k > 0.0\n  ensures result == 1.0 / (k * k)\n  decreases 0\n=\n  return 1.0 / (k * k)",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_ch_014.li#L3",
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            "language": "toml",
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            "content": "theorem st_lm_bc_clt_005 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  rfl\n",
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            "content": "def clt_standardized_sum(z: float) -> float\n  requires true\n  ensures result == z\n  decreases 0\n=\n  return z",
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            "content": "theorem st_lm_bc_clt_015 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n\n/-- CLT normalized sum statement (ST-LM-MC-005). -/",
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            "content": "theorem st_lm_bc_clt_025 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def clt_standardized_sum(z: float) -> float\n  requires true\n  ensures result == z\n  decreases 0\n=\n  return z",
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            "start_line": 546,
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      "id": "ST-LM-BC-CLT-035",
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      "domain": "probability",
      "statement": "Central limit theorem statement (normalized sum, case 35).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Probability.CentralLimitTheorem",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_035",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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      "bench_refs": [],
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_clt_035.li",
      "notes": "standardizedSum scaffold; literature: ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum_sub",
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            "symbol": "st_lm_bc_clt_035",
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            "highlight_line": 296,
            "content": "theorem st_lm_bc_clt_035 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_clt_035.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def clt_standardized_sum(z: float) -> float\n  requires true\n  ensures result == z\n  decreases 0\n=\n  return z",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_clt_035.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 562,
            "highlight_line": 563,
            "content": "[[entry]]\nid = \"ST-LM-BC-CLT-035\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Central limit theorem statement (normalized sum, case 35).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_035\"\nmathlib_ref = \"Mathlib.Probability.CentralLimitTheorem\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_clt_035.li\"\ncontent_tier = \"raw\"\nnotes = \"standardizedSum scaffold; literature: ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum_sub\"\nlast_verified_lic_commit = \"708c67b8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L562",
            "role": "catalog",
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    },
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      "id": "ST-LM-BC-CLT-045",
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      "statement": "Central limit theorem statement (normalized sum, case 45).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Probability.CentralLimitTheorem",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_045",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_clt_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_clt_045.li",
      "notes": "standardizedSum scaffold; literature: ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum_sub",
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            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_045",
            "start_line": 301,
            "highlight_line": 301,
            "content": "theorem st_lm_bc_clt_045 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_clt_045.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def clt_standardized_sum(z: float) -> float\n  requires true\n  ensures result == z\n  decreases 0\n=\n  return z",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_clt_045.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 578,
            "highlight_line": 579,
            "content": "[[entry]]\nid = \"ST-LM-BC-CLT-045\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Central limit theorem statement (normalized sum, case 45).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_045\"\nmathlib_ref = \"Mathlib.Probability.CentralLimitTheorem\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_clt_045.li\"\ncontent_tier = \"raw\"\nnotes = \"standardizedSum scaffold; literature: ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum_sub\"\nlast_verified_lic_commit = \"708c67b8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L578",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
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    },
    {
      "id": "ST-LM-BC-COV-007",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Covariance Cov(X,Y) = E[XY]-E[X]E[Y] (case 7).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
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      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_007",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_cov_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_cov_007.li",
      "notes": "phase8-basic-corpus tranche=1",
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            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_007",
            "start_line": 116,
            "highlight_line": 116,
            "content": "theorem st_lm_bc_cov_007 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  rfl\n",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_cov_007.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def covariance(exy: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == exy - ex * ey\n  decreases 0\n=\n  return exy - ex * ey",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_cov_007.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 96,
            "highlight_line": 97,
            "content": "[[entry]]\nid = \"ST-LM-BC-COV-007\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Covariance Cov(X,Y) = E[XY]-E[X]E[Y] (case 7).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_cov_007\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_cov_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"a9441d8e\"",
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            "role": "catalog",
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    },
    {
      "id": "ST-LM-BC-COV-017",
      "kind": "lemma",
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      "domain": "probability",
      "statement": "Covariance Cov(X,Y) = E[XY]-E[X]E[Y] (case 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_017",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_cov_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_cov_017.li",
      "notes": "phase8-basic-corpus tranche=2",
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            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_017",
            "start_line": 120,
            "highlight_line": 120,
            "content": "theorem st_lm_bc_cov_017 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  st_lm_bc_cov_007 \u03a9 X Y\n",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_cov_017.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def covariance(exy: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == exy - ex * ey\n  decreases 0\n=\n  return exy - ex * ey",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_cov_017.li#L3",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 252,
            "highlight_line": 253,
            "content": "[[entry]]\nid = \"ST-LM-BC-COV-017\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Covariance Cov(X,Y) = E[XY]-E[X]E[Y] (case 17).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_cov_017\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_cov_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L252",
            "role": "catalog",
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        ]
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    },
    {
      "id": "ST-LM-BC-COV-027",
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      "domain": "probability",
      "statement": "Covariance Cov(X,Y) = E[XY]-E[X]E[Y] (case 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_027",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_cov_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_cov_027.li",
      "notes": "phase8-basic-corpus tranche=3",
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          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_027",
            "start_line": 124,
            "highlight_line": 124,
            "content": "theorem st_lm_bc_cov_027 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  st_lm_bc_cov_007 \u03a9 X Y\n",
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_cov_027.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def covariance(exy: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == exy - ex * ey\n  decreases 0\n=\n  return exy - ex * ey",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_cov_027.li#L3",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 594,
            "highlight_line": 595,
            "content": "[[entry]]\nid = \"ST-LM-BC-COV-027\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Covariance Cov(X,Y) = E[XY]-E[X]E[Y] (case 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_cov_027\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_cov_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
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            "role": "catalog",
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        ]
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    },
    {
      "id": "ST-LM-BC-COV-037",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Covariance Cov(X,Y) = E[XY]-E[X]E[Y] (case 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_037",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_cov_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_cov_037.li",
      "notes": "phase8-basic-corpus tranche=3",
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          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_037",
            "start_line": 128,
            "highlight_line": 128,
            "content": "theorem st_lm_bc_cov_037 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  st_lm_bc_cov_007 \u03a9 X Y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L128",
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          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_cov_037.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def covariance(exy: float, ex: float, ey: float) -> float\n  requires true\n  ensures result == exy - ex * ey\n  decreases 0\n=\n  return exy - ex * ey",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_cov_037.li#L3",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
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      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_gau_020",
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            "content": "theorem st_lm_bc_gau_020 (\u03c3 x \u03bc : \u211d) :\n    normalPdf \u03c3 x \u03bc =\n      (1 / (\u03c3 * Real.sqrt (2 * Real.pi))) * Real.exp (-(x - \u03bc) ^ 2 / (2 * \u03c3 ^ 2)) :=\n  rfl\n",
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            "content": "theorem st_lm_bc_gau_030 (\u03c3 x \u03bc : \u211d) :\n    normalPdf \u03c3 x \u03bc =\n      (1 / (\u03c3 * Real.sqrt (2 * Real.pi))) * Real.exp (-(x - \u03bc) ^ 2 / (2 * \u03c3 ^ 2)) :=\n  rfl\n",
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            "content": "theorem st_lm_bc_gau_040 (\u03c3 x \u03bc : \u211d) :\n    normalPdf \u03c3 x \u03bc =\n      (1 / (\u03c3 * Real.sqrt (2 * Real.pi))) * Real.exp (-(x - \u03bc) ^ 2 / (2 * \u03c3 ^ 2)) :=\n  rfl\n",
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            "content": "theorem st_lm_bc_gau_050 (\u03c3 x \u03bc : \u211d) :\n    normalPdf \u03c3 x \u03bc =\n      (1 / (\u03c3 * Real.sqrt (2 * Real.pi))) * Real.exp (-(x - \u03bc) ^ 2 / (2 * \u03c3 ^ 2)) :=\n  rfl\n\n/-- Chebyshev bound form 1/k\u00b2 (ST-LM-BC-CH-*). -/\nnoncomputable def chebyshevBound (k : \u211d) : \u211d := 1 / (k * k)\n",
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    {
      "id": "ST-LM-BC-MGF-008",
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      "priority_tier": null,
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            "content": "theorem st_lm_bc_mgf_008 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 := by\n  dsimp [mgf, expect]\n  simp\n",
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            "content": "def mgf_at_zero(m0: float) -> float\n  requires true\n  ensures result == m0\n  decreases 0\n=\n  return m0",
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    },
    {
      "id": "ST-LM-BC-MGF-018",
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      "statement": "MGF moment generating function definition (case 18).",
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      "erdos_id": null,
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      "convergence_class": null,
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      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Exp",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_018",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=2",
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            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_mgf_018",
            "start_line": 241,
            "highlight_line": 241,
            "content": "theorem st_lm_bc_mgf_018 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 :=\n  st_lm_bc_mgf_008 \u03a9 X\n",
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            "symbol": null,
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            "content": "def mgf_at_zero(m0: float) -> float\n  requires true\n  ensures result == m0\n  decreases 0\n=\n  return m0",
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            "content": "[[entry]]\nid = \"ST-LM-BC-MGF-018\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"MGF moment generating function definition (case 18).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_018\"\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_mgf_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
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    },
    {
      "id": "ST-LM-BC-MGF-028",
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      "domain": "probability",
      "statement": "MGF moment generating function definition (case 28).",
      "catalog_status": "proved",
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      "diverges": false,
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      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Analysis.SpecialFunctions.Exp",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_028",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
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      "notes": "phase8-basic-corpus tranche=3",
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            "symbol": "st_lm_bc_mgf_028",
            "start_line": 245,
            "highlight_line": 245,
            "content": "theorem st_lm_bc_mgf_028 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 :=\n  st_lm_bc_mgf_008 \u03a9 X\n",
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            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def mgf_at_zero(m0: float) -> float\n  requires true\n  ensures result == m0\n  decreases 0\n=\n  return m0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_mgf_028.li#L3",
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            "language": "toml",
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            "start_line": 687,
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            "content": "[[entry]]\nid = \"ST-LM-BC-MGF-028\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"MGF moment generating function definition (case 28).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_028\"\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_mgf_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
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    },
    {
      "id": "ST-LM-BC-MGF-038",
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      "domain": "probability",
      "statement": "MGF moment generating function definition (case 38).",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_038",
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      "notes": "phase8-basic-corpus tranche=3",
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            "symbol": "st_lm_bc_mgf_038",
            "start_line": 249,
            "highlight_line": 249,
            "content": "theorem st_lm_bc_mgf_038 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 :=\n  st_lm_bc_mgf_008 \u03a9 X\n",
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            "content": "def mgf_at_zero(m0: float) -> float\n  requires true\n  ensures result == m0\n  decreases 0\n=\n  return m0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_mgf_038.li#L3",
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            "content": "[[entry]]\nid = \"ST-LM-BC-MGF-038\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"MGF moment generating function definition (case 38).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_038\"\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_mgf_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
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    },
    {
      "id": "ST-LM-BC-MGF-048",
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      "domain": "probability",
      "statement": "MGF moment generating function definition (case 48).",
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      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_048",
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      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_mgf_048.li",
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            "symbol": "st_lm_bc_mgf_048",
            "start_line": 253,
            "highlight_line": 253,
            "content": "theorem st_lm_bc_mgf_048 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 :=\n  st_lm_bc_mgf_008 \u03a9 X\n\n/-!\n## Central limit theorem scaffold (ST-LM-BC-CLT-*)\n\nNormalized sum Z_n = (\u2211 X_i \u2212 n\u03bc) / (\u03c3\u221an). Convergence in distribution to Gaussian\nis Mathlib `ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum_sub`\n(`Mathlib.Probability.CentralLimitTheorem`).\n-/\n\n/-- Standardized sum for finite samples (CLT statement LHS; ST-LM-BC-CLT-005). -/\nnoncomputable def standardizedSum {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) : \u211d :=\n  ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d))\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L253",
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            "path": "proof-db/statistics/basic-corpus/st_lm_bc_mgf_048.li",
            "symbol": null,
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            "content": "def mgf_at_zero(m0: float) -> float\n  requires true\n  ensures result == m0\n  decreases 0\n=\n  return m0",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_mgf_048.li#L3",
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            "language": "toml",
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            "start_line": 719,
            "highlight_line": 720,
            "content": "[[entry]]\nid = \"ST-LM-BC-MGF-048\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"MGF moment generating function definition (case 48).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_048\"\nmathlib_ref = \"Mathlib.Analysis.SpecialFunctions.Exp\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_mgf_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L719",
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        ]
      }
    },
    {
      "id": "ST-LM-BC-VR-003",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Variance Var(X) = E[X^2] - E[X]^2 (case 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
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      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_003",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_vr_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_003.li",
      "notes": "phase8-basic-corpus tranche=1",
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          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_003",
            "start_line": 73,
            "highlight_line": 73,
            "content": "theorem st_lm_bc_vr_003 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  rfl\n\n/-- Variance identity (ST-LM-MC-003). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L73",
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            "path": "proof-db/statistics/basic-corpus/st_lm_bc_vr_003.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def variance_from_moments(e_x2: float, e_x: float) -> float\n  requires true\n  ensures result == e_x2 - e_x * e_x\n  decreases 0\n=\n  return e_x2 - e_x * e_x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_003.li#L3",
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            "language": "toml",
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            "start_line": 143,
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            "content": "[[entry]]\nid = \"ST-LM-BC-VR-003\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Variance Var(X) = E[X^2] - E[X]^2 (case 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_003\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_vr_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=1\"\nlast_verified_lic_commit = \"a9441d8e\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L143",
            "role": "catalog",
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        ]
      }
    },
    {
      "id": "ST-LM-BC-VR-013",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Variance Var(X) = E[X^2] - E[X]^2 (case 13).",
      "catalog_status": "proved",
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      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_013",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_vr_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_013.li",
      "notes": "phase8-basic-corpus tranche=2",
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            "symbol": "st_lm_bc_vr_013",
            "start_line": 82,
            "highlight_line": 82,
            "content": "theorem st_lm_bc_vr_013 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n",
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            "path": "proof-db/statistics/basic-corpus/st_lm_bc_vr_013.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def variance_from_moments(e_x2: float, e_x: float) -> float\n  requires true\n  ensures result == e_x2 - e_x * e_x\n  decreases 0\n=\n  return e_x2 - e_x * e_x",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 299,
            "highlight_line": 300,
            "content": "[[entry]]\nid = \"ST-LM-BC-VR-013\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Variance Var(X) = E[X^2] - E[X]^2 (case 13).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_013\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_vr_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=2\"\nlast_verified_lic_commit = \"b673a562\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L299",
            "role": "catalog",
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        ]
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    },
    {
      "id": "ST-LM-BC-VR-023",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Variance Var(X) = E[X^2] - E[X]^2 (case 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_023",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_vr_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_023.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_023",
            "start_line": 86,
            "highlight_line": 86,
            "content": "theorem st_lm_bc_vr_023 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity (ST-LM-MC-023). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L86",
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          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_vr_023.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def variance_from_moments(e_x2: float, e_x: float) -> float\n  requires true\n  ensures result == e_x2 - e_x * e_x\n  decreases 0\n=\n  return e_x2 - e_x * e_x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_023.li#L3",
            "role": "li_specimen",
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          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 735,
            "highlight_line": 736,
            "content": "[[entry]]\nid = \"ST-LM-BC-VR-023\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Variance Var(X) = E[X^2] - E[X]^2 (case 23).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_023\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_vr_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L735",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-BC-VR-033",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Variance Var(X) = E[X^2] - E[X]^2 (case 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_033",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_vr_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_033.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_033",
            "start_line": 95,
            "highlight_line": 95,
            "content": "theorem st_lm_bc_vr_033 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity (ST-LM-MC-033). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L95",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_vr_033.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def variance_from_moments(e_x2: float, e_x: float) -> float\n  requires true\n  ensures result == e_x2 - e_x * e_x\n  decreases 0\n=\n  return e_x2 - e_x * e_x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_033.li#L3",
            "role": "li_specimen",
            "label": "Li specimen / contract"
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 751,
            "highlight_line": 752,
            "content": "[[entry]]\nid = \"ST-LM-BC-VR-033\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Variance Var(X) = E[X^2] - E[X]^2 (case 33).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_033\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_vr_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L751",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-BC-VR-043",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Variance Var(X) = E[X^2] - E[X]^2 (case 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": "Mathlib.Algebra.BigOperators.Finset.Basic",
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_043",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_bc_vr_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_043.li",
      "notes": "phase8-basic-corpus tranche=3",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_043",
            "start_line": 104,
            "highlight_line": 104,
            "content": "theorem st_lm_bc_vr_043 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L104",
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          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_bc_vr_043.li",
            "symbol": null,
            "start_line": 3,
            "highlight_line": 3,
            "content": "def variance_from_moments(e_x2: float, e_x: float) -> float\n  requires true\n  ensures result == e_x2 - e_x * e_x\n  decreases 0\n=\n  return e_x2 - e_x * e_x",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_bc_vr_043.li#L3",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml",
            "start_line": 767,
            "highlight_line": 768,
            "content": "[[entry]]\nid = \"ST-LM-BC-VR-043\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Variance Var(X) = E[X^2] - E[X]^2 (case 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_043\"\nmathlib_ref = \"Mathlib.Algebra.BigOperators.Finset.Basic\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_bc_vr_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase8-basic-corpus tranche=3\"\nlast_verified_lic_commit = \"708c67b8\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-basic-corpus.toml#L767",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-002",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 2).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_007",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_002.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_002.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_007",
            "start_line": 116,
            "highlight_line": 116,
            "content": "theorem st_lm_bc_cov_007 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L116",
            "role": "lean_formal",
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          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_002.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-002: Expectation linearity corollary (mega 2).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_002.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 83,
            "highlight_line": 84,
            "content": "[[entry]]\nid = \"ST-LM-MC-002\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 2).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_cov_007\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_002.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L83",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-003",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 3).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_003",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_003.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_003.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_003",
            "start_line": 73,
            "highlight_line": 73,
            "content": "theorem st_lm_bc_vr_003 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  rfl\n\n/-- Variance identity (ST-LM-MC-003). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L73",
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          },
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_003.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-003: Variance identity E[X^2]-E[X]^2 (mega 3).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_003.li#L1",
            "role": "li_specimen",
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          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 99,
            "highlight_line": 100,
            "content": "[[entry]]\nid = \"ST-LM-MC-003\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 3).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_003\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_003.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L99",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-004",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 4).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_009",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_004.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_004.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_ber_009",
            "start_line": 205,
            "highlight_line": 205,
            "content": "theorem st_lm_bc_ber_009 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) := by\n  dsimp [variance, expect, bernoulliSpace, bernoulliIndicator]\n  simp [Fin.sum_univ_two, pow_two]\n  ring\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L205",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_004.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-004: Markov inequality (mega 4).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_004.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 115,
            "highlight_line": 116,
            "content": "[[entry]]\nid = \"ST-LM-MC-004\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 4).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_ber_009\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_004.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L115",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-005",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 5).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_005",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_005.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_005.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_005",
            "start_line": 269,
            "highlight_line": 269,
            "content": "theorem st_lm_bc_clt_005 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  rfl\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L269",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_005.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-005: CLT normalized sum statement (mega 5).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_005.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 131,
            "highlight_line": 132,
            "content": "[[entry]]\nid = \"ST-LM-MC-005\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 5).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_005\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_005.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L131",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-006",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 6).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_006",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_006.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_006.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_006",
            "start_line": 137,
            "highlight_line": 137,
            "content": "theorem st_lm_bc_bn_006 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B := by\n  field_simp\n\n/-- Bayes rule finite case (ST-LM-MC-006). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L137",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_006.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-006: Bayes rule finite case (mega 6).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_006.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 147,
            "highlight_line": 148,
            "content": "[[entry]]\nid = \"ST-LM-MC-006\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 6).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_006\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_006.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L147",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-007",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 7).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_007.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_007.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_007.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-007: MLE consistency sketch (mega 7).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_007.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 163,
            "highlight_line": 164,
            "content": "[[entry]]\nid = \"ST-LM-MC-007\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 7).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_007.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L163",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-008",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 8).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_008_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_008.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_008.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_008_bootstrap",
            "start_line": 342,
            "highlight_line": 342,
            "content": "theorem st_lm_mc_008_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- Markov inequality anchor (inequality) \u2014 still open (measure-theoretic). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L342",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_008.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-008: Bootstrap percentile interval (mega 8).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_008.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 178,
            "highlight_line": 179,
            "content": "[[entry]]\nid = \"ST-LM-MC-008\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 8).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_008_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_008.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L178",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-009",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 9).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_009_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_009.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_009.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_009_ols",
            "start_line": 393,
            "highlight_line": 393,
            "content": "theorem st_lm_mc_009_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Expectation linearity corollary (ST-LM-MC-012). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L393",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_009.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-009: OLS normal equations (mega 9).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_009.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 194,
            "highlight_line": 195,
            "content": "[[entry]]\nid = \"ST-LM-MC-009\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 9).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_009_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_009.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L194",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-010",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 10).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_010_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_010.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_010.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_010_neyman",
            "start_line": 1076,
            "highlight_line": 1076,
            "content": "theorem st_lm_mc_010_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1076",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_010.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-010: Neyman-Pearson lemma sketch (mega 10).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_010.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 210,
            "highlight_line": 211,
            "content": "[[entry]]\nid = \"ST-LM-MC-010\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 10).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_010_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_010.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L210",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-012",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 12).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_017",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_012.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_012.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_017",
            "start_line": 120,
            "highlight_line": 120,
            "content": "theorem st_lm_bc_cov_017 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  st_lm_bc_cov_007 \u03a9 X Y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L120",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_012.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-012: Expectation linearity corollary (mega 12).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_012.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 226,
            "highlight_line": 227,
            "content": "[[entry]]\nid = \"ST-LM-MC-012\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 12).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_cov_017\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_012.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L226",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-013",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 13).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_013",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_013.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_013.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_013",
            "start_line": 82,
            "highlight_line": 82,
            "content": "theorem st_lm_bc_vr_013 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L82",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_013.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-013: Variance identity E[X^2]-E[X]^2 (mega 13).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_013.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 242,
            "highlight_line": 243,
            "content": "[[entry]]\nid = \"ST-LM-MC-013\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 13).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_013\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_013.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L242",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-014",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 14).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.bernoulliSpace",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_014.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_014.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_014.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-014: Markov inequality (mega 14).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_014.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 258,
            "highlight_line": 259,
            "content": "[[entry]]\nid = \"ST-LM-MC-014\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 14).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.bernoulliSpace\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_014.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L258",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-015",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 15).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_015",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_015.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_015.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_015",
            "start_line": 274,
            "highlight_line": 274,
            "content": "theorem st_lm_bc_clt_015 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n\n/-- CLT normalized sum statement (ST-LM-MC-005). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L274",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_015.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-015: CLT normalized sum statement (mega 15).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_015.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 274,
            "highlight_line": 275,
            "content": "[[entry]]\nid = \"ST-LM-MC-015\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 15).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_015\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_015.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L274",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-016",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 16).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_016",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_016.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_016.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_016",
            "start_line": 150,
            "highlight_line": 150,
            "content": "theorem st_lm_bc_bn_016 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bayes rule finite case (ST-LM-MC-016). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L150",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_016.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-016: Bayes rule finite case (mega 16).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_016.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 290,
            "highlight_line": 291,
            "content": "[[entry]]\nid = \"ST-LM-MC-016\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 16).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_016\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_016.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L290",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-017",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 17).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_017_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_017.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_017.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_017_mle",
            "start_line": 418,
            "highlight_line": 418,
            "content": "theorem st_lm_mc_017_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- Bayes rule finite case (ST-LM-MC-026). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L418",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_017.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-017: MLE consistency sketch (mega 17).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_017.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 306,
            "highlight_line": 307,
            "content": "[[entry]]\nid = \"ST-LM-MC-017\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 17).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_017_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_017.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L306",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-018",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 18).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_018_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_018.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_018.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_018_bootstrap",
            "start_line": 408,
            "highlight_line": 408,
            "content": "theorem st_lm_mc_018_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- MLE sketch: maximizing likelihood \u2261 maximizing log-likelihood (strict mono of log). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L408",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_018.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-018: Bootstrap percentile interval (mega 18).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_018.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 322,
            "highlight_line": 323,
            "content": "[[entry]]\nid = \"ST-LM-MC-018\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 18).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_018_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_018.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L322",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-019",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 19).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_019_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_019.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_019.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_019_ols",
            "start_line": 435,
            "highlight_line": 435,
            "content": "theorem st_lm_mc_019_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Expectation linearity corollary (ST-LM-MC-052). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L435",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_019.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-019: OLS normal equations (mega 19).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_019.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 338,
            "highlight_line": 339,
            "content": "[[entry]]\nid = \"ST-LM-MC-019\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 19).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_019_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_019.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L338",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-020",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 20).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_020_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_020.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_020.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_020_neyman",
            "start_line": 1080,
            "highlight_line": 1080,
            "content": "theorem st_lm_mc_020_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1080",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_020.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-020: Neyman-Pearson lemma sketch (mega 20).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_020.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 354,
            "highlight_line": 355,
            "content": "[[entry]]\nid = \"ST-LM-MC-020\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 20).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_020_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_020.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L354",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-022",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 22).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_027",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_022.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_022.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_027",
            "start_line": 124,
            "highlight_line": 124,
            "content": "theorem st_lm_bc_cov_027 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  st_lm_bc_cov_007 \u03a9 X Y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L124",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_022.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-022: Expectation linearity corollary (mega 22).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_022.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 370,
            "highlight_line": 371,
            "content": "[[entry]]\nid = \"ST-LM-MC-022\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 22).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_cov_027\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_022.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L370",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-023",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 23).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_023",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_023.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_023.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_023",
            "start_line": 86,
            "highlight_line": 86,
            "content": "theorem st_lm_bc_vr_023 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity (ST-LM-MC-023). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L86",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_023.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-023: Variance identity E[X^2]-E[X]^2 (mega 23).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_023.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 386,
            "highlight_line": 387,
            "content": "[[entry]]\nid = \"ST-LM-MC-023\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 23).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_023\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_023.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L386",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-024",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 24).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_029",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_024.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_024.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_ber_029",
            "start_line": 217,
            "highlight_line": 217,
            "content": "theorem st_lm_bc_ber_029 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) :=\n  st_lm_bc_ber_009 p hp0 hp1\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L217",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_024.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-024: Markov inequality (mega 24).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_024.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 402,
            "highlight_line": 403,
            "content": "[[entry]]\nid = \"ST-LM-MC-024\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 24).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_ber_029\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_024.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L402",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-025",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 25).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_025",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_025.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_025.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_025",
            "start_line": 291,
            "highlight_line": 291,
            "content": "theorem st_lm_bc_clt_025 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L291",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_025.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-025: CLT normalized sum statement (mega 25).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_025.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 418,
            "highlight_line": 419,
            "content": "[[entry]]\nid = \"ST-LM-MC-025\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 25).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_025\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_025.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L418",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-026",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 26).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_026",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_026.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_026.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_026",
            "start_line": 163,
            "highlight_line": 163,
            "content": "theorem st_lm_bc_bn_026 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L163",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_026.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-026: Bayes rule finite case (mega 26).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_026.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 434,
            "highlight_line": 435,
            "content": "[[entry]]\nid = \"ST-LM-MC-026\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 26).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_026\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_026.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L434",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-027",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 27).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_027_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_027.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_027.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_027_mle",
            "start_line": 450,
            "highlight_line": 450,
            "content": "theorem st_lm_mc_027_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- MLE consistency sketch (ST-LM-MC-037). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L450",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_027.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-027: MLE consistency sketch (mega 27).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_027.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 450,
            "highlight_line": 451,
            "content": "[[entry]]\nid = \"ST-LM-MC-027\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 27).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_027_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_027.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L450",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-028",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 28).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_038_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_028.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_028.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_038_bootstrap",
            "start_line": 477,
            "highlight_line": 477,
            "content": "theorem st_lm_mc_038_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- Bootstrap percentile interval (ST-LM-MC-058). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L477",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_028.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-028: Bootstrap percentile interval (mega 28).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_028.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 466,
            "highlight_line": 467,
            "content": "[[entry]]\nid = \"ST-LM-MC-028\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 28).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_038_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_028.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L466",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-029",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 29).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_029_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_029.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_029.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_029_ols",
            "start_line": 467,
            "highlight_line": 467,
            "content": "theorem st_lm_mc_029_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Expectation linearity corollary (ST-LM-MC-032). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L467",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_029.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-029: OLS normal equations (mega 29).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_029.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 482,
            "highlight_line": 483,
            "content": "[[entry]]\nid = \"ST-LM-MC-029\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 29).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_029_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_029.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L482",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-030",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 30).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_030_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_030.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_030.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_030_neyman",
            "start_line": 1084,
            "highlight_line": 1084,
            "content": "theorem st_lm_mc_030_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1084",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_030.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-030: Neyman-Pearson lemma sketch (mega 30).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_030.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 498,
            "highlight_line": 499,
            "content": "[[entry]]\nid = \"ST-LM-MC-030\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 30).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_030_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_030.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L498",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-032",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 32).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_037",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_032.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_032.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_037",
            "start_line": 128,
            "highlight_line": 128,
            "content": "theorem st_lm_bc_cov_037 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  st_lm_bc_cov_007 \u03a9 X Y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L128",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_032.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-032: Expectation linearity corollary (mega 32).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_032.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 514,
            "highlight_line": 515,
            "content": "[[entry]]\nid = \"ST-LM-MC-032\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 32).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_cov_037\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_032.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L514",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-033",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 33).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_033",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_033.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_033.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_033",
            "start_line": 95,
            "highlight_line": 95,
            "content": "theorem st_lm_bc_vr_033 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity (ST-LM-MC-033). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L95",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_033.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-033: Variance identity E[X^2]-E[X]^2 (mega 33).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_033.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 530,
            "highlight_line": 531,
            "content": "[[entry]]\nid = \"ST-LM-MC-033\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 33).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_033\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_033.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L530",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-034",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 34).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_039",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_034.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_034.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_ber_039",
            "start_line": 222,
            "highlight_line": 222,
            "content": "theorem st_lm_bc_ber_039 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) :=\n  st_lm_bc_ber_009 p hp0 hp1\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L222",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_034.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-034: Markov inequality (mega 34).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_034.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 546,
            "highlight_line": 547,
            "content": "[[entry]]\nid = \"ST-LM-MC-034\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 34).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_ber_039\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_034.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L546",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-035",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 35).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_035.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_035.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_035.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-035: CLT normalized sum statement (mega 35).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_035.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 562,
            "highlight_line": 563,
            "content": "[[entry]]\nid = \"ST-LM-MC-035\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 35).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_035.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L562",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-036",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 36).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_036",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_036.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_036.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_036",
            "start_line": 169,
            "highlight_line": 169,
            "content": "theorem st_lm_bc_bn_036 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L169",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_036.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-036: Bayes rule finite case (mega 36).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_036.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 577,
            "highlight_line": 578,
            "content": "[[entry]]\nid = \"ST-LM-MC-036\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 36).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_036\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_036.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L577",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-037",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 37).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_047_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_037.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_037.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_047_mle",
            "start_line": 738,
            "highlight_line": 738,
            "content": "theorem st_lm_mc_047_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- CLT normalized sum statement (ST-LM-MC-065). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L738",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_037.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-037: MLE consistency sketch (mega 37).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_037.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 593,
            "highlight_line": 594,
            "content": "[[entry]]\nid = \"ST-LM-MC-037\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 37).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_047_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_037.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L593",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-038",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 38).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_038_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_038.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_038.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_038_bootstrap",
            "start_line": 477,
            "highlight_line": 477,
            "content": "theorem st_lm_mc_038_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- Bootstrap percentile interval (ST-LM-MC-058). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L477",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_038.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-038: Bootstrap percentile interval (mega 38).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_038.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 609,
            "highlight_line": 610,
            "content": "[[entry]]\nid = \"ST-LM-MC-038\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 38).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_038_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_038.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L609",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-039",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 39).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_039_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_039.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_039.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_039_ols",
            "start_line": 388,
            "highlight_line": 388,
            "content": "theorem st_lm_mc_039_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- OLS normal equations (ST-LM-MC-009). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L388",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_039.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-039: OLS normal equations (mega 39).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_039.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 625,
            "highlight_line": 626,
            "content": "[[entry]]\nid = \"ST-LM-MC-039\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 39).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_039_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_039.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L625",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-040",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 40).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_040_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_040.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_040.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_040_neyman",
            "start_line": 1088,
            "highlight_line": 1088,
            "content": "theorem st_lm_mc_040_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1088",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_040.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-040: Neyman-Pearson lemma sketch (mega 40).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_040.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 641,
            "highlight_line": 642,
            "content": "[[entry]]\nid = \"ST-LM-MC-040\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 40).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_040_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_040.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L641",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-042",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 42).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_042.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_042.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_042.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-042: Expectation linearity corollary (mega 42).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_042.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 657,
            "highlight_line": 658,
            "content": "[[entry]]\nid = \"ST-LM-MC-042\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 42).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_042.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L657",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-043",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 43).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_043",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_043.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_043.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_043",
            "start_line": 104,
            "highlight_line": 104,
            "content": "theorem st_lm_bc_vr_043 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L104",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_043.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-043: Variance identity E[X^2]-E[X]^2 (mega 43).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_043.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 672,
            "highlight_line": 673,
            "content": "[[entry]]\nid = \"ST-LM-MC-043\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 43).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_043\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_043.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L672",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-044",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 44).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_049",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_044.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_044.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_ber_049",
            "start_line": 227,
            "highlight_line": 227,
            "content": "theorem st_lm_bc_ber_049 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) :=\n  st_lm_bc_ber_009 p hp0 hp1\n\n/-- Moment generating function M_X(t) = E[e^{tX}] (ST-LM-BC-MGF-008). -/\nnoncomputable def mgf {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) (t : \u211d) : \u211d :=\n  expect \u03a9 (fun i => Real.exp (t * X i))\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L227",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_044.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-044: Markov inequality (mega 44).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_044.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 688,
            "highlight_line": 689,
            "content": "[[entry]]\nid = \"ST-LM-MC-044\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 44).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_ber_049\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_044.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L688",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-045",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 45).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_045",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_045.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_045.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_045",
            "start_line": 301,
            "highlight_line": 301,
            "content": "theorem st_lm_bc_clt_045 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L301",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_045.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-045: CLT normalized sum statement (mega 45).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_045.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 704,
            "highlight_line": 705,
            "content": "[[entry]]\nid = \"ST-LM-MC-045\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 45).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_045\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_045.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L704",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-046",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 46).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_046",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_046.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_046.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_046",
            "start_line": 175,
            "highlight_line": 175,
            "content": "theorem st_lm_bc_bn_046 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L175",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_046.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-046: Bayes rule finite case (mega 46).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_046.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 720,
            "highlight_line": 721,
            "content": "[[entry]]\nid = \"ST-LM-MC-046\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 46).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_046\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_046.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L720",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-047",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 47).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_057_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_047.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_047.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_057_mle",
            "start_line": 685,
            "highlight_line": 685,
            "content": "theorem st_lm_mc_057_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- OLS normal equations (ST-LM-MC-079). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L685",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_047.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-047: MLE consistency sketch (mega 47).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_047.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 736,
            "highlight_line": 737,
            "content": "[[entry]]\nid = \"ST-LM-MC-047\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 47).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_057_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_047.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L736",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-048",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 48).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_048_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_048.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_048.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_048_bootstrap",
            "start_line": 487,
            "highlight_line": 487,
            "content": "theorem st_lm_mc_048_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- CLT normalized sum statement (ST-LM-MC-045). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L487",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_048.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-048: Bootstrap percentile interval (mega 48).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_048.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 752,
            "highlight_line": 753,
            "content": "[[entry]]\nid = \"ST-LM-MC-048\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 48).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_048_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_048.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L752",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-049",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 49).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_059_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_049.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_049.li",
      "notes": "phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_059_ols",
            "start_line": 445,
            "highlight_line": 445,
            "content": "theorem st_lm_mc_059_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- MLE consistency sketch (ST-LM-MC-027). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L445",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_049.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-049: OLS normal equations (mega 49).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_049.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 768,
            "highlight_line": 769,
            "content": "[[entry]]\nid = \"ST-LM-MC-049\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 49).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_059_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_049.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=1; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L768",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-050",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 50).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_050_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_050.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_050.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_050_neyman",
            "start_line": 1092,
            "highlight_line": 1092,
            "content": "theorem st_lm_mc_050_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1092",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_050.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-050: Neyman-Pearson lemma sketch (mega 50).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_050.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 784,
            "highlight_line": 785,
            "content": "[[entry]]\nid = \"ST-LM-MC-050\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 50).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_050_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_050.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L784",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-052",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 52).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_008",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_052.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_052.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_mgf_008",
            "start_line": 236,
            "highlight_line": 236,
            "content": "theorem st_lm_bc_mgf_008 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 := by\n  dsimp [mgf, expect]\n  simp\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L236",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_052.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-052: Expectation linearity corollary (mega 52).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_052.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 880,
            "highlight_line": 881,
            "content": "[[entry]]\nid = \"ST-LM-MC-052\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 52).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_008\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_052.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L880",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-053",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 53).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_053",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_053.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_053.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_053",
            "start_line": 108,
            "highlight_line": 108,
            "content": "theorem st_lm_bc_vr_053 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Covariance: Cov(X,Y) = E[XY] \u2212 E[X]E[Y] (ST-LM-BC-COV-007). -/\nnoncomputable def covariance {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) : \u211d :=\n  expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L108",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_053.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-053: Variance identity E[X^2]-E[X]^2 (mega 53).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_053.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 896,
            "highlight_line": 897,
            "content": "[[entry]]\nid = \"ST-LM-MC-053\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 53).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_053\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_053.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L896",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-054",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 54).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_054.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_054.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_054.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-054: Markov inequality (mega 54).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_054.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 912,
            "highlight_line": 913,
            "content": "[[entry]]\nid = \"ST-LM-MC-054\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 54).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_054.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L912",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-055",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 55).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_055",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_055.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_055.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_055",
            "start_line": 306,
            "highlight_line": 306,
            "content": "theorem st_lm_bc_clt_055 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L306",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_055.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-055: CLT normalized sum statement (mega 55).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_055.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 927,
            "highlight_line": 928,
            "content": "[[entry]]\nid = \"ST-LM-MC-055\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 55).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_055\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_055.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L927",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-056",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 56).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_066",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_056.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_056.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_066",
            "start_line": 187,
            "highlight_line": 187,
            "content": "theorem st_lm_bc_bn_066 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bernoulli({0,1}) with success probability `p` (ST-LM-BC-BER-009). -/\nnoncomputable def bernoulliSpace (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) : FiniteProbSpace 2 where\n  p := fun i => if i = 0 then p else 1 - p\n  p_nonneg := by\n    intro i\n    fin_cases i <;> simp [sub_nonneg, hp0, hp1]\n  p_sum_one := by\n    simp [Fin.sum_univ_two]\n\n/-- Bernoulli success indicator. -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L187",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_056.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-056: Bayes rule finite case (mega 56).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_056.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 943,
            "highlight_line": 944,
            "content": "[[entry]]\nid = \"ST-LM-MC-056\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 56).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_066\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_056.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L943",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-057",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 57).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_067_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_057.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_057.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_067_mle",
            "start_line": 515,
            "highlight_line": 515,
            "content": "theorem st_lm_mc_067_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- Bayes rule finite case (ST-LM-MC-066). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L515",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_057.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-057: MLE consistency sketch (mega 57).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_057.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 959,
            "highlight_line": 960,
            "content": "[[entry]]\nid = \"ST-LM-MC-057\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 57).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_067_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_057.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L959",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-058",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 58).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_068_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_058.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_058.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_068_bootstrap",
            "start_line": 591,
            "highlight_line": 591,
            "content": "theorem st_lm_mc_068_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L591",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_058.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-058: Bootstrap percentile interval (mega 58).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_058.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 975,
            "highlight_line": 976,
            "content": "[[entry]]\nid = \"ST-LM-MC-058\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 58).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_068_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_058.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L975",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-059",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 59).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_059_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_059.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_059.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_059_ols",
            "start_line": 445,
            "highlight_line": 445,
            "content": "theorem st_lm_mc_059_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- MLE consistency sketch (ST-LM-MC-027). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L445",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_059.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-059: OLS normal equations (mega 59).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_059.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 991,
            "highlight_line": 992,
            "content": "[[entry]]\nid = \"ST-LM-MC-059\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 59).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_059_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_059.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L991",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-060",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 60).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_060_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_060.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_060.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_060_neyman",
            "start_line": 1096,
            "highlight_line": 1096,
            "content": "theorem st_lm_mc_060_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1096",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_060.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-060: Neyman-Pearson lemma sketch (mega 60).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_060.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1007,
            "highlight_line": 1008,
            "content": "[[entry]]\nid = \"ST-LM-MC-060\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 60).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_060_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_060.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1007",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-062",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 62).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_018",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_062.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_062.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_mgf_018",
            "start_line": 241,
            "highlight_line": 241,
            "content": "theorem st_lm_bc_mgf_018 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 :=\n  st_lm_bc_mgf_008 \u03a9 X\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L241",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_062.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-062: Expectation linearity corollary (mega 62).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_062.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1023,
            "highlight_line": 1024,
            "content": "[[entry]]\nid = \"ST-LM-MC-062\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 62).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_018\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_062.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1023",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-063",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 63).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_073",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_063.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_063.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_073",
            "start_line": 671,
            "highlight_line": 671,
            "content": "theorem st_lm_bc_vr_073 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L671",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_063.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-063: Variance identity E[X^2]-E[X]^2 (mega 63).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_063.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1039,
            "highlight_line": 1040,
            "content": "[[entry]]\nid = \"ST-LM-MC-063\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 63).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_073\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_063.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1039",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-064",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 64).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_ber_019",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_064.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_064.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_ber_019",
            "start_line": 212,
            "highlight_line": 212,
            "content": "theorem st_lm_bc_ber_019 (p : \u211d) (hp0 : 0 \u2264 p) (hp1 : p \u2264 1) :\n    let \u03a9 := bernoulliSpace p hp0 hp1\n    variance \u03a9 bernoulliIndicator = p * (1 - p) :=\n  st_lm_bc_ber_009 p hp0 hp1\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L212",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_064.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-064: Markov inequality (mega 64).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_064.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1055,
            "highlight_line": 1056,
            "content": "[[entry]]\nid = \"ST-LM-MC-064\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 64).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_ber_019\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_064.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1055",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-065",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 65).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_065",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_065.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_065.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_065",
            "start_line": 311,
            "highlight_line": 311,
            "content": "theorem st_lm_bc_clt_065 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L311",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_065.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-065: CLT normalized sum statement (mega 65).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_065.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1071,
            "highlight_line": 1072,
            "content": "[[entry]]\nid = \"ST-LM-MC-065\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 65).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_065\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_065.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1071",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-066",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 66).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_076",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_066.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_066.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_076",
            "start_line": 181,
            "highlight_line": 181,
            "content": "theorem st_lm_bc_bn_076 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L181",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_066.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-066: Bayes rule finite case (mega 66).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_066.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1087,
            "highlight_line": 1088,
            "content": "[[entry]]\nid = \"ST-LM-MC-066\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 66).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_076\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_066.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1087",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-067",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 67).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_067_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_067.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_067.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_067_mle",
            "start_line": 515,
            "highlight_line": 515,
            "content": "theorem st_lm_mc_067_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- Bayes rule finite case (ST-LM-MC-066). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L515",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_067.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-067: MLE consistency sketch (mega 67).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_067.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1103,
            "highlight_line": 1104,
            "content": "[[entry]]\nid = \"ST-LM-MC-067\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 67).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_067_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_067.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1103",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-068",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 68).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_078_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_068.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_068.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_078_bootstrap",
            "start_line": 715,
            "highlight_line": 715,
            "content": "theorem st_lm_mc_078_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- MLE consistency sketch (ST-LM-MC-087). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L715",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_068.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-068: Bootstrap percentile interval (mega 68).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_068.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1119,
            "highlight_line": 1120,
            "content": "[[entry]]\nid = \"ST-LM-MC-068\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 68).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_078_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_068.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1119",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-069",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 69).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_069_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_069.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_069.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_069_ols",
            "start_line": 705,
            "highlight_line": 705,
            "content": "theorem st_lm_mc_069_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Expectation linearity corollary (ST-LM-MC-072). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L705",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_069.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-069: OLS normal equations (mega 69).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_069.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1135,
            "highlight_line": 1136,
            "content": "[[entry]]\nid = \"ST-LM-MC-069\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 69).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_069_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_069.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1135",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-070",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 70).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_070_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_070.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_070.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_070_neyman",
            "start_line": 1100,
            "highlight_line": 1100,
            "content": "theorem st_lm_mc_070_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1100",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_070.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-070: Neyman-Pearson lemma sketch (mega 70).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_070.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1151,
            "highlight_line": 1152,
            "content": "[[entry]]\nid = \"ST-LM-MC-070\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 70).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_070_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_070.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1151",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-072",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 72).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_028",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_072.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_072.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_mgf_028",
            "start_line": 245,
            "highlight_line": 245,
            "content": "theorem st_lm_bc_mgf_028 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 :=\n  st_lm_bc_mgf_008 \u03a9 X\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L245",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_072.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-072: Expectation linearity corollary (mega 72).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_072.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1167,
            "highlight_line": 1168,
            "content": "[[entry]]\nid = \"ST-LM-MC-072\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 72).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_028\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_072.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1167",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-073",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 73).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_083",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_073.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_073.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_083",
            "start_line": 797,
            "highlight_line": 797,
            "content": "theorem st_lm_bc_vr_083 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity E[X^2]-E[X]^2 (ST-LM-MC-083). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L797",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_073.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-073: Variance identity E[X^2]-E[X]^2 (mega 73).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_073.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1183,
            "highlight_line": 1184,
            "content": "[[entry]]\nid = \"ST-LM-MC-073\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 73).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_083\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_073.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1183",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-074",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 74).",
      "catalog_status": "proved",
      "lean_status": "open",
      "diverges": true,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.bernoulliIndicator",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_074.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_074.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "bernoulliIndicator",
            "start_line": 203,
            "highlight_line": 203,
            "content": "def bernoulliIndicator : Fin 2 \u2192 \u211d := fun i => if i = 0 then 1 else 0\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L203",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_074.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-074: Markov inequality (mega 74).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_074.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1199,
            "highlight_line": 1200,
            "content": "[[entry]]\nid = \"ST-LM-MC-074\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 74).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.bernoulliIndicator\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_074.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1199",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-075",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 75).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_075",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_075.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_075.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_075",
            "start_line": 316,
            "highlight_line": 316,
            "content": "theorem st_lm_bc_clt_075 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L316",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_075.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-075: CLT normalized sum statement (mega 75).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_075.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1215,
            "highlight_line": 1216,
            "content": "[[entry]]\nid = \"ST-LM-MC-075\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 75).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_075\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_075.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1215",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-076",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 76).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_086",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_076.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_076.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_086",
            "start_line": 595,
            "highlight_line": 595,
            "content": "theorem st_lm_bc_bn_086 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bayes rule finite case (ST-LM-MC-086). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L595",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_076.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-076: Bayes rule finite case (mega 76).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_076.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1231,
            "highlight_line": 1232,
            "content": "[[entry]]\nid = \"ST-LM-MC-076\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 76).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_086\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_076.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1231",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-077",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 77).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_097_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_077.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_077.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_097_mle",
            "start_line": 817,
            "highlight_line": 817,
            "content": "theorem st_lm_mc_097_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- Markov inequality (ST-LM-MC-104). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L817",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_077.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-077: MLE consistency sketch (mega 77).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_077.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1247,
            "highlight_line": 1248,
            "content": "[[entry]]\nid = \"ST-LM-MC-077\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 77).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_097_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_077.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1247",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-078",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 78).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_088_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_078.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_078.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_088_bootstrap",
            "start_line": 700,
            "highlight_line": 700,
            "content": "theorem st_lm_mc_088_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- OLS normal equations (ST-LM-MC-069). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L700",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_078.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-078: Bootstrap percentile interval (mega 78).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_078.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1263,
            "highlight_line": 1264,
            "content": "[[entry]]\nid = \"ST-LM-MC-078\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 78).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_088_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_078.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1263",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-079",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 79).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_089_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_079.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_079.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_089_ols",
            "start_line": 807,
            "highlight_line": 807,
            "content": "theorem st_lm_mc_089_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Expectation linearity corollary (ST-LM-MC-092). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L807",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_079.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-079: OLS normal equations (mega 79).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_079.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1279,
            "highlight_line": 1280,
            "content": "[[entry]]\nid = \"ST-LM-MC-079\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 79).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_089_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_079.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1279",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-080",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 80).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_080_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_080.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_080.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_080_neyman",
            "start_line": 1104,
            "highlight_line": 1104,
            "content": "theorem st_lm_mc_080_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1104",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_080.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-080: Neyman-Pearson lemma sketch (mega 80).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_080.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1295,
            "highlight_line": 1296,
            "content": "[[entry]]\nid = \"ST-LM-MC-080\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 80).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_080_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_080.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1295",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-082",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 82).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_038",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_082.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_082.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_mgf_038",
            "start_line": 249,
            "highlight_line": 249,
            "content": "theorem st_lm_bc_mgf_038 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 :=\n  st_lm_bc_mgf_008 \u03a9 X\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L249",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_082.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-082: Expectation linearity corollary (mega 82).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_082.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1311,
            "highlight_line": 1312,
            "content": "[[entry]]\nid = \"ST-LM-MC-082\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 82).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_038\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_082.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1311",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-083",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 83).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_093",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_083.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_083.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_093",
            "start_line": 675,
            "highlight_line": 675,
            "content": "theorem st_lm_bc_vr_093 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity E[X^2]-E[X]^2 (ST-LM-MC-073). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L675",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_083.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-083: Variance identity E[X^2]-E[X]^2 (mega 83).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_083.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1327,
            "highlight_line": 1328,
            "content": "[[entry]]\nid = \"ST-LM-MC-083\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 83).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_093\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_083.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1327",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-084",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 84).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_084.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_084.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_084.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-084: Markov inequality (mega 84).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_084.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1343,
            "highlight_line": 1344,
            "content": "[[entry]]\nid = \"ST-LM-MC-084\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 84).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_084.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1343",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-085",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 85).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_035",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_085.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_085.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_035",
            "start_line": 296,
            "highlight_line": 296,
            "content": "theorem st_lm_bc_clt_035 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L296",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_085.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-085: CLT normalized sum statement (mega 85).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_085.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1358,
            "highlight_line": 1359,
            "content": "[[entry]]\nid = \"ST-LM-MC-085\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 85).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_035\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_085.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1358",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-086",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 86).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_096",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_086.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_086.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_096",
            "start_line": 724,
            "highlight_line": 724,
            "content": "theorem st_lm_bc_bn_096 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bayes rule finite case (ST-LM-MC-096). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L724",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_086.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-086: Bayes rule finite case (mega 86).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_086.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1374,
            "highlight_line": 1375,
            "content": "[[entry]]\nid = \"ST-LM-MC-086\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 86).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_096\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_086.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1374",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-087",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 87).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_107_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_087.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_087.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_107_mle",
            "start_line": 620,
            "highlight_line": 620,
            "content": "theorem st_lm_mc_107_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- Bootstrap percentile interval (ST-LM-MC-148). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L620",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_087.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-087: MLE consistency sketch (mega 87).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_087.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1390,
            "highlight_line": 1391,
            "content": "[[entry]]\nid = \"ST-LM-MC-087\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 87).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_107_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_087.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1390",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-088",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 88).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_108_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_088.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_088.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_108_bootstrap",
            "start_line": 773,
            "highlight_line": 773,
            "content": "theorem st_lm_mc_108_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- MLE consistency sketch (ST-LM-MC-117). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L773",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_088.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-088: Bootstrap percentile interval (mega 88).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_088.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1406,
            "highlight_line": 1407,
            "content": "[[entry]]\nid = \"ST-LM-MC-088\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 88).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_108_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_088.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1406",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-089",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 89).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_099_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_089.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_089.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_099_ols",
            "start_line": 884,
            "highlight_line": 884,
            "content": "theorem st_lm_mc_099_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L884",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_089.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-089: OLS normal equations (mega 89).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_089.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1422,
            "highlight_line": 1423,
            "content": "[[entry]]\nid = \"ST-LM-MC-089\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 89).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_099_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_089.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1422",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-090",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 90).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_090_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_090.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_090.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_090_neyman",
            "start_line": 1108,
            "highlight_line": 1108,
            "content": "theorem st_lm_mc_090_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1108",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_090.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-090: Neyman-Pearson lemma sketch (mega 90).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_090.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1438,
            "highlight_line": 1439,
            "content": "[[entry]]\nid = \"ST-LM-MC-090\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 90).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_090_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_090.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1438",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-092",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 92).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_mgf_048",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_092.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_092.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_mgf_048",
            "start_line": 253,
            "highlight_line": 253,
            "content": "theorem st_lm_bc_mgf_048 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    mgf \u03a9 X 0 = 1 :=\n  st_lm_bc_mgf_008 \u03a9 X\n\n/-!\n## Central limit theorem scaffold (ST-LM-BC-CLT-*)\n\nNormalized sum Z_n = (\u2211 X_i \u2212 n\u03bc) / (\u03c3\u221an). Convergence in distribution to Gaussian\nis Mathlib `ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum_sub`\n(`Mathlib.Probability.CentralLimitTheorem`).\n-/\n\n/-- Standardized sum for finite samples (CLT statement LHS; ST-LM-BC-CLT-005). -/\nnoncomputable def standardizedSum {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) : \u211d :=\n  ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d))\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L253",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_092.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-092: Expectation linearity corollary (mega 92).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_092.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1454,
            "highlight_line": 1455,
            "content": "[[entry]]\nid = \"ST-LM-MC-092\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 92).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_mgf_048\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_092.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1454",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-093",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 93).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_103",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_093.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_093.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_103",
            "start_line": 825,
            "highlight_line": 825,
            "content": "theorem st_lm_bc_vr_103 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity E[X^2]-E[X]^2 (ST-LM-MC-103). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L825",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_093.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-093: Variance identity E[X^2]-E[X]^2 (mega 93).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_093.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1470,
            "highlight_line": 1471,
            "content": "[[entry]]\nid = \"ST-LM-MC-093\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 93).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_103\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_093.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1470",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-094",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 94).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_094.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_094.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_094.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-094: Markov inequality (mega 94).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_094.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1486,
            "highlight_line": 1487,
            "content": "[[entry]]\nid = \"ST-LM-MC-094\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 94).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_094.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1486",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-095",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 95).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_095",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_095.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_095.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_095",
            "start_line": 608,
            "highlight_line": 608,
            "content": "theorem st_lm_bc_clt_095 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n\n/-- CLT normalized sum statement (ST-LM-MC-095). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L608",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_095.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-095: CLT normalized sum statement (mega 95).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_095.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1501,
            "highlight_line": 1502,
            "content": "[[entry]]\nid = \"ST-LM-MC-095\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 95).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_095\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_095.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1501",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-096",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 96).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_106",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_096.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_096.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_106",
            "start_line": 547,
            "highlight_line": 547,
            "content": "theorem st_lm_bc_bn_106 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bayes rule finite case (ST-LM-MC-106). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L547",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_096.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-096: Bayes rule finite case (mega 96).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_096.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1517,
            "highlight_line": 1518,
            "content": "[[entry]]\nid = \"ST-LM-MC-096\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 96).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_106\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_096.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1517",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-097",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 97).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_097_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_097.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_097.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_097_mle",
            "start_line": 817,
            "highlight_line": 817,
            "content": "theorem st_lm_mc_097_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- Markov inequality (ST-LM-MC-104). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L817",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_097.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-097: MLE consistency sketch (mega 97).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_097.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1533,
            "highlight_line": 1534,
            "content": "[[entry]]\nid = \"ST-LM-MC-097\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 97).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_097_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_097.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1533",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-098",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 98).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_118_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_098.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_098.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_118_bootstrap",
            "start_line": 840,
            "highlight_line": 840,
            "content": "theorem st_lm_mc_118_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- MLE consistency sketch (ST-LM-MC-127). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L840",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_098.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-098: Bootstrap percentile interval (mega 98).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_098.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1549,
            "highlight_line": 1550,
            "content": "[[entry]]\nid = \"ST-LM-MC-098\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 98).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_118_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_098.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1549",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-099",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 99).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_109_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_099.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_099.li",
      "notes": "phase18-mega-corpus tranche=2; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_109_ols",
            "start_line": 835,
            "highlight_line": 835,
            "content": "theorem st_lm_mc_109_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Bootstrap percentile interval (ST-LM-MC-118). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L835",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_099.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-099: OLS normal equations (mega 99).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_099.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1565,
            "highlight_line": 1566,
            "content": "[[entry]]\nid = \"ST-LM-MC-099\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 99).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_109_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_099.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=2; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1565",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-100",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 100).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_100_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_100.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_100.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_100_neyman",
            "start_line": 1112,
            "highlight_line": 1112,
            "content": "theorem st_lm_mc_100_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1112",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_100.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-100: Neyman-Pearson lemma sketch (mega 100).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_100.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1581,
            "highlight_line": 1582,
            "content": "[[entry]]\nid = \"ST-LM-MC-100\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 100).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_100_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_100.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1581",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-102",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 102).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_102.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_102.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_102.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-102: Expectation linearity corollary (mega 102).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_102.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1677,
            "highlight_line": 1678,
            "content": "[[entry]]\nid = \"ST-LM-MC-102\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 102).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_102.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1677",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-103",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 103).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_123",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_103.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_103.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_123",
            "start_line": 888,
            "highlight_line": 888,
            "content": "theorem st_lm_bc_vr_123 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity E[X^2]-E[X]^2 (ST-LM-MC-123). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L888",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_103.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-103: Variance identity E[X^2]-E[X]^2 (mega 103).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_103.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1692,
            "highlight_line": 1693,
            "content": "[[entry]]\nid = \"ST-LM-MC-103\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 103).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_123\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_103.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1692",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-104",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 104).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_104.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_104.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_104.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-104: Markov inequality (mega 104).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_104.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1708,
            "highlight_line": 1709,
            "content": "[[entry]]\nid = \"ST-LM-MC-104\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 104).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_104.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1708",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-105",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 105).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_115",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_105.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_105.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_115",
            "start_line": 560,
            "highlight_line": 560,
            "content": "theorem st_lm_bc_clt_115 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n\n/-- CLT normalized sum statement (ST-LM-MC-115). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L560",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_105.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-105: CLT normalized sum statement (mega 105).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_105.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1723,
            "highlight_line": 1724,
            "content": "[[entry]]\nid = \"ST-LM-MC-105\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 105).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_115\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_105.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1723",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-106",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 106).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_116",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_106.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_106.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_116",
            "start_line": 629,
            "highlight_line": 629,
            "content": "theorem st_lm_bc_bn_116 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bayes rule finite case (ST-LM-MC-116). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L629",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_106.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-106: Bayes rule finite case (mega 106).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_106.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1739,
            "highlight_line": 1740,
            "content": "[[entry]]\nid = \"ST-LM-MC-106\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 106).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_116\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_106.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1739",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-107",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 107).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_107_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_107.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_107.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_107_mle",
            "start_line": 620,
            "highlight_line": 620,
            "content": "theorem st_lm_mc_107_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n\n/-- Bootstrap percentile interval (ST-LM-MC-148). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L620",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_107.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-107: MLE consistency sketch (mega 107).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_107.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1755,
            "highlight_line": 1756,
            "content": "[[entry]]\nid = \"ST-LM-MC-107\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 107).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_107_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_107.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1755",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-108",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 108).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_108_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_108.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_108.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_108_bootstrap",
            "start_line": 773,
            "highlight_line": 773,
            "content": "theorem st_lm_mc_108_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- MLE consistency sketch (ST-LM-MC-117). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L773",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_108.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-108: Bootstrap percentile interval (mega 108).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_108.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1771,
            "highlight_line": 1772,
            "content": "[[entry]]\nid = \"ST-LM-MC-108\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 108).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_108_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_108.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1771",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-109",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 109).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_129_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_109.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_109.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_129_ols",
            "start_line": 898,
            "highlight_line": 898,
            "content": "theorem st_lm_mc_129_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Bootstrap percentile interval (ST-LM-MC-128). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L898",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_109.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-109: OLS normal equations (mega 109).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_109.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1787,
            "highlight_line": 1788,
            "content": "[[entry]]\nid = \"ST-LM-MC-109\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 109).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_129_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_109.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1787",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-110",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 110).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_110_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_110.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_110.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_110_neyman",
            "start_line": 1116,
            "highlight_line": 1116,
            "content": "theorem st_lm_mc_110_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1116",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_110.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-110: Neyman-Pearson lemma sketch (mega 110).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_110.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1803,
            "highlight_line": 1804,
            "content": "[[entry]]\nid = \"ST-LM-MC-110\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 110).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_110_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_110.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1803",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-112",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 112).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_112.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_112.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_112.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-112: Expectation linearity corollary (mega 112).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_112.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1819,
            "highlight_line": 1820,
            "content": "[[entry]]\nid = \"ST-LM-MC-112\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 112).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_112.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1819",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-113",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 113).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_vr_143",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_113.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_113.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_vr_143",
            "start_line": 657,
            "highlight_line": 657,
            "content": "theorem st_lm_bc_vr_143 {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) :\n    variance \u03a9 X = expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2 :=\n  st_lm_bc_vr_003 \u03a9 X\n\n/-- Variance identity E[X^2]-E[X]^2 (ST-LM-MC-143). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L657",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_113.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-113: Variance identity E[X^2]-E[X]^2 (mega 113).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_113.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1834,
            "highlight_line": 1835,
            "content": "[[entry]]\nid = \"ST-LM-MC-113\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 113).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_vr_143\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_113.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1834",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-114",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 114).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_114.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_114.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_114.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-114: Markov inequality (mega 114).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_114.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1850,
            "highlight_line": 1851,
            "content": "[[entry]]\nid = \"ST-LM-MC-114\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 114).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_114.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1850",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-115",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 115).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_125",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_115.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_115.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_125",
            "start_line": 642,
            "highlight_line": 642,
            "content": "theorem st_lm_bc_clt_125 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n\n/-- CLT normalized sum statement (ST-LM-MC-125). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L642",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_115.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-115: CLT normalized sum statement (mega 115).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_115.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1865,
            "highlight_line": 1866,
            "content": "[[entry]]\nid = \"ST-LM-MC-115\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 115).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_125\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_115.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1865",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-116",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 116).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_136",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_116.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_116.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_136",
            "start_line": 849,
            "highlight_line": 849,
            "content": "theorem st_lm_bc_bn_136 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bayes rule finite case (ST-LM-MC-136). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L849",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_116.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-116: Bayes rule finite case (mega 116).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_116.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1881,
            "highlight_line": 1882,
            "content": "[[entry]]\nid = \"ST-LM-MC-116\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 116).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_136\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_116.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1881",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-117",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 117).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_117_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_117.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_117.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_117_mle",
            "start_line": 778,
            "highlight_line": 778,
            "content": "theorem st_lm_mc_117_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L778",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_117.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-117: MLE consistency sketch (mega 117).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_117.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1897,
            "highlight_line": 1898,
            "content": "[[entry]]\nid = \"ST-LM-MC-117\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 117).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_117_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_117.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1897",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-118",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 118).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_138_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_118.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_118.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_138_bootstrap",
            "start_line": 931,
            "highlight_line": 931,
            "content": "theorem st_lm_mc_138_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-! Statistics-basic AX (ST-AX-BC-*) \u2014 Kolmogorov + expectation linearity (iter10). -/\n\n/-- Non-negativity: P(A) \u2265 0 (ST-AX-BC-PR-031 / Kolmogorov K1). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L931",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_118.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-118: Bootstrap percentile interval (mega 118).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_118.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1913,
            "highlight_line": 1914,
            "content": "[[entry]]\nid = \"ST-LM-MC-118\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 118).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_138_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_118.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1913",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-119",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 119).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_139_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_119.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_119.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_139_ols",
            "start_line": 576,
            "highlight_line": 576,
            "content": "theorem st_lm_mc_139_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Expectation linearity corollary (ST-LM-MC-142). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L576",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_119.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-119: OLS normal equations (mega 119).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_119.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1929,
            "highlight_line": 1930,
            "content": "[[entry]]\nid = \"ST-LM-MC-119\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 119).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_139_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_119.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1929",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-120",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 120).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_120_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_120.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_120.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_120_neyman",
            "start_line": 1120,
            "highlight_line": 1120,
            "content": "theorem st_lm_mc_120_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1120",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_120.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-120: Neyman-Pearson lemma sketch (mega 120).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_120.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1945,
            "highlight_line": 1946,
            "content": "[[entry]]\nid = \"ST-LM-MC-120\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 120).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_120_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_120.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1945",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-122",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 122).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.covariance",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_122.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_122.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_122.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-122: Expectation linearity corollary (mega 122).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_122.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1961,
            "highlight_line": 1962,
            "content": "[[entry]]\nid = \"ST-LM-MC-122\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 122).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.covariance\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_122.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1961",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-123",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 123).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_123.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_123.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_123.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-123: Variance identity E[X^2]-E[X]^2 (mega 123).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_123.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1977,
            "highlight_line": 1978,
            "content": "[[entry]]\nid = \"ST-LM-MC-123\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 123).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_123.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1977",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-124",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 124).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_124.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_124.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_124.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-124: Markov inequality (mega 124).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_124.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 1992,
            "highlight_line": 1993,
            "content": "[[entry]]\nid = \"ST-LM-MC-124\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 124).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_124.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L1992",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-125",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 125).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_135",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_125.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_125.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_135",
            "start_line": 782,
            "highlight_line": 782,
            "content": "theorem st_lm_bc_clt_135 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n\n/-- CLT normalized sum statement (ST-LM-MC-135). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L782",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_125.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-125: CLT normalized sum statement (mega 125).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_125.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2007,
            "highlight_line": 2008,
            "content": "[[entry]]\nid = \"ST-LM-MC-125\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 125).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_135\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_125.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2007",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-126",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 126).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_bn_146",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_126.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_126.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_bn_146",
            "start_line": 912,
            "highlight_line": 912,
            "content": "theorem st_lm_bc_bn_146 {n : Nat} (\u03a9 : FiniteProbSpace n) (A B : Finset (Fin n))\n    (hA : 0 < eventProb \u03a9 A) (hB : 0 < eventProb \u03a9 B) :\n    eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 B =\n      (eventProb \u03a9 (A \u2229 B) / eventProb \u03a9 A) * eventProb \u03a9 A / eventProb \u03a9 B :=\n  st_lm_bc_bn_006 \u03a9 A B hA hB\n\n/-- Bayes rule finite case (ST-LM-MC-146). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L912",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_126.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-126: Bayes rule finite case (mega 126).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_126.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2023,
            "highlight_line": 2024,
            "content": "[[entry]]\nid = \"ST-LM-MC-126\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 126).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_bn_146\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_126.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2023",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-127",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 127).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_127_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_127.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_127.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_127_mle",
            "start_line": 845,
            "highlight_line": 845,
            "content": "theorem st_lm_mc_127_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L845",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_127.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-127: MLE consistency sketch (mega 127).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_127.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2039,
            "highlight_line": 2040,
            "content": "[[entry]]\nid = \"ST-LM-MC-127\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 127).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_127_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_127.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2039",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-128",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 128).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_128_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_128.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_128.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_128_bootstrap",
            "start_line": 903,
            "highlight_line": 903,
            "content": "theorem st_lm_mc_128_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-- MLE consistency sketch (ST-LM-MC-137). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L903",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_128.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-128: Bootstrap percentile interval (mega 128).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_128.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2055,
            "highlight_line": 2056,
            "content": "[[entry]]\nid = \"ST-LM-MC-128\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 128).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_128_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_128.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2055",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-129",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 129).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_149_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_129.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_129.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_149_ols",
            "start_line": 667,
            "highlight_line": 667,
            "content": "theorem st_lm_mc_149_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L667",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_129.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-129: OLS normal equations (mega 129).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_129.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2071,
            "highlight_line": 2072,
            "content": "[[entry]]\nid = \"ST-LM-MC-129\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 129).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_149_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_129.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2071",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-130",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 130).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_130_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_130.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_130.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_130_neyman",
            "start_line": 1124,
            "highlight_line": 1124,
            "content": "theorem st_lm_mc_130_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1124",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_130.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-130: Neyman-Pearson lemma sketch (mega 130).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_130.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2087,
            "highlight_line": 2088,
            "content": "[[entry]]\nid = \"ST-LM-MC-130\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 130).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_130_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_130.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2087",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-132",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 132).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.mgf",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_132.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_132.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_132.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-132: Expectation linearity corollary (mega 132).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_132.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2103,
            "highlight_line": 2104,
            "content": "[[entry]]\nid = \"ST-LM-MC-132\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 132).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.mgf\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_132.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2103",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-133",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 133).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_133.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_133.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_133.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-133: Variance identity E[X^2]-E[X]^2 (mega 133).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_133.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2119,
            "highlight_line": 2120,
            "content": "[[entry]]\nid = \"ST-LM-MC-133\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 133).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_133.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2119",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-134",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 134).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_134.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_134.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_134.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-134: Markov inequality (mega 134).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_134.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2134,
            "highlight_line": 2135,
            "content": "[[entry]]\nid = \"ST-LM-MC-134\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 134).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_134.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2134",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-135",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 135).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_clt_145",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_135.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_135.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_clt_145",
            "start_line": 862,
            "highlight_line": 862,
            "content": "theorem st_lm_bc_clt_145 {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n\n/-- CLT normalized sum statement (ST-LM-MC-145). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L862",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_135.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-135: CLT normalized sum statement (mega 135).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_135.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2149,
            "highlight_line": 2150,
            "content": "[[entry]]\nid = \"ST-LM-MC-135\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 135).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_clt_145\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_135.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2149",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-136",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 136).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_136.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_136.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_136.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-136: Bayes rule finite case (mega 136).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_136.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2165,
            "highlight_line": 2166,
            "content": "[[entry]]\nid = \"ST-LM-MC-136\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 136).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_136.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2165",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-137",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 137).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_137_mle",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_137.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_137.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_137_mle",
            "start_line": 908,
            "highlight_line": 908,
            "content": "theorem st_lm_mc_137_mle {a b : \u211d} (ha : 0 < a) (hb : 0 < b) :\n    a \u2264 b \u2194 Real.log a \u2264 Real.log b :=\n  mle_loglik_mono ha hb\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L908",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_137.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-137: MLE consistency sketch (mega 137).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_137.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2180,
            "highlight_line": 2181,
            "content": "[[entry]]\nid = \"ST-LM-MC-137\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 137).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_137_mle\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_137.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2180",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-138",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 138).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_138_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_138.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_138.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_138_bootstrap",
            "start_line": 931,
            "highlight_line": 931,
            "content": "theorem st_lm_mc_138_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n\n/-! Statistics-basic AX (ST-AX-BC-*) \u2014 Kolmogorov + expectation linearity (iter10). -/\n\n/-- Non-negativity: P(A) \u2265 0 (ST-AX-BC-PR-031 / Kolmogorov K1). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L931",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_138.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-138: Bootstrap percentile interval (mega 138).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_138.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2196,
            "highlight_line": 2197,
            "content": "[[entry]]\nid = \"ST-LM-MC-138\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 138).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_138_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_138.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2196",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-139",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 139).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_139_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_139.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_139.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_139_ols",
            "start_line": 576,
            "highlight_line": 576,
            "content": "theorem st_lm_mc_139_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n\n/-- Expectation linearity corollary (ST-LM-MC-142). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L576",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_139.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-139: OLS normal equations (mega 139).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_139.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2212,
            "highlight_line": 2213,
            "content": "[[entry]]\nid = \"ST-LM-MC-139\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 139).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_139_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_139.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2212",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-140",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 140).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_140_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_140.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_140.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_140_neyman",
            "start_line": 1128,
            "highlight_line": 1128,
            "content": "theorem st_lm_mc_140_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1128",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_140.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-140: Neyman-Pearson lemma sketch (mega 140).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_140.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2228,
            "highlight_line": 2229,
            "content": "[[entry]]\nid = \"ST-LM-MC-140\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 140).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_140_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_140.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2228",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-142",
      "kind": "lemma",
      "field": "statistics",
      "domain": "probability",
      "statement": "Expectation linearity corollary (mega 142).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_bc_cov_047",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_142.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_142.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_bc_cov_047",
            "start_line": 132,
            "highlight_line": 132,
            "content": "theorem st_lm_bc_cov_047 {n : Nat} (\u03a9 : FiniteProbSpace n) (X Y : Fin n \u2192 \u211d) :\n    covariance \u03a9 X Y = expect \u03a9 (fun i => X i * Y i) - expect \u03a9 X * expect \u03a9 Y :=\n  st_lm_bc_cov_007 \u03a9 X Y\n\n/-- Bayes: P(A|B) = P(B|A)P(A)/P(B) when P(A), P(B) > 0 (ST-LM-BC-BN-006). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L132",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_142.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-142: Expectation linearity corollary (mega 142).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_142.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2244,
            "highlight_line": 2245,
            "content": "[[entry]]\nid = \"ST-LM-MC-142\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"probability\"\nstatement = \"Expectation linearity corollary (mega 142).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_bc_cov_047\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_142.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2244",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-143",
      "kind": "lemma",
      "field": "statistics",
      "domain": "variance",
      "statement": "Variance identity E[X^2]-E[X]^2 (mega 143).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_143.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_143.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_143.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-143: Variance identity E[X^2]-E[X]^2 (mega 143).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_143.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2260,
            "highlight_line": 2261,
            "content": "[[entry]]\nid = \"ST-LM-MC-143\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"variance\"\nstatement = \"Variance identity E[X^2]-E[X]^2 (mega 143).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_143.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2260",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-144",
      "kind": "lemma",
      "field": "statistics",
      "domain": "inequality",
      "statement": "Markov inequality (mega 144).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_144.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_144.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_144.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-144: Markov inequality (mega 144).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_144.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2275,
            "highlight_line": 2276,
            "content": "[[entry]]\nid = \"ST-LM-MC-144\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"inequality\"\nstatement = \"Markov inequality (mega 144).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_144.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2275",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-145",
      "kind": "lemma",
      "field": "statistics",
      "domain": "clt",
      "statement": "CLT normalized sum statement (mega 145).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_005_clt",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_145.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_145.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_005_clt",
            "start_line": 280,
            "highlight_line": 280,
            "content": "theorem st_lm_mc_005_clt {n : Nat} (hn : 0 < n) (\u03bc \u03c3 : \u211d) (samples : Fin n \u2192 \u211d) :\n    standardizedSum hn \u03bc \u03c3 samples =\n      ((\u2211 i, samples i) - (n : \u211d) * \u03bc) / (\u03c3 * Real.sqrt (n : \u211d)) :=\n  st_lm_bc_clt_005 hn \u03bc \u03c3 samples\n\n/-- CLT normalized sum statement (ST-LM-MC-015). -/",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L280",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_145.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-145: CLT normalized sum statement (mega 145).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_145.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2290,
            "highlight_line": 2291,
            "content": "[[entry]]\nid = \"ST-LM-MC-145\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"clt\"\nstatement = \"CLT normalized sum statement (mega 145).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_005_clt\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_145.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2290",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-146",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bayes",
      "statement": "Bayes rule finite case (mega 146).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_146.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_146.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_146.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-146: Bayes rule finite case (mega 146).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_146.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2306,
            "highlight_line": 2307,
            "content": "[[entry]]\nid = \"ST-LM-MC-146\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bayes\"\nstatement = \"Bayes rule finite case (mega 146).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_146.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 open-discharge; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2306",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-147",
      "kind": "lemma",
      "field": "statistics",
      "domain": "mle",
      "statement": "MLE consistency sketch (mega 147).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "modeling_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": null,
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_147.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_147.li",
      "notes": "phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge",
      "drilldown": {
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        "release_pin": null,
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_147.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-147: MLE consistency sketch (mega 147).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_147.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2321,
            "highlight_line": 2322,
            "content": "[[entry]]\nid = \"ST-LM-MC-147\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"mle\"\nstatement = \"MLE consistency sketch (mega 147).\"\nproof_status = \"proved\"\ngap_kind = \"modeling_gap\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_147.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus tranche=3; phase19 stub-proved reclass; phase19 discrepancy-discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a13e0bf9\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2321",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-148",
      "kind": "lemma",
      "field": "statistics",
      "domain": "bootstrap",
      "statement": "Bootstrap percentile interval (mega 148).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_148_bootstrap",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_148.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_148.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_148_bootstrap",
            "start_line": 625,
            "highlight_line": 625,
            "content": "theorem st_lm_mc_148_bootstrap {n k : Nat} (hn : 0 < n) (hk : k \u2264 n) :\n    0 \u2264 (k : \u211d) / n \u2227 (k : \u211d) / n \u2264 1 :=\n  bootstrap_percentile_ecdf_bounded hn hk\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L625",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_148.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-148: Bootstrap percentile interval (mega 148).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_148.li#L1",
            "role": "li_specimen",
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2336,
            "highlight_line": 2337,
            "content": "[[entry]]\nid = \"ST-LM-MC-148\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"bootstrap\"\nstatement = \"Bootstrap percentile interval (mega 148).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_148_bootstrap\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_148.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2336",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-149",
      "kind": "lemma",
      "field": "statistics",
      "domain": "regression",
      "statement": "OLS normal equations (mega 149).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_149_ols",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_149.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_149.li",
      "notes": "phase18-mega-corpus; iter12 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_149_ols",
            "start_line": 667,
            "highlight_line": 667,
            "content": "theorem st_lm_mc_149_ols {m n : Nat} (X : Matrix (Fin m) (Fin n) \u211d) (\u03b2 : Fin n \u2192 \u211d) :\n    X\u1d40 *\u1d65 (X *\u1d65 \u03b2) = (X\u1d40 * X) *\u1d65 \u03b2 :=\n  ols_normal_equations X \u03b2\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L667",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
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            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_149.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-149: OLS normal equations (mega 149).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_149.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2352,
            "highlight_line": 2353,
            "content": "[[entry]]\nid = \"ST-LM-MC-149\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"regression\"\nstatement = \"OLS normal equations (mega 149).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_149_ols\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_149.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter12 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2352",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-MC-150",
      "kind": "lemma",
      "field": "statistics",
      "domain": "hypothesis",
      "statement": "Neyman-Pearson lemma sketch (mega 150).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "axiom_layer",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.st_lm_mc_150_neyman",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": "proof-db/statistics/basic-corpus/st_lm_mc_150.li",
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_150.li",
      "notes": "phase18-mega-corpus; iter13 Mathlib/Init discharge",
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "st_lm_mc_150_neyman",
            "start_line": 1132,
            "highlight_line": 1132,
            "content": "theorem st_lm_mc_150_neyman {L0 L1 \u03b7 : \u211d} (h0 : 0 < L0) (h1 : 0 < L1) (h\u03b7 : 0 < \u03b7) :\n    \u03b7 \u2264 L1 / L0 \u2194 Real.log \u03b7 \u2264 Real.log L1 - Real.log L0 :=\n  neyman_pearson_lr_threshold h0 h1 h\u03b7\n\n/-- Normal PDF (ST-LM-BC-GAU-*): (1/(\u03c3 \u221a(2\u03c0))) exp(\u2212(x\u2212\u03bc)\u00b2/(2\u03c3\u00b2)). -/\nnoncomputable def normalPdf (\u03c3 x \u03bc : \u211d) : \u211d :=\n  (1 / (\u03c3 * Real.sqrt (2 * Real.pi))) * Real.exp (-(x - \u03bc) ^ 2 / (2 * \u03c3 ^ 2))\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L1132",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "li",
            "path": "proof-db/statistics/basic-corpus/st_lm_mc_150.li",
            "symbol": null,
            "start_line": 1,
            "highlight_line": 1,
            "content": "# ST-LM-MC-150: Neyman-Pearson lemma sketch (mega 150).",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/basic-corpus/st_lm_mc_150.li#L1",
            "role": "li_specimen",
            "label": "Li specimen / contract"
          },
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            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml",
            "start_line": 2368,
            "highlight_line": 2369,
            "content": "[[entry]]\nid = \"ST-LM-MC-150\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"hypothesis\"\nstatement = \"Neyman-Pearson lemma sketch (mega 150).\"\nproof_status = \"proved\"\ngap_kind = \"axiom_layer\"\ngap_id = \"G-stats\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.st_lm_mc_150_neyman\"\nli_specimen = \"proof-db/statistics/basic-corpus/st_lm_mc_150.li\"\ncontent_tier = \"raw\"\nnotes = \"phase18-mega-corpus; iter13 Mathlib/Init discharge\"\nregister_source = \"proof-db/statistics/register.json\"\nlast_verified_lic_commit = \"a849effef4\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-mega-corpus.toml#L2368",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "ST-LM-SAMPLE-MEAN-UNBIASED",
      "kind": "lemma",
      "field": "statistics",
      "domain": "estimation",
      "statement": "Sample mean is an unbiased estimator of population mean (finite variance target).",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": "G-stats",
      "gap_kind": "proof_gap",
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDb.Statistics.sample_mean_unbiased",
      "lean_module": "proof-db/statistics/StatsAxioms.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "docs\\verification\\proof-database\\entries\\statistics-lemmas.toml",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
        "backlog_ref": null,
        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/statistics/StatsAxioms.lean",
            "symbol": "sample_mean_unbiased",
            "start_line": 58,
            "highlight_line": 58,
            "content": "theorem sample_mean_unbiased {n : Nat} (hn : 0 < n) (\u03bc : \u211d) (draws : Fin n \u2192 \u211d)\n    (h : expect (uniformSpace n hn) draws = \u03bc) :\n    expect (uniformSpace n hn) (fun _ => sampleMean hn draws) = \u03bc := by\n  have hmean : sampleMean hn draws = \u03bc := by\n    rw [\u2190 sample_mean_eq_expect_uniform hn draws]; exact h\n  rw [expect_const, hmean]\n\n/-- Event probability on a finite outcome space. -/\nnoncomputable def eventProb {n : Nat} (\u03a9 : FiniteProbSpace n) (S : Finset (Fin n)) : \u211d :=\n  \u2211 i in S, \u03a9.p i\n\n/-- Variance via second moment: Var(X) = E[X\u00b2] \u2212 E[X]\u00b2 (ST-LM-BC-VR-003). -/\nnoncomputable def variance {n : Nat} (\u03a9 : FiniteProbSpace n) (X : Fin n \u2192 \u211d) : \u211d :=\n  expect \u03a9 (fun i => X i ^ 2) - (expect \u03a9 X) ^ 2\n",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/statistics/StatsAxioms.lean#L58",
            "role": "lean_formal",
            "label": "Lean formalization"
          },
          {
            "language": "toml",
            "path": "docs\\verification\\proof-database\\entries\\statistics-lemmas.toml",
            "start_line": 3,
            "highlight_line": 4,
            "content": "[[entry]]\nid = \"ST-LM-SAMPLE-MEAN-UNBIASED\"\nkind = \"lemma\"\nfield = \"statistics\"\ndomain = \"estimation\"\nstatement = \"Sample mean is an unbiased estimator of population mean (finite variance target).\"\nproof_status = \"proved\"\ngap_id = \"G-stats\"\ngap_kind = \"proof_gap\"\nlean_module = \"proof-db/statistics/StatsAxioms.lean\"\nlean_thm = \"Li.ProofDb.Statistics.sample_mean_unbiased\"\nlast_verified_lic_commit = \"f242981d0a\"",
            "github_url": "https://github.com/li-langverse/lic/blob/main/docs\\verification\\proof-database\\entries\\statistics-lemmas.toml#L3",
            "role": "catalog",
            "label": "Catalog entry (TOML)"
          }
        ]
      }
    },
    {
      "id": "std_add_comm",
      "kind": "lemma",
      "field": "stdlib",
      "domain": null,
      "statement": "\u2200 a b \u2208 \u2124, a + b = b + a",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": null,
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDB.std_add_comm_int",
      "lean_module": "proof-db/lean/ProofDB.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "proof-db/index.json",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/lean/ProofDB.lean",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/lean/ProofDB.lean",
            "symbol": "std_add_comm_int",
            "start_line": 7,
            "highlight_line": 7,
            "content": "theorem std_add_comm_int (a b : Int) : a + b = b + a := Int.add_comm a b",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/lean/ProofDB.lean#L7",
            "role": "lean_formal",
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          }
        ]
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    },
    {
      "id": "std_dot4_bilinear_right",
      "kind": "lemma",
      "field": "stdlib",
      "domain": null,
      "statement": "\u27e8a,b+c\u27e9 = \u27e8a,b\u27e9+\u27e8a,c\u27e9",
      "catalog_status": "open",
      "lean_status": "open",
      "diverges": false,
      "gap_id": "P-linalg",
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDB.std_dot4_bilinear_right",
      "lean_module": "proof-db/lean/ProofDB.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "proof-db/index.json",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/lean/ProofDB.lean",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/lean/ProofDB.lean",
            "symbol": "std_dot4_bilinear_right",
            "start_line": 11,
            "highlight_line": 11,
            "content": "theorem std_dot4_bilinear_right (a b c : LiArray Int 4) :\n    dot4_eval a (add4_int b c) = dot4_eval a b + dot4_eval a c := by sorry",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/lean/ProofDB.lean#L11",
            "role": "lean_formal",
            "label": "Lean formalization"
          }
        ]
      }
    },
    {
      "id": "std_dot4_comm",
      "kind": "lemma",
      "field": "stdlib",
      "domain": null,
      "statement": "\u27e8a,b\u27e9 = \u27e8b,a\u27e9",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": null,
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDB.std_dot4_comm",
      "lean_module": "proof-db/lean/ProofDB.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "proof-db/index.json",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/lean/ProofDB.lean",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/lean/ProofDB.lean",
            "symbol": "std_dot4_comm",
            "start_line": 13,
            "highlight_line": 13,
            "content": "theorem std_dot4_comm (a b : LiArray Int 4) : dot4_eval a b = dot4_eval b a := by\n  unfold dot4_eval\n  simp only [Int.mul_comm, Int.add_assoc, Int.add_comm, Int.add_left_comm]",
            "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/lean/ProofDB.lean#L13",
            "role": "lean_formal",
            "label": "Lean formalization"
          }
        ]
      }
    },
    {
      "id": "std_mul_assoc",
      "kind": "lemma",
      "field": "stdlib",
      "domain": null,
      "statement": "\u2200 a b c \u2208 \u2124, (a\u00b7b)\u00b7c = a\u00b7(b\u00b7c)",
      "catalog_status": "proved",
      "lean_status": "proved",
      "diverges": false,
      "gap_id": null,
      "gap_kind": null,
      "erdos_id": null,
      "erdos_status": null,
      "convergence_class": null,
      "benchmark_ref": null,
      "mathlib_ref": null,
      "priority_tier": null,
      "lean_theorem": "Li.ProofDB.std_mul_assoc_int",
      "lean_module": "proof-db/lean/ProofDB.lean",
      "li_specimen": null,
      "bench_refs": [],
      "source": "proof-db/index.json",
      "github_url": "https://github.com/li-langverse/lic/blob/main/proof-db/lean/ProofDB.lean",
      "notes": null,
      "drilldown": {
        "discharged_from": [],
        "release_pin": null,
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        "implementations": [
          {
            "language": "lean",
            "path": "proof-db/lean/ProofDB.lean",
            "symbol": "std_mul_assoc_int",
            "start_line": 8,
            "highlight_line": 8,
            "content": "theorem std_mul_assoc_int (a b c : Int) : (a * b) * c = a * (b * c) := Int.mul_assoc a b c",
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